3.32 \(\int \frac{\left (c+d x^3\right )^{2/3}}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=241 \[ \frac{2 c \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{9 a^{5/3} \sqrt [3]{b c-a d}}-\frac{2 c \tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b c-a d}}-\frac{c \log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{9 a^{5/3} \sqrt [3]{b c-a d}}+\frac{x \left (c+d x^3\right )^{2/3}}{3 a \left (a+b x^3\right )} \]

[Out]

(x*(c + d*x^3)^(2/3))/(3*a*(a + b*x^3)) - (2*c*ArcTan[(a^(1/3) - (2*(b*c - a*d)^
(1/3)*x)/(c + d*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(b*c - a*d)^(
1/3)) + (2*c*Log[a^(1/3) + ((b*c - a*d)^(1/3)*x)/(c + d*x^3)^(1/3)])/(9*a^(5/3)*
(b*c - a*d)^(1/3)) - (c*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)])/(9*a^(5/3)*(b*c - a*d)^(1/3)
)

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Rubi [A]  time = 0.384953, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{2 c \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\sqrt [3]{a}\right )}{9 a^{5/3} \sqrt [3]{b c-a d}}-\frac{2 c \tan ^{-1}\left (\frac{\sqrt [3]{a}-\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} \sqrt [3]{b c-a d}}-\frac{c \log \left (a^{2/3}-\frac{\sqrt [3]{a} x \sqrt [3]{b c-a d}}{\sqrt [3]{c+d x^3}}+\frac{x^2 (b c-a d)^{2/3}}{\left (c+d x^3\right )^{2/3}}\right )}{9 a^{5/3} \sqrt [3]{b c-a d}}+\frac{x \left (c+d x^3\right )^{2/3}}{3 a \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x^3)^(2/3))/(3*a*(a + b*x^3)) - (2*c*ArcTan[(a^(1/3) - (2*(b*c - a*d)^
(1/3)*x)/(c + d*x^3)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(b*c - a*d)^(
1/3)) + (2*c*Log[a^(1/3) + ((b*c - a*d)^(1/3)*x)/(c + d*x^3)^(1/3)])/(9*a^(5/3)*
(b*c - a*d)^(1/3)) - (c*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)])/(9*a^(5/3)*(b*c - a*d)^(1/3)
)

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Rubi in Sympy [A]  time = 44.9982, size = 216, normalized size = 0.9 \[ \frac{x \left (c + d x^{3}\right )^{\frac{2}{3}}}{3 a \left (a + b x^{3}\right )} - \frac{2 c \log{\left (\sqrt [3]{a} - \frac{x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{a d - b c}} + \frac{c \log{\left (a^{\frac{2}{3}} + \frac{\sqrt [3]{a} x \sqrt [3]{a d - b c}}{\sqrt [3]{c + d x^{3}}} + \frac{x^{2} \left (a d - b c\right )^{\frac{2}{3}}}{\left (c + d x^{3}\right )^{\frac{2}{3}}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{a d - b c}} + \frac{2 \sqrt{3} c \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 x \sqrt [3]{a d - b c}}{3 \sqrt [3]{c + d x^{3}}}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \sqrt [3]{a d - b c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c + d*x**3)**(2/3)/(3*a*(a + b*x**3)) - 2*c*log(a**(1/3) - x*(a*d - b*c)**(1/
3)/(c + d*x**3)**(1/3))/(9*a**(5/3)*(a*d - b*c)**(1/3)) + c*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))/(9*a**(5/3)*(a*d - b*c)**(1/3)) + 2*sqrt(3)*c*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))/(9*a**(5/3)*(a*d
- b*c)**(1/3))

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Mathematica [A]  time = 0.5817, size = 198, normalized size = 0.82 \[ \frac{c \left (\log \left (a^{2/3}+\frac{\sqrt [3]{a} x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}+\frac{x^2 (a d-b c)^{2/3}}{\left (c x^3+d\right )^{2/3}}\right )-2 \log \left (\sqrt [3]{a}-\frac{x \sqrt [3]{a d-b c}}{\sqrt [3]{c x^3+d}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a d-b c}}{\sqrt [3]{a} \sqrt [3]{c x^3+d}}+1}{\sqrt{3}}\right )\right )}{9 a^{5/3} \sqrt [3]{a d-b c}}+\frac{x \left (c+d x^3\right )^{2/3}}{3 a \left (a+b x^3\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*(c + d*x^3)^(2/3))/(3*a*(a + b*x^3)) + (c*(2*Sqrt[3]*ArcTan[(1 + (2*(-(b*c) +
a*d)^(1/3)*x)/(a^(1/3)*(d + c*x^3)^(1/3)))/Sqrt[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)]))/(9*a^(5/
3)*(-(b*c) + a*d)^(1/3))

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ( d{x}^{3}+c \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((d*x^3+c)^(2/3)/(b*x^3+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{2}{3}}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(2/3)/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)^(2/3)/(b*x^3 + a)^2, 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((d*x^3 + c)^(2/3)/(b*x^3 + a)^2,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((d*x**3+c)**(2/3)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{2}{3}}}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(2/3)/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^(2/3)/(b*x^3 + a)^2, x)