Integrand size = 19, antiderivative size = 1480 \[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx =\text {Too large to display} \] Output:
2/3*(d*x^3+c)^(1/2)/b-2*a*d^(1/3)*(d*x^3+c)^(1/2)/b^2/((1+3^(1/2))*c^(1/3) +d^(1/3)*x)-c^(1/6)*(b*c^(1/3)-a*d^(1/3))^(1/2)*(b^2*c^(2/3)+a*b*c^(1/3)*d ^(1/3)+a^2*d^(2/3))^(1/2)*(c^(1/3)+d^(1/3)*x)*(c^(2/3)*(1-d^(1/3)*x/c^(1/3 )+d^(2/3)*x^2/c^(2/3))/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*arctanh(1/ 3*(1/2*6^(1/2)-1/2*2^(1/2))*(b^2*c^(2/3)+a*b*c^(1/3)*d^(1/3)+a^2*d^(2/3))^ (1/2)*(1-((1-3^(1/2))*c^(1/3)+d^(1/3)*x)^2/((1+3^(1/2))*c^(1/3)+d^(1/3)*x) ^2)^(1/2)*3^(3/4)/b^(1/2)/c^(1/6)/(b*c^(1/3)-a*d^(1/3))^(1/2)/(7-4*3^(1/2) +((1-3^(1/2))*c^(1/3)+d^(1/3)*x)^2/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2 ))/b^(5/2)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2) ^(1/2)/(d*x^3+c)^(1/2)+3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a*c^(1/3)*d^(1/3) *(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2)) *c^(1/3)+d^(1/3)*x)^2)^(1/2)*EllipticE(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1 +3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^(1/2)+2*I)/b^2/(c^(1/3)*(c^(1/3)+d^(1/3)* x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)+2/3*(1/2*6^(1/ 2)+1/2*2^(1/2))*a*((1-3^(1/2))*b*c^(1/3)+a*d^(1/3))*d^(1/3)*(c^(1/3)+d^(1/ 3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3 )*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/ 3)+d^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/b^3/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+ 3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)-2/3*(1/2*6^(1/2)+1/2* 2^(1/2))*(-a^3*d+b^3*c)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^(1/3)*d^(1/3)*x...
Result contains complex when optimal does not.
Time = 8.81 (sec) , antiderivative size = 877, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[c + d*x^3]/(a + b*x),x]
Output:
(2*(c + d*x^3 + (3*Sqrt[2]*a*c^(1/3)*d^(1/3)*((-1)^(1/3)*c^(1/3) - d^(1/3) *x)*Sqrt[((-1)^(1/3)*c^(1/3) - (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^( 1/3))]*Sqrt[(I*(1 + (d^(1/3)*x)/c^(1/3)))/(3*I + Sqrt[3])]*((-1 + (-1)^(2/ 3))*EllipticE[ArcSin[Sqrt[-(((-1)^(2/3)*((-1)^(2/3)*c^(1/3) + d^(1/3)*x))/ ((1 + (-1)^(1/3))*c^(1/3)))]], (-1)^(1/3)/(-1 + (-1)^(1/3))] + EllipticF[A rcSin[Sqrt[-(((-1)^(2/3)*((-1)^(2/3)*c^(1/3) + d^(1/3)*x))/((1 + (-1)^(1/3 ))*c^(1/3)))]], (-1)^(1/3)/(-1 + (-1)^(1/3))]))/(b*Sqrt[(c^(1/3) + (-1)^(2 /3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]) - (3*a^2*d^(2/3)*((-1)^(1/3)*c ^(1/3) - d^(1/3)*x)*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))] *Sqrt[((-1)^(1/3)*c^(1/3) - (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3 ))]*EllipticF[ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3 ))*c^(1/3))]], (-1)^(1/3)])/(b^2*Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]) - ((3*I)*b*c^(4/3)*Sqrt[(c^(1/3) + d^(1/3)*x)/(( 1 + (-1)^(1/3))*c^(1/3))]*Sqrt[1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^( 2/3)]*EllipticPi[(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)), ArcSin[Sqrt[(c^(1/3) + (-1)^(2/3)*d^(1/3)*x)/((1 + (-1)^(1/3))*c^(1/3))]] , (-1)^(1/3)])/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)) + ((-1)^(1/3)*Sqrt[3]*(1 + (-1)^(1/3))*a^3*c^(1/3)*d*Sqrt[(c^(1/3) + d^(1/3)*x)/((1 + (-1)^(1/3))* c^(1/3))]*Sqrt[1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)]*EllipticPi [(I*Sqrt[3]*b*c^(1/3))/((-1)^(1/3)*b*c^(1/3) + a*d^(1/3)), ArcSin[Sqrt[...
Time = 4.81 (sec) , antiderivative size = 1361, normalized size of antiderivative = 0.92, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {2573, 793, 2417, 759, 2416, 2561, 27, 759, 2567, 2538, 412, 435, 104, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 2573 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {d x^3+c}}dx+\frac {a d \int \frac {a-b x}{\sqrt {d x^3+c}}dx}{b^3}+\frac {d \int \frac {x^2}{\sqrt {d x^3+c}}dx}{b}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {d x^3+c}}dx+\frac {a d \int \frac {a-b x}{\sqrt {d x^3+c}}dx}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 2417 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {d x^3+c}}dx+\frac {a d \left (\left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \int \frac {1}{\sqrt {d x^3+c}}dx-\frac {b \int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}\right )}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {d x^3+c}}dx+\frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {b \int \frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt {d x^3+c}}dx}{\sqrt [3]{d}}\right )}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 2416 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \int \frac {1}{(a+b x) \sqrt {d x^3+c}}dx+\frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {b \left (\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 2561 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {b \int \frac {\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{c} (a+b x) \sqrt {d x^3+c}}dx}{-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}+\sqrt {3} b+b}-\frac {\sqrt [3]{d} \int \frac {1}{\sqrt {d x^3+c}}dx}{\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}}\right )+\frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {b \left (\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {b \int \frac {\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}{(a+b x) \sqrt {d x^3+c}}dx}{\sqrt [3]{c} \left (-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}+\sqrt {3} b+b\right )}-\frac {\sqrt [3]{d} \int \frac {1}{\sqrt {d x^3+c}}dx}{\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}}\right )+\frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}-\frac {b \left (\frac {2 \sqrt {c+d x^3}}{\sqrt [3]{d} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\frac {2 \sqrt {c+d x^3}}{3 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {b \int \frac {\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}{(a+b x) \sqrt {d x^3+c}}dx}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right )}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 2567 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \int \frac {1}{\sqrt {1-\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )}d\left (-\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \left (\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b\right ) \int \frac {1}{\sqrt {1-\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7} \left (\left (\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}-\left (1-\sqrt {3}\right ) b\right )^2-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right )^2 \left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}\right )}d\left (-\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )-\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \int -\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right ) \sqrt {1-\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7} \left (\left (\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}-\left (1-\sqrt {3}\right ) b\right )^2-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right )^2 \left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}\right )}d\left (-\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \left (-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b\right ) \operatorname {EllipticPi}\left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2},\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2}-\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \int -\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right ) \sqrt {1-\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7} \left (\left (\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}-\left (1-\sqrt {3}\right ) b\right )^2-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right )^2 \left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}\right )}d\left (-\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \left (-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b\right ) \operatorname {EllipticPi}\left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2},\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2}-\frac {1}{2} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \int \frac {1}{\sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}+1} \left (\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right ) \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right )^2}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}+\left (\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}-\left (1-\sqrt {3}\right ) b\right )^2\right )}d\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \left (-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b\right ) \operatorname {EllipticPi}\left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2},\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2}-\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \int \frac {1}{\frac {4 \sqrt {3} b \left (b \sqrt [3]{c}-a \sqrt [3]{d}\right )}{\sqrt [3]{c}}-\frac {4 \left (2-\sqrt {3}\right ) \left (d^{2/3} a^2+b \sqrt [3]{c} \sqrt [3]{d} a+b^2 c^{2/3}\right ) \sqrt {\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}+1}}{c^{2/3} \sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7}}}d\frac {\sqrt {\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}+1}}{\sqrt {\frac {\left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )^2}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}-4 \sqrt {3}+7}}\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {a d \left (\frac {2 \sqrt {2+\sqrt {3}} \left (a+\frac {\left (1-\sqrt {3}\right ) b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {b \left (\frac {2 \sqrt {d x^3+c}}{\sqrt [3]{d} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )}{\sqrt [3]{d}}\right )}{b^3}+\left (c-\frac {a^3 d}{b^3}\right ) \left (\frac {4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {c^{2/3} \left (\frac {d^{2/3} x^2}{c^{2/3}}-\frac {\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \left (\frac {\sqrt {c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \text {arctanh}\left (\frac {\sqrt {2-\sqrt {3}} \sqrt {d^{2/3} a^2+b \sqrt [3]{c} \sqrt [3]{d} a+b^2 c^{2/3}} \left (\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}\right )}{\sqrt [4]{3} \sqrt {b} \sqrt [6]{c} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )}\right )}{4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt {b} \sqrt {b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt {d^{2/3} a^2+b \sqrt [3]{c} \sqrt [3]{d} a+b^2 c^{2/3}}}-\frac {\left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}-\sqrt {3} b+b\right ) \operatorname {EllipticPi}\left (\frac {\left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2}{\left (\left (1-\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right )^2},\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \left (\left (1-\sqrt {3}\right ) b-\frac {a \sqrt [3]{d}}{\sqrt [3]{c}}\right )^2}\right )}{\sqrt [3]{c} \left (-\frac {\sqrt [3]{d} a}{\sqrt [3]{c}}+\sqrt {3} b+b\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}-\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right ) \sqrt {\frac {d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (\left (1+\sqrt {3}\right ) b \sqrt [3]{c}-a \sqrt [3]{d}\right ) \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\left (\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}\right )^2}} \sqrt {d x^3+c}}\right )+\frac {2 \sqrt {d x^3+c}}{3 b}\) |
Input:
Int[Sqrt[c + d*x^3]/(a + b*x),x]
Output:
(2*Sqrt[c + d*x^3])/(3*b) + (a*d*(-((b*((2*Sqrt[c + d*x^3])/(d^(1/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(1/3)*(c^(1 /3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sq rt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d ^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(d^(1/3)* Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2 ]*Sqrt[c + d*x^3])))/d^(1/3)) + (2*Sqrt[2 + Sqrt[3]]*(a + ((1 - Sqrt[3])*b *c^(1/3))/d^(1/3))*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[(( 1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*d^(1/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])))/b^3 + (c - (a^3*d)/b^ 3)*((-2*Sqrt[2 + Sqrt[3]]*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^ (1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[Ar cSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)* x)], -7 - 4*Sqrt[3]])/(3^(1/4)*((1 + Sqrt[3])*b*c^(1/3) - a*d^(1/3))*Sqrt[ (c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqr t[c + d*x^3]) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*b*(c^(1/3) + d^(1/3)*x)*Sqrt[ (c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + Sqrt[3]) *c^(1/3) + d^(1/3)*x)^2]*((Sqrt[c]*(b + Sqrt[3]*b - (a*d^(1/3))/c^(1/3)...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt [3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) *s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 - Sqrt[3])*d*s)/r Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r Int[((1 - Sqrt[3])*s + r*x)/Sq rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[-q/((1 + Sqrt[3])*d - c*q) Int[1/Sqrt[a + b*x^3], x] , x] + Simp[d/((1 + Sqrt[3])*d - c*q) Int[(1 + Sqrt[3] + q*x)/((c + d*x)* Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b *c^3*d^3 - 8*a^2*d^6, 0]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{q = Simplify[(1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/(q* Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])) Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sq rt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt [3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^3]/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[b/d Int[x^2/Sqrt[a + b*x^3], x], x] + (-Simp[(b*c^3 - a*d^3)/d^3 Int[1/((c + d*x)*Sqrt[a + b*x^3]), x], x] + Simp[b*(c/d^3) Int[(c - d*x)/Sqrt[a + b*x ^3], x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^3 - a*d^3, 0]
Time = 1.56 (sec) , antiderivative size = 1126, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1126\) |
| elliptic | \(\text {Expression too large to display}\) | \(1126\) |
| risch | \(\text {Expression too large to display}\) | \(1137\) |
Input:
int((d*x^3+c)^(1/2)/(b*x+a),x,method=_RETURNVERBOSE)
Output:
2/3*(d*x^3+c)^(1/2)/b-2/3*I*a^2/b^3*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c *d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2 )*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^ (1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))* 3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*EllipticF(1/3*3^(1/2)*(I*( x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^ (1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1 /2)/d*(-c*d^2)^(1/3)))^(1/2))+2/3*I*a/b^2*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2 /d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3) )^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c *d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^( 1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1 /3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c* d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2) ,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^ 2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c *d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2 ),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d ^2)^(1/3)))^(1/2)))+2/3*I*(a^3*d-b^3*c)/b^4*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x +1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2...
\[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{b x + a} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="fricas")
Output:
integral(sqrt(d*x^3 + c)/(b*x + a), x)
\[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\int \frac {\sqrt {c + d x^{3}}}{a + b x}\, dx \] Input:
integrate((d*x**3+c)**(1/2)/(b*x+a),x)
Output:
Integral(sqrt(c + d*x**3)/(a + b*x), x)
\[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{b x + a} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/(b*x + a), x)
\[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{b x + a} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)/(b*x + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\int \frac {\sqrt {d\,x^3+c}}{a+b\,x} \,d x \] Input:
int((c + d*x^3)^(1/2)/(a + b*x),x)
Output:
int((c + d*x^3)^(1/2)/(a + b*x), x)
\[ \int \frac {\sqrt {c+d x^3}}{a+b x} \, dx=\frac {2 \sqrt {d \,x^{3}+c}+3 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{4}+a d \,x^{3}+b c x +a c}d x \right ) b c -3 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{2}}{b d \,x^{4}+a d \,x^{3}+b c x +a c}d x \right ) a d}{3 b} \] Input:
int((d*x^3+c)^(1/2)/(b*x+a),x)
Output:
(2*sqrt(c + d*x**3) + 3*int(sqrt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x + b*d *x**4),x)*b*c - 3*int((sqrt(c + d*x**3)*x**2)/(a*c + a*d*x**3 + b*c*x + b* d*x**4),x)*a*d)/(3*b)