Integrand size = 29, antiderivative size = 324 \[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=-\frac {c}{2 \left (a c^2-d^2\right ) x^2}+\frac {d \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 \left (a c^2-d^2\right ) x^2 \sqrt {a+b x^3}}+\frac {b^{2/3} c^{7/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{5/3}}-\frac {b^{2/3} c^{7/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{5/3}}+\frac {b^{2/3} c^{7/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{5/3}} \] Output:
-1/2*c/(a*c^2-d^2)/x^2+1/2*d*(1+b*x^3/a)^(1/2)*AppellF1(-2/3,1/2,1,1/3,-b* x^3/a,-b*c^2*x^3/(a*c^2-d^2))/(a*c^2-d^2)/x^2/(b*x^3+a)^(1/2)+1/3*b^(2/3)* c^(7/3)*arctan(1/3*(1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))*3^(1/2))*3^(1 /2)/(a*c^2-d^2)^(5/3)-1/3*b^(2/3)*c^(7/3)*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^( 2/3)*x)/(a*c^2-d^2)^(5/3)+1/6*b^(2/3)*c^(7/3)*ln((a*c^2-d^2)^(2/3)-b^(1/3) *c^(2/3)*(a*c^2-d^2)^(1/3)*x+b^(2/3)*c^(4/3)*x^2)/(a*c^2-d^2)^(5/3)
Time = 15.26 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.86 \[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\frac {b^2 c^2 d x^4 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{16 a \left (-a c^2+d^2\right )^2 \sqrt {a+b x^3}}+\frac {2 b d \left (-5 a c^2+d^2\right ) x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\sqrt {a+b x^3} \left (a c^2-d^2+b c^2 x^3\right ) \left (8 a \left (-a c^2+d^2\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )+3 b x^3 \left (2 a c^2 \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )+\left (a c^2-d^2\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )\right )\right )}+\frac {-3 a c \left (a c^2-d^2\right )^{2/3}+3 d \left (a c^2-d^2\right )^{2/3} \sqrt {a+b x^3}-2 \sqrt {3} a b^{2/3} c^{7/3} x^2 \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 a b^{2/3} c^{7/3} x^2 \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+a b^{2/3} c^{7/3} x^2 \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 a \left (a c^2-d^2\right )^{5/3} x^2} \] Input:
Integrate[1/(x^3*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]
Output:
(b^2*c^2*d*x^4*Sqrt[1 + (b*x^3)/a]*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a) , -((b*c^2*x^3)/(a*c^2 - d^2))])/(16*a*(-(a*c^2) + d^2)^2*Sqrt[a + b*x^3]) + (2*b*d*(-5*a*c^2 + d^2)*x*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b *c^2*x^3)/(a*c^2 - d^2))])/(Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(8*a *(-(a*c^2) + d^2)*AppellF1[1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2*x^3)/( a*c^2 - d^2))] + 3*b*x^3*(2*a*c^2*AppellF1[4/3, 1/2, 2, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + (a*c^2 - d^2)*AppellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))]))) + (-3*a*c*(a*c^2 - d^2)^(2/ 3) + 3*d*(a*c^2 - d^2)^(2/3)*Sqrt[a + b*x^3] - 2*Sqrt[3]*a*b^(2/3)*c^(7/3) *x^2*ArcTan[(-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2* a*b^(2/3)*c^(7/3)*x^2*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + a*b^( 2/3)*c^(7/3)*x^2*Log[(a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^( 1/3)*x + b^(2/3)*c^(4/3)*x^2])/(6*a*(a*c^2 - d^2)^(5/3)*x^2)
Time = 1.21 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {2587, 27, 847, 750, 16, 1013, 1012, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (d \sqrt {a+b x^3}+a c+b c x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2587 |
| \(\displaystyle a c \int \frac {1}{a x^3 \left (b c^2 x^3+a c^2-d^2\right )}dx-a d \int \frac {1}{a x^3 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \int \frac {1}{x^3 \left (b c^2 x^3+a c^2-d^2\right )}dx-d \int \frac {1}{x^3 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle c \left (-\frac {b c^2 \int \frac {1}{b c^2 x^3+a c^2-d^2}dx}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^3 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}dx}{3 \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^3 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^3 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )-\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{x^3 \sqrt {\frac {b x^3}{a}+1} \left (b c^2 x^3+a c^2-d^2\right )}dx}{\sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {2 \sqrt [3]{a c^2-d^2}-\sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},1,\frac {1}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt {a+b x^3} \left (a c^2-d^2\right )}+c \left (-\frac {b c^2 \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}-\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{2 \sqrt [3]{b} c^{2/3}}}{3 \left (a c^2-d^2\right )^{2/3}}+\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \sqrt [3]{b} c^{2/3} \left (a c^2-d^2\right )^{2/3}}\right )}{a c^2-d^2}-\frac {1}{2 x^2 \left (a c^2-d^2\right )}\right )\) |
Input:
Int[1/(x^3*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]
Output:
(d*Sqrt[1 + (b*x^3)/a]*AppellF1[-2/3, 1/2, 1, 1/3, -((b*x^3)/a), -((b*c^2* x^3)/(a*c^2 - d^2))])/(2*(a*c^2 - d^2)*x^2*Sqrt[a + b*x^3]) + c*(-1/2*1/(( a*c^2 - d^2)*x^2) - (b*c^2*(Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x]/( 3*b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3 )*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]])/(b^(1/3)*c^(2/3))) - Log[(a*c^ 2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x ^2]/(2*b^(1/3)*c^(2/3)))/(3*(a*c^2 - d^2)^(2/3))))/(a*c^2 - d^2))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ n/a))^FracPart[p]) Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_ Symbol] :> Simp[c Int[u/(c^2 - a*e^2 + c*d*x^n), x], x] - Simp[a*e Int[ u/((c^2 - a*e^2 + c*d*x^n)*Sqrt[a + b*x^n]), x], x] /; FreeQ[{a, b, c, d, e , n}, x] && EqQ[b*c - a*d, 0]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 0.51 (sec) , antiderivative size = 1049, normalized size of antiderivative = 3.24
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1049\) |
| default | \(\text {Expression too large to display}\) | \(1948\) |
Input:
int(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x,method=_RETURNVERBOSE)
Output:
(b*x^3+a)^(1/2)*(d+c*(b*x^3+a)^(1/2))/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2))*(c*( -(1/3/b/c^2/((a*c^2-d^2)/b/c^2)^(2/3)*ln(x+((a*c^2-d^2)/b/c^2)^(1/3))-1/6/ b/c^2/((a*c^2-d^2)/b/c^2)^(2/3)*ln(x^2-((a*c^2-d^2)/b/c^2)^(1/3)*x+((a*c^2 -d^2)/b/c^2)^(2/3))+1/3/b/c^2/((a*c^2-d^2)/b/c^2)^(2/3)*3^(1/2)*arctan(1/3 *3^(1/2)*(2/((a*c^2-d^2)/b/c^2)^(1/3)*x-1)))*b*c^2/(a*c^2-d^2)-1/2/(a*c^2- d^2)/x^2)+1/2*d/a/(a*c^2-d^2)*(b*x^3+a)^(1/2)/x^2-1/6*I/a/(a*c^2-d^2)*d*3^ (1/2)*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^( 1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a* b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1 /3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3 +a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b *(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/ 3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/3*I/d/ b^2*c^2*2^(1/2)*sum(1/_alpha^2/(a*c^2-d^2)*(-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/ b*(-I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x- 1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(- 1/2*I*b*(2*x+1/b*(I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3) )^(1/2)/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3 )*3^(1/2)+2*_alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticP i(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)...
Timed out. \[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)
Output:
Timed out
\[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int { \frac {1}{{\left (b c x^{3} + a c + \sqrt {b x^{3} + a} d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3), x)
\[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int { \frac {1}{{\left (b c x^{3} + a c + \sqrt {b x^{3} + a} d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x, algorithm="giac")
Output:
integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3), x)
Timed out. \[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int \frac {1}{x^3\,\left (a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3\right )} \,d x \] Input:
int(1/(x^3*(a*c + d*(a + b*x^3)^(1/2) + b*c*x^3)),x)
Output:
int(1/(x^3*(a*c + d*(a + b*x^3)^(1/2) + b*c*x^3)), x)
\[ \int \frac {1}{x^3 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\frac {2 \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}}-2 c^{\frac {2}{3}} b^{\frac {1}{3}} x}{\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \sqrt {3}}\right ) b \,c^{3} x^{2}+\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \mathrm {log}\left (\left (a \,c^{2}-d^{2}\right )^{\frac {2}{3}}-c^{\frac {2}{3}} b^{\frac {1}{3}} \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} x +c^{\frac {4}{3}} b^{\frac {2}{3}} x^{2}\right ) b \,c^{3} x^{2}-2 \left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}} \mathrm {log}\left (\left (a \,c^{2}-d^{2}\right )^{\frac {1}{3}}+c^{\frac {2}{3}} b^{\frac {1}{3}} x \right ) b \,c^{3} x^{2}-6 c^{\frac {14}{3}} b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} c^{2} x^{9}+2 a b \,c^{2} x^{6}-b \,d^{2} x^{6}+a^{2} c^{2} x^{3}-a \,d^{2} x^{3}}d x \right ) a^{2} d \,x^{2}+12 c^{\frac {8}{3}} b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} c^{2} x^{9}+2 a b \,c^{2} x^{6}-b \,d^{2} x^{6}+a^{2} c^{2} x^{3}-a \,d^{2} x^{3}}d x \right ) a \,d^{3} x^{2}-6 c^{\frac {2}{3}} b^{\frac {1}{3}} \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} c^{2} x^{9}+2 a b \,c^{2} x^{6}-b \,d^{2} x^{6}+a^{2} c^{2} x^{3}-a \,d^{2} x^{3}}d x \right ) d^{5} x^{2}-3 c^{\frac {11}{3}} b^{\frac {1}{3}} a +3 c^{\frac {5}{3}} b^{\frac {1}{3}} d^{2}}{6 c^{\frac {2}{3}} b^{\frac {1}{3}} x^{2} \left (a^{2} c^{4}-2 a \,c^{2} d^{2}+d^{4}\right )} \] Input:
int(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)
Output:
(2*(a*c**2 - d**2)**(1/3)*sqrt(3)*atan(((a*c**2 - d**2)**(1/3) - 2*c**(2/3 )*b**(1/3)*x)/((a*c**2 - d**2)**(1/3)*sqrt(3)))*b*c**3*x**2 + (a*c**2 - d* *2)**(1/3)*log((a*c**2 - d**2)**(2/3) - c**(2/3)*b**(1/3)*(a*c**2 - d**2)* *(1/3)*x + c**(1/3)*b**(2/3)*c*x**2)*b*c**3*x**2 - 2*(a*c**2 - d**2)**(1/3 )*log((a*c**2 - d**2)**(1/3) + c**(2/3)*b**(1/3)*x)*b*c**3*x**2 - 6*c**(2/ 3)*b**(1/3)*int(sqrt(a + b*x**3)/(a**2*c**2*x**3 + 2*a*b*c**2*x**6 - a*d** 2*x**3 + b**2*c**2*x**9 - b*d**2*x**6),x)*a**2*c**4*d*x**2 + 12*c**(2/3)*b **(1/3)*int(sqrt(a + b*x**3)/(a**2*c**2*x**3 + 2*a*b*c**2*x**6 - a*d**2*x* *3 + b**2*c**2*x**9 - b*d**2*x**6),x)*a*c**2*d**3*x**2 - 6*c**(2/3)*b**(1/ 3)*int(sqrt(a + b*x**3)/(a**2*c**2*x**3 + 2*a*b*c**2*x**6 - a*d**2*x**3 + b**2*c**2*x**9 - b*d**2*x**6),x)*d**5*x**2 - 3*c**(2/3)*b**(1/3)*a*c**3 + 3*c**(2/3)*b**(1/3)*c*d**2)/(6*c**(2/3)*b**(1/3)*x**2*(a**2*c**4 - 2*a*c** 2*d**2 + d**4))