3.400 \(\int \frac{1}{x^3 \left (a c+b c x^3+d \sqrt{a+b x^3}\right )} \, dx\)

Optimal. Leaf size=324 \[ \frac{d \sqrt{\frac{b x^3}{a}+1} F_1\left (-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{b^{2/3} c^{7/3} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{5/3}}-\frac{b^{2/3} c^{7/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{5/3}}+\frac{b^{2/3} c^{7/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} \left (a c^2-d^2\right )^{5/3}}-\frac{c}{2 x^2 \left (a c^2-d^2\right )} \]

[Out]

-c/(2*(a*c^2 - d^2)*x^2) + (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])
+ (b^(2/3)*c^(7/3)*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[
3]])/(Sqrt[3]*(a*c^2 - d^2)^(5/3)) - (b^(2/3)*c^(7/3)*Log[(a*c^2 - d^2)^(1/3) +
b^(1/3)*c^(2/3)*x])/(3*(a*c^2 - d^2)^(5/3)) + (b^(2/3)*c^(7/3)*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])/(6*(a*c^2
- d^2)^(5/3))

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Rubi [A]  time = 0.955046, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.345 \[ \frac{d \sqrt{\frac{b x^3}{a}+1} F_1\left (-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )}{2 x^2 \sqrt{a+b x^3} \left (a c^2-d^2\right )}+\frac{b^{2/3} c^{7/3} \log \left (-\sqrt [3]{b} c^{2/3} x \sqrt [3]{a c^2-d^2}+\left (a c^2-d^2\right )^{2/3}+b^{2/3} c^{4/3} x^2\right )}{6 \left (a c^2-d^2\right )^{5/3}}-\frac{b^{2/3} c^{7/3} \log \left (\sqrt [3]{a c^2-d^2}+\sqrt [3]{b} c^{2/3} x\right )}{3 \left (a c^2-d^2\right )^{5/3}}+\frac{b^{2/3} c^{7/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} c^{2/3} x}{\sqrt [3]{a c^2-d^2}}}{\sqrt{3}}\right )}{\sqrt{3} \left (a c^2-d^2\right )^{5/3}}-\frac{c}{2 x^2 \left (a c^2-d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]

[Out]

-c/(2*(a*c^2 - d^2)*x^2) + (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])
+ (b^(2/3)*c^(7/3)*ArcTan[(1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/Sqrt[
3]])/(Sqrt[3]*(a*c^2 - d^2)^(5/3)) - (b^(2/3)*c^(7/3)*Log[(a*c^2 - d^2)^(1/3) +
b^(1/3)*c^(2/3)*x])/(3*(a*c^2 - d^2)^(5/3)) + (b^(2/3)*c^(7/3)*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])/(6*(a*c^2
- d^2)^(5/3))

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Rubi in Sympy [A]  time = 92.3923, size = 292, normalized size = 0.9 \[ \frac{b^{\frac{2}{3}} c^{\frac{7}{3}} \log{\left (\sqrt [3]{b} c^{\frac{2}{3}} x - \sqrt [3]{- a c^{2} + d^{2}} \right )}}{3 \left (- a c^{2} + d^{2}\right )^{\frac{5}{3}}} - \frac{b^{\frac{2}{3}} c^{\frac{7}{3}} \log{\left (a^{\frac{2}{3}} b^{\frac{2}{3}} c^{\frac{4}{3}} x^{2} + a^{\frac{2}{3}} \sqrt [3]{b} c^{\frac{2}{3}} x \sqrt [3]{- a c^{2} + d^{2}} + a^{\frac{2}{3}} \left (- a c^{2} + d^{2}\right )^{\frac{2}{3}} \right )}}{6 \left (- a c^{2} + d^{2}\right )^{\frac{5}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} c^{\frac{7}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} c^{\frac{2}{3}} x}{3 \sqrt [3]{- a c^{2} + d^{2}}} + \frac{1}{3}\right ) \right )}}{3 \left (- a c^{2} + d^{2}\right )^{\frac{5}{3}}} + \frac{c}{2 x^{2} \left (- a c^{2} + d^{2}\right )} - \frac{d \sqrt{a + b x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{1}{2},1,\frac{1}{3},- \frac{b x^{3}}{a},- \frac{b c^{2} x^{3}}{a c^{2} - d^{2}} \right )}}{2 a x^{2} \sqrt{1 + \frac{b x^{3}}{a}} \left (- a c^{2} + d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

b**(2/3)*c**(7/3)*log(b**(1/3)*c**(2/3)*x - (-a*c**2 + d**2)**(1/3))/(3*(-a*c**2
+ d**2)**(5/3)) - b**(2/3)*c**(7/3)*log(a**(2/3)*b**(2/3)*c**(4/3)*x**2 + a**(2
/3)*b**(1/3)*c**(2/3)*x*(-a*c**2 + d**2)**(1/3) + a**(2/3)*(-a*c**2 + d**2)**(2/
3))/(6*(-a*c**2 + d**2)**(5/3)) - sqrt(3)*b**(2/3)*c**(7/3)*atan(sqrt(3)*(2*b**(
1/3)*c**(2/3)*x/(3*(-a*c**2 + d**2)**(1/3)) + 1/3))/(3*(-a*c**2 + d**2)**(5/3))
+ c/(2*x**2*(-a*c**2 + d**2)) - d*sqrt(a + b*x**3)*appellf1(-2/3, 1/2, 1, 1/3, -
b*x**3/a, -b*c**2*x**3/(a*c**2 - d**2))/(2*a*x**2*sqrt(1 + b*x**3/a)*(-a*c**2 +
d**2))

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Mathematica [B]  time = 6.67882, size = 1044, normalized size = 3.22 \[ \frac{7 b^2 c^2 d F_1\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 ) x^4}{8 \sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (14 a \left (a c^2-d^2\right ) F_1\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 )-3 b x^3 \left (2 a F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right ) c^2+\left (a c^2-d^2\right ) F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{b x^3}{a},-\frac{b c^2 x^3}{a c^2-d^2}\right )\right )\right )}-\frac{2 b d^3 F_1\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 ) x}{\sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (8 a \left (a c^2-d^2\right ) F_1\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 F_1\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 ) c^2+\left (a c^2-d^2\right ) F_1\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{10 a b c^2 d F_1\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 ) x}{\sqrt{b x^3+a} \left (b c^2 x^3+a c^2-d^2\right ) \left (8 a \left (a c^2-d^2\right ) F_1\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 F_1\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 ) c^2+\left (a c^2-d^2\right ) F_1\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{b^{2/3} c^{7/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} c^{2/3} x-\sqrt [3]{a c^2-d^2}}{\sqrt{3} \sqrt [3]{a c^2-d^2}}\right )}{\sqrt{3} \left (a c^2-d^2\right )^{5/3}}-\frac{b^{2/3} c^{7/3} \log \left (\sqrt [3]{b} c^{2/3} x+\sqrt [3]{a c^2-d^2}\right )}{3 \left (a c^2-d^2\right )^{5/3}}+\frac{b^{2/3} c^{7/3} \log \left (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}\right )}{6 \left (a c^2-d^2\right )^{5/3}}+\frac{d \sqrt{b x^3+a}}{2 a \left (a c^2-d^2\right ) x^2}-\frac{c}{2 \left (a c^2-d^2\right ) x^2} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x^3*(a*c + b*c*x^3 + d*Sqrt[a + b*x^3])),x]

[Out]

-c/(2*(a*c^2 - d^2)*x^2) + (d*Sqrt[a + b*x^3])/(2*a*(a*c^2 - d^2)*x^2) + (10*a*b
*c^2*d*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*AppellF
1[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))]))) - (2*b
*d^3*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)*Ap
pellF1[4/3, 3/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))]))) + (7*b^2
*c^2*d*x^4*AppellF1[4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))
])/(8*Sqrt[a + b*x^3]*(a*c^2 - d^2 + b*c^2*x^3)*(14*a*(a*c^2 - d^2)*AppellF1[4/3
, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] - 3*b*x^3*(2*a*c^2*Ap
pellF1[7/3, 1/2, 2, 10/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))] + (a*c^2 -
d^2)*AppellF1[7/3, 3/2, 1, 10/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))])))
- (b^(2/3)*c^(7/3)*ArcTan[(-(a*c^2 - d^2)^(1/3) + 2*b^(1/3)*c^(2/3)*x)/(Sqrt[3]
*(a*c^2 - d^2)^(1/3))])/(Sqrt[3]*(a*c^2 - d^2)^(5/3)) - (b^(2/3)*c^(7/3)*Log[(a*
c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x])/(3*(a*c^2 - d^2)^(5/3)) + (b^(2/3)*c^(7/3
)*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])/(6*(a*c^2 - d^2)^(5/3))

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Maple [C]  time = 0.052, size = 1789, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)

[Out]

1/3*c/d^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*ln(x+(1/c^2/b*(a*c^2-d^2))^(1/3))-1/6*c/d^
2/(1/c^2/b*(a*c^2-d^2))^(2/3)*ln(x^2-x*(1/c^2/b*(a*c^2-d^2))^(1/3)+(1/c^2/b*(a*c
^2-d^2))^(2/3))+1/3*c/d^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)
*(2/(1/c^2/b*(a*c^2-d^2))^(1/3)*x-1))-1/3*a*c^3/(a*c^2-d^2)/d^2/(1/c^2/b*(a*c^2-
d^2))^(2/3)*ln(x+(1/c^2/b*(a*c^2-d^2))^(1/3))+1/6*a*c^3/(a*c^2-d^2)/d^2/(1/c^2/b
*(a*c^2-d^2))^(2/3)*ln(x^2-x*(1/c^2/b*(a*c^2-d^2))^(1/3)+(1/c^2/b*(a*c^2-d^2))^(
2/3))-1/3*a*c^3/(a*c^2-d^2)/d^2/(1/c^2/b*(a*c^2-d^2))^(2/3)*3^(1/2)*arctan(1/3*3
^(1/2)*(2/(1/c^2/b*(a*c^2-d^2))^(1/3)*x-1))-1/2*c/(a*c^2-d^2)/x^2+1/2*d/a/(a*c^2
-d^2)/x^2*(b*x^3+a)^(1/2)+1/2*I*d/a/(a*c^2-d^2)*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))+
2/3*I/a/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)*El
lipticF(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))-2/3*I/(a*c^2-d^2)*c^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/(a*c^2-d^2)/b^2*c^2/d*2^(1/2)*sum(1/_alpha^2*(-a*b^2)^(1
/3)*(1/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-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*((-a*b^2)^(1/3)+I*3^(1/2)*(-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))*EllipticPi(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),-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^
2)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^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)),_alp
ha=RootOf(_Z^3*b*c^2+a*c^2-d^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*c*x^3 + a*c + sqrt(b*x^3 + a)*d)*x^3), x)