Integrand size = 24, antiderivative size = 546 \[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\frac {d \left (5 b c d^2-6 b^2 e^2+18 a c e^2\right ) x \sqrt {a+b x^2+c x^4}}{15 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e \left (6 c d^2-b e^2\right ) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{16 c^2}+\frac {d x \left (5 c d^2+3 b e^2+9 c e^2 x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}+\frac {e^3 \left (a+b x^2+c x^4\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) e \left (6 c d^2-b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{32 c^{5/2}}-\frac {\sqrt [4]{a} d \left (5 b c d^2-6 b^2 e^2+18 a c e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{15 c^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) d \left (5 c d^2-6 b e^2+9 \sqrt {a} \sqrt {c} e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{30 c^{7/4} \sqrt {a+b x^2+c x^4}} \] Output:
1/15*d*(18*a*c*e^2-6*b^2*e^2+5*b*c*d^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(a ^(1/2)+c^(1/2)*x^2)+1/16*e*(-b*e^2+6*c*d^2)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(1 /2)/c^2+1/15*d*x*(9*c*e^2*x^2+3*b*e^2+5*c*d^2)*(c*x^4+b*x^2+a)^(1/2)/c+1/6 *e^3*(c*x^4+b*x^2+a)^(3/2)/c-1/32*(-4*a*c+b^2)*e*(-b*e^2+6*c*d^2)*arctanh( 1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(5/2)-1/15*a^(1/4)*d*(18* a*c*e^2-6*b^2*e^2+5*b*c*d^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/ 2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2 -b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)+1/30*a^(1/4)*(b+2 *a^(1/2)*c^(1/2))*d*(5*c*d^2-6*b*e^2+9*a^(1/2)*c^(1/2)*e^2)*(a^(1/2)+c^(1/ 2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2* arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+ b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 14.29 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\frac {2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-15 b^2 e^3+2 b c e \left (45 d^2+24 d e x+5 e^2 x^2\right )+4 c \left (10 a e^3+c x \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )\right )\right )+\frac {-8 i \sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) d \left (-5 b c d^2+6 b^2 e^2-18 a c e^2\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+8 i \sqrt {2} \sqrt {c} d \left (-6 b^3 e^2+b c \left (-5 \sqrt {b^2-4 a c} d^2+24 a e^2\right )+b^2 \left (5 c d^2+6 \sqrt {b^2-4 a c} e^2\right )-2 a c \left (10 c d^2+9 \sqrt {b^2-4 a c} e^2\right )\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-15 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \left (-6 c d^2+b e^2\right ) \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{480 c^{5/2} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(d + e*x)^3*Sqrt[a + b*x^2 + c*x^4],x]
Output:
(2*Sqrt[c]*(a + b*x^2 + c*x^4)*(-15*b^2*e^3 + 2*b*c*e*(45*d^2 + 24*d*e*x + 5*e^2*x^2) + 4*c*(10*a*e^3 + c*x*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10 *e^3*x^3))) + ((-8*I)*Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*d*(-5*b*c*d ^2 + 6*b^2*e^2 - 18*a*c*e^2)*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - S qrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], ( b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + (8*I)*Sqrt[2]*Sqrt[c]*d* (-6*b^3*e^2 + b*c*(-5*Sqrt[b^2 - 4*a*c]*d^2 + 24*a*e^2) + b^2*(5*c*d^2 + 6 *Sqrt[b^2 - 4*a*c]*e^2) - 2*a*c*(10*c*d^2 + 9*Sqrt[b^2 - 4*a*c]*e^2))*Sqrt [(b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt [b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt [2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[ b^2 - 4*a*c])] - 15*(b^2 - 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*e*(-6*c* d^2 + b*e^2)*Sqrt[a + b*x^2 + c*x^4]*Log[b + 2*c*x^2 - 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(480*c^(5/2)*Sqrt[a + b* x^2 + c*x^4])
Time = 1.50 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.96, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2202, 1490, 27, 1511, 27, 1416, 1509, 1576, 27, 1160, 1087, 1092, 219}
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.
| \(\displaystyle \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \left (d^3+3 e^2 x^2 d\right ) \sqrt {c x^4+b x^2+a}dx+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\int \frac {d \left (\left (5 b c d^2-6 b^2 e^2+18 a c e^2\right ) x^2+a \left (10 c d^2-3 b e^2\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {\left (5 b c d^2-6 b^2 e^2+18 a c e^2\right ) x^2+a \left (10 c d^2-3 b e^2\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{15 c}+\int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \int x \left (x^2 e^3+3 d^2 e\right ) \sqrt {c x^4+b x^2+a}dx+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int e \left (3 d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx^2+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} e \int \left (3 d^2+e^2 x^2\right ) \sqrt {c x^4+b x^2+a}dx^2+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {d \left (\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (9 \sqrt {a} \sqrt {c} e^2-6 b e^2+5 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (18 a c e^2-6 b^2 e^2+5 b c d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{15 c}+\frac {1}{2} e \left (\frac {\left (6 c d^2-b e^2\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{2 c}+\frac {e^2 \left (a+b x^2+c x^4\right )^{3/2}}{3 c}\right )+\frac {d x \sqrt {a+b x^2+c x^4} \left (3 b e^2+5 c d^2+9 c e^2 x^2\right )}{15 c}\) |
Input:
Int[(d + e*x)^3*Sqrt[a + b*x^2 + c*x^4],x]
Output:
(d*x*(5*c*d^2 + 3*b*e^2 + 9*c*e^2*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*c) + ( e*((e^2*(a + b*x^2 + c*x^4)^(3/2))/(3*c) + ((6*c*d^2 - b*e^2)*(((b + 2*c*x ^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/ (2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(2*c)))/2 + (d*(-(((5* b*c*d^2 - 6*b^2*e^2 + 18*a*c*e^2)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4 )/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + ( a^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(5*c*d^2 - 6*b*e^2 + 9*Sqrt[a]*Sqrt[c]*e^2 )*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2) ^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4]) /(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4])))/(15*c)
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 4.05 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {e^{3} x^{4} \sqrt {c \,x^{4}+b \,x^{2}+a}}{6}+\frac {3 d \,e^{2} x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {\left (\frac {1}{6} b \,e^{3}+3 d^{2} e c \right ) x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (\frac {3}{5} b d \,e^{2}+c \,d^{3}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (\frac {e^{3} a}{3}+3 b \,d^{2} e -\frac {3 b \left (\frac {1}{6} b \,e^{3}+3 d^{2} e c \right )}{4 c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{2 c}+\frac {\left (d^{3} a -\frac {a \left (\frac {3}{5} b d \,e^{2}+c \,d^{3}\right )}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {\left (3 d^{2} e a -\frac {a \left (\frac {1}{6} b \,e^{3}+3 d^{2} e c \right )}{2 c}-\frac {b \left (\frac {e^{3} a}{3}+3 b \,d^{2} e -\frac {3 b \left (\frac {1}{6} b \,e^{3}+3 d^{2} e c \right )}{4 c}\right )}{2 c}\right ) \ln \left (\frac {2 c \,x^{2}+b}{\sqrt {c}}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {\left (\frac {6 a d \,e^{2}}{5}+b \,d^{3}-\frac {2 b \left (\frac {3}{5} b d \,e^{2}+c \,d^{3}\right )}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(687\) |
| risch | \(\frac {\left (40 x^{4} e^{3} c^{2}+144 c^{2} d \,e^{2} x^{3}+10 b c \,e^{3} x^{2}+180 c^{2} d^{2} e \,x^{2}+48 b d \,e^{2} x c +80 c^{2} d^{3} x +40 a c \,e^{3}-15 b^{2} e^{3}+90 b c \,d^{2} e \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{240 c^{2}}-\frac {\frac {8 c d \left (18 a c \,e^{2}-6 b^{2} e^{2}+5 b c \,d^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {15 e \left (4 a b c \,e^{2}-24 a \,c^{2} d^{2}-b^{3} e^{2}+6 b^{2} c \,d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 \sqrt {c}}-\frac {40 a \,c^{2} d^{3} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {12 a b c d \,e^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{240 c^{2}}\) | \(724\) |
| default | \(d^{3} \left (\frac {x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+e^{3} \left (\frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {b \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+3 d \,e^{2} \left (\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5}+\frac {b x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{60 c \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {2 a}{5}-\frac {2 b^{2}}{15 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+3 d^{2} e \left (\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\right )\) | \(986\) |
Input:
int((e*x+d)^3*(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*e^3*x^4*(c*x^4+b*x^2+a)^(1/2)+3/5*d*e^2*x^3*(c*x^4+b*x^2+a)^(1/2)+1/4* (1/6*b*e^3+3*d^2*e*c)/c*x^2*(c*x^4+b*x^2+a)^(1/2)+1/3*(3/5*b*d*e^2+c*d^3)/ c*x*(c*x^4+b*x^2+a)^(1/2)+1/2*(1/3*e^3*a+3*b*d^2*e-3/4*b/c*(1/6*b*e^3+3*d^ 2*e*c))/c*(c*x^4+b*x^2+a)^(1/2)+1/4*(d^3*a-1/3*a/c*(3/5*b*d*e^2+c*d^3))*2^ (1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2 )^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*Ell ipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4* a*c+b^2)^(1/2))/a/c)^(1/2))+1/2*(3*d^2*e*a-1/2*a/c*(1/6*b*e^3+3*d^2*e*c)-1 /2*b/c*(1/3*e^3*a+3*b*d^2*e-3/4*b/c*(1/6*b*e^3+3*d^2*e*c)))*ln((2*c*x^2+b) /c^(1/2)+2*(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-1/2*(6/5*a*d*e^2+b*d^3-2/3*b/c*( 3/5*b*d*e^2+c*d^3))*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+( -4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/( c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+ (-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2) )-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b +(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.14 \[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\frac {32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, b c^{2} d^{3} - 6 \, {\left (b^{2} c - 3 \, a c^{2}\right )} d e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, b^{2} c d^{3} - 6 \, {\left (b^{3} - 3 \, a b c\right )} d e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 32 \, \sqrt {\frac {1}{2}} {\left ({\left (5 \, {\left (b c^{2} - 2 \, c^{3}\right )} d^{3} - 3 \, {\left (2 \, b^{2} c - {\left (6 \, a + b\right )} c^{2}\right )} d e^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (5 \, {\left (b^{2} c + 2 \, b c^{2}\right )} d^{3} - 3 \, {\left (2 \, b^{3} - {\left (6 \, a b - b^{2}\right )} c\right )} d e^{2}\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 15 \, {\left (6 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} e - {\left (b^{3} - 4 \, a b c\right )} e^{3}\right )} \sqrt {c} x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) + 4 \, {\left (40 \, c^{3} e^{3} x^{5} + 144 \, c^{3} d e^{2} x^{4} + 80 \, b c^{2} d^{3} - 96 \, {\left (b^{2} c - 3 \, a c^{2}\right )} d e^{2} + 10 \, {\left (18 \, c^{3} d^{2} e + b c^{2} e^{3}\right )} x^{3} + 16 \, {\left (5 \, c^{3} d^{3} + 3 \, b c^{2} d e^{2}\right )} x^{2} + 5 \, {\left (18 \, b c^{2} d^{2} e - {\left (3 \, b^{2} c - 8 \, a c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{2} + a}}{960 \, c^{3} x} \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
1/960*(32*sqrt(1/2)*((5*b*c^2*d^3 - 6*(b^2*c - 3*a*c^2)*d*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (5*b^2*c*d^3 - 6*(b^3 - 3*a*b*c)*d*e^2)*x)*sqrt(c)*sqrt(( c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt ((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2 *a*c)/(a*c)) - 32*sqrt(1/2)*((5*(b*c^2 - 2*c^3)*d^3 - 3*(2*b^2*c - (6*a + b)*c^2)*d*e^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (5*(b^2*c + 2*b*c^2)*d^3 - 3*(2 *b^3 - (6*a*b - b^2)*c)*d*e^2)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c )/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 15*(6*(b^2* c - 4*a*c^2)*d^2*e - (b^3 - 4*a*b*c)*e^3)*sqrt(c)*x*log(8*c^2*x^4 + 8*b*c* x^2 + b^2 - 4*sqrt(c*x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) + 4*( 40*c^3*e^3*x^5 + 144*c^3*d*e^2*x^4 + 80*b*c^2*d^3 - 96*(b^2*c - 3*a*c^2)*d *e^2 + 10*(18*c^3*d^2*e + b*c^2*e^3)*x^3 + 16*(5*c^3*d^3 + 3*b*c^2*d*e^2)* x^2 + 5*(18*b*c^2*d^2*e - (3*b^2*c - 8*a*c^2)*e^3)*x)*sqrt(c*x^4 + b*x^2 + a))/(c^3*x)
\[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\int \left (d + e x\right )^{3} \sqrt {a + b x^{2} + c x^{4}}\, dx \] Input:
integrate((e*x+d)**3*(c*x**4+b*x**2+a)**(1/2),x)
Output:
Integral((d + e*x)**3*sqrt(a + b*x**2 + c*x**4), x)
\[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(e*x + d)^3, x)
\[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{2} + a} {\left (e x + d\right )}^{3} \,d x } \] Input:
integrate((e*x+d)^3*(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(c*x^4 + b*x^2 + a)*(e*x + d)^3, x)
Timed out. \[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx=\int {\left (d+e\,x\right )}^3\,\sqrt {c\,x^4+b\,x^2+a} \,d x \] Input:
int((d + e*x)^3*(a + b*x^2 + c*x^4)^(1/2),x)
Output:
int((d + e*x)^3*(a + b*x^2 + c*x^4)^(1/2), x)
\[ \int (d+e x)^3 \sqrt {a+b x^2+c x^4} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3*(c*x^4+b*x^2+a)^(1/2),x)
Output:
(80*sqrt(a + b*x**2 + c*x**4)*a*c**2*e**3 - 30*sqrt(a + b*x**2 + c*x**4)*b **2*c*e**3 + 180*sqrt(a + b*x**2 + c*x**4)*b*c**2*d**2*e + 96*sqrt(a + b*x **2 + c*x**4)*b*c**2*d*e**2*x + 20*sqrt(a + b*x**2 + c*x**4)*b*c**2*e**3*x **2 + 160*sqrt(a + b*x**2 + c*x**4)*c**3*d**3*x + 360*sqrt(a + b*x**2 + c* x**4)*c**3*d**2*e*x**2 + 288*sqrt(a + b*x**2 + c*x**4)*c**3*d*e**2*x**3 + 80*sqrt(a + b*x**2 + c*x**4)*c**3*e**3*x**4 + 60*sqrt(c)*log(sqrt(a + b*x* *2 + c*x**4) - sqrt(c)*x**2)*a*b*c*e**3 - 360*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*a*c**2*d**2*e - 15*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) - sqrt(c)*x**2)*b**3*e**3 + 90*sqrt(c)*log(sqrt(a + b*x**2 + c*x* *4) - sqrt(c)*x**2)*b**2*c*d**2*e - 60*sqrt(c)*log(sqrt(a + b*x**2 + c*x** 4) + sqrt(c)*x**2)*a*b*c*e**3 + 360*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*a*c**2*d**2*e + 15*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt(c)*x**2)*b**3*e**3 - 90*sqrt(c)*log(sqrt(a + b*x**2 + c*x**4) + sqrt (c)*x**2)*b**2*c*d**2*e - 96*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x **2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a**2*b*c**2*d*e**2 + 320*int(sqr t(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x** 6),x)*a**2*c**3*d**3 + 576*int((sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 2* a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*a*b*c**3*d*e**2 - 192*int(( sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2 + 2*a*b*x**2 + a*c*x**4 + b**2*x**4 + b*c*x**6),x)*b**3*c**2*d*e**2 + 160*int((sqrt(a + b*x**2 + c*x**4)*x*...