Integrand size = 29, antiderivative size = 319 \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=-\frac {c}{\left (a c^2-d^2\right ) x}+\frac {d \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) x \sqrt {a+b x^3}}+\frac {\sqrt [3]{b} c^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt {3} \left (a c^2-d^2\right )^{4/3}}+\frac {\sqrt [3]{b} c^{5/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{4/3}}-\frac {\sqrt [3]{b} c^{5/3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{4/3}} \]
-c/(a*c^2-d^2)/x+1/3*b^(1/3)*c^(5/3)*ln((a*c^2-d^2)^(1/3)+b^(1/3)*c^(2/3)* x)/(a*c^2-d^2)^(4/3)-1/6*b^(1/3)*c^(5/3)*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)^(4/3)+1/3*b^(1/3) *c^(5/3)*arctan(1/3*(1-2*b^(1/3)*c^(2/3)*x/(a*c^2-d^2)^(1/3))*3^(1/2))/(a* c^2-d^2)^(4/3)*3^(1/2)+d*AppellF1(-1/3,1/2,1,2/3,-b*x^3/a,-b*c^2*x^3/(a*c^ 2-d^2))*(1+b*x^3/a)^(1/2)/(a*c^2-d^2)/x/(b*x^3+a)^(1/2)
Time = 10.52 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.55 \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\frac {15 b d \sqrt [3]{a c^2-d^2} \left (a c^2+d^2\right ) x^3 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-6 b^2 c^2 d \sqrt [3]{a c^2-d^2} x^6 \sqrt {1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )-10 \left (a c^2-d^2\right ) \left (-6 a d \sqrt [3]{a c^2-d^2}-6 b d \sqrt [3]{a c^2-d^2} x^3+6 a c \sqrt [3]{a c^2-d^2} \sqrt {a+b x^3}+2 \sqrt {3} a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \arctan \left (\frac {-1+\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )-2 a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )+a \sqrt [3]{b} c^{5/3} x \sqrt {a+b x^3} \log \left (\left (a c^2-d^2\right )^{2/3}-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+b^{2/3} c^{4/3} x^2\right )\right )}{60 a \left (a c^2-d^2\right )^{7/3} x \sqrt {a+b x^3}} \]
(15*b*d*(a*c^2 - d^2)^(1/3)*(a*c^2 + d^2)*x^3*Sqrt[1 + (b*x^3)/a]*AppellF1 [2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 6*b^2*c^2 *d*(a*c^2 - d^2)^(1/3)*x^6*Sqrt[1 + (b*x^3)/a]*AppellF1[5/3, 1/2, 1, 8/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 10*(a*c^2 - d^2)*(-6*a*d*(a* c^2 - d^2)^(1/3) - 6*b*d*(a*c^2 - d^2)^(1/3)*x^3 + 6*a*c*(a*c^2 - d^2)^(1/ 3)*Sqrt[a + b*x^3] + 2*Sqrt[3]*a*b^(1/3)*c^(5/3)*x*Sqrt[a + b*x^3]*ArcTan[ (-1 + (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[3]] - 2*a*b^(1/3)*c^ (5/3)*x*Sqrt[a + b*x^3]*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x] + a*b ^(1/3)*c^(5/3)*x*Sqrt[a + b*x^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]))/(60*a*(a*c^2 - d^2)^(7/3)* x*Sqrt[a + b*x^3])
Time = 0.74 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.07, 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, 821, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (d \sqrt {a+b x^3}+a c+b c x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2587 |
| \(\displaystyle a c \int \frac {1}{a x^2 \left (b c^2 x^3+a c^2-d^2\right )}dx-a d \int \frac {1}{a x^2 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \int \frac {1}{x^2 \left (b c^2 x^3+a c^2-d^2\right )}dx-d \int \frac {1}{x^2 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle c \left (-\frac {b c^2 \int \frac {x}{b c^2 x^3+a c^2-d^2}dx}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^2 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\int \frac {1}{\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^2 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )-d \int \frac {1}{x^2 \sqrt {b x^3+a} \left (b c^2 x^3+a c^2-d^2\right )}dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )-\frac {d \sqrt {\frac {b x^3}{a}+1} \int \frac {1}{x^2 \sqrt {\frac {b x^3}{a}+1} \left (b c^2 x^3+a c^2-d^2\right )}dx}{\sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\int \frac {\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} c^{2/3} \left (\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x\right )}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{2 \sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3}{2} \sqrt [3]{a c^2-d^2} \int \frac {1}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}\right )}{\sqrt [3]{b} c^{2/3}}-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle c \left (-\frac {b c^2 \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a c^2-d^2}-2 \sqrt [3]{b} c^{2/3} x}{b^{2/3} c^{4/3} x^2-\sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2} x+\left (a c^2-d^2\right )^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )+\frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {d \sqrt {\frac {b x^3}{a}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b x^3}{a},-\frac {b c^2 x^3}{a c^2-d^2}\right )}{x \sqrt {a+b x^3} \left (a c^2-d^2\right )}+c \left (-\frac {b c^2 \left (\frac {\frac {\log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{2 \sqrt [3]{b} c^{2/3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt {3}}\right )}{\sqrt [3]{b} c^{2/3}}}{3 \sqrt [3]{b} c^{2/3} \sqrt [3]{a c^2-d^2}}-\frac {\log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 b^{2/3} c^{4/3} \sqrt [3]{a c^2-d^2}}\right )}{a c^2-d^2}-\frac {1}{x \left (a c^2-d^2\right )}\right )\) |
(d*Sqrt[1 + (b*x^3)/a]*AppellF1[-1/3, 1/2, 1, 2/3, -((b*x^3)/a), -((b*c^2* x^3)/(a*c^2 - d^2))])/((a*c^2 - d^2)*x*Sqrt[a + b*x^3]) + c*(-(1/((a*c^2 - d^2)*x)) - (b*c^2*(-1/3*Log[(a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x]/(b^( 2/3)*c^(4/3)*(a*c^2 - d^2)^(1/3)) + (-((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*b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3))))/(a*c^2 - d^2 ))
3.6.60.3.1 Defintions of rubi rules used
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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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 = 0.83 (sec) , antiderivative size = 1200, normalized size of antiderivative = 3.76
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1200\) |
| default | \(\text {Expression too large to display}\) | \(2404\) |
(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/(a*c^2-d^2)/x-(-1/3/b/c^2/((a*c^2-d^2)/b/c^2)^(1/3)*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)^(1/3)*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*3^(1/2)/b/c^2/((a*c^2-d^2)/b/c^2) ^(1/3)*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))+d/a/(a*c^2-d^2)*(b*x^3+a)^(1/2)/x+1/3*I*d/a/(a*c^2-d^2)*3^(1/2)*(-b ^2*a)^(1/3)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^( 1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x-1/b*(-b^2*a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3 )+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-b^2*a)^(1/3)+1/2*I *3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)/(b*x^3+a)^(1/2) *((-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*EllipticE(1/3*3^( 1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/ (-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1 /2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2))+1/b*(-b^2*a)^(1/3)*EllipticF(1/3*3^ (1/2)*(I*(x+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b /(-b^2*a)^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+ 1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)))+1/3*I/d/b^2*c^2*2^(1/2)*sum(1/(a* c^2-d^2)/_alpha*(-b^2*a)^(1/3)*(1/2*I*b*(2*x+1/b*((-b^2*a)^(1/3)-I*3^(1/2) *(-b^2*a)^(1/3)))/(-b^2*a)^(1/3))^(1/2)*(b*(x-1/b*(-b^2*a)^(1/3))/(-3*(-b^ 2*a)^(1/3)+I*3^(1/2)*(-b^2*a)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+1/b*((-b^2*a...
Timed out. \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int \frac {1}{x^{2} \left (a c + b c x^{3} + d \sqrt {a + b x^{3}}\right )}\, dx \]
\[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int { \frac {1}{{\left (b c x^{3} + a c + \sqrt {b x^{3} + a} d\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int { \frac {1}{{\left (b c x^{3} + a c + \sqrt {b x^{3} + a} d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \left (a c+b c x^3+d \sqrt {a+b x^3}\right )} \, dx=\int \frac {1}{x^2\,\left (a\,c+d\,\sqrt {b\,x^3+a}+b\,c\,x^3\right )} \,d x \]