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

Optimal. Leaf size=208 \[ \frac{\log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{3 a^{2/3} \sqrt [3]{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b c-a d}}-\frac{\log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{6 a^{2/3} \sqrt [3]{b c-a d}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.457942, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{3 a^{2/3} \sqrt [3]{b c-a d}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b c-a d}}-\frac{\log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{6 a^{2/3} \sqrt [3]{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.6177, size = 185, normalized size = 0.89 \[ - \frac{\log{\left (\sqrt [3]{a} - \frac{x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} + \frac{\log{\left (a^{\frac{2}{3}} + \frac{\sqrt [3]{a} x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} + \frac{x^{2} \left (a d - b c\right )^{\frac{2}{3}}}{\left (c + d x^{3}\right )^{\frac{2}{3}}} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{a d - b c}}{3 \sqrt [3]{c + d x^{3}}}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(a**(1/3) - x*(a*d - b*c)**(1/3)/(c + d*x**3)**(1/3))/(3*a**(2/3)*(a*d - b*c
)**(1/3)) + log(a**(2/3) + a**(1/3)*x*(a*d - b*c)**(1/3)/(c + d*x**3)**(1/3) + x
**2*(a*d - b*c)**(2/3)/(c + d*x**3)**(2/3))/(6*a**(2/3)*(a*d - b*c)**(1/3)) + sq
rt(3)*atan(sqrt(3)*(a**(1/3)/3 + 2*x*(a*d - b*c)**(1/3)/(3*(c + d*x**3)**(1/3)))
/a**(1/3))/(3*a**(2/3)*(a*d - b*c)**(1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.344175, size = 168, normalized size = 0.81 \[ \frac{\log \left (a^{2/3}+\frac{\sqrt [3]{a} x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}+\frac{x^2 (a d-b c)^{2/3}}{\left (c x^3+d\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{a}-\frac{x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a d-b c}}{\sqrt [3]{a} \sqrt [3]{c x^3+d}}+1}{\sqrt{3}}\right )}{6 a^{2/3} \sqrt [3]{a d-b c}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{3}+a}{\frac{1}{\sqrt [3]{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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*x^3 + a)*(d*x^3 + c)^(1/3)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x**3)*(c + d*x**3)**(1/3)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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