3.358 \(\int \frac{x^3}{a-b x^3} \, dx\)

Optimal. Leaf size=120 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3}}-\frac{x}{b} \]

[Out]

-(x/b) + (a^(1/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^
(4/3)) - (a^(1/3)*Log[a^(1/3) - b^(1/3)*x])/(3*b^(4/3)) + (a^(1/3)*Log[a^(2/3) +
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(4/3))

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Rubi [A]  time = 0.13795, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3}}-\frac{x}{b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a - b*x^3),x]

[Out]

-(x/b) + (a^(1/3)*ArcTan[(a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^
(4/3)) - (a^(1/3)*Log[a^(1/3) - b^(1/3)*x])/(3*b^(4/3)) + (a^(1/3)*Log[a^(2/3) +
a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(4/3))

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Rubi in Sympy [A]  time = 28.1144, size = 112, normalized size = 0.93 \[ - \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} - \sqrt [3]{b} x \right )}}{3 b^{\frac{4}{3}}} + \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{4}{3}}} - \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(-b*x**3+a),x)

[Out]

-a**(1/3)*log(a**(1/3) - b**(1/3)*x)/(3*b**(4/3)) + a**(1/3)*log(a**(2/3) + a**(
1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(4/3)) + sqrt(3)*a**(1/3)*atan(sqrt(3)*(a
**(1/3)/3 + 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(4/3)) - x/b

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Mathematica [A]  time = 0.0291933, size = 108, normalized size = 0.9 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-6 \sqrt [3]{b} x}{6 b^{4/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a - b*x^3),x]

[Out]

(-6*b^(1/3)*x + 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] -
2*a^(1/3)*Log[a^(1/3) - b^(1/3)*x] + a^(1/3)*Log[a^(2/3) + a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(6*b^(4/3))

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Maple [A]  time = 0.004, size = 101, normalized size = 0.8 \[ -{\frac{x}{b}}-{\frac{a}{3\,{b}^{2}}\ln \left ( x-\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,{b}^{2}}\ln \left ({x}^{2}+x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-b*x^3+a),x)

[Out]

-x/b-1/3*a/b^2/(a/b)^(2/3)*ln(x-(a/b)^(1/3))+1/6*a/b^2/(a/b)^(2/3)*ln(x^2+x*(a/b
)^(1/3)+(a/b)^(2/3))+1/3*a/b^2/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(
1/3)*x+1))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(b*x^3 - a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243661, size = 157, normalized size = 1.31 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 6 \, \sqrt{3} x + 6 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(b*x^3 - a),x, algorithm="fricas")

[Out]

-1/18*sqrt(3)*(sqrt(3)*(-a/b)^(1/3)*log(x^2 - x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 2
*sqrt(3)*(-a/b)^(1/3)*log(x + (-a/b)^(1/3)) + 6*sqrt(3)*x + 6*(-a/b)^(1/3)*arcta
n(-1/3*(2*sqrt(3)*x - sqrt(3)*(-a/b)^(1/3))/(-a/b)^(1/3)))/b

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Sympy [A]  time = 1.25044, size = 24, normalized size = 0.2 \[ - \operatorname{RootSum}{\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log{\left (- 3 t b + x \right )} \right )\right )} - \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(-b*x**3+a),x)

[Out]

-RootSum(27*_t**3*b**4 - a, Lambda(_t, _t*log(-3*_t*b + x))) - x/b

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GIAC/XCAS [A]  time = 0.231407, size = 140, normalized size = 1.17 \[ -\frac{\left (\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} - \frac{x}{b} + \frac{\sqrt{3} \left (a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{2}} + \frac{\left (a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3/(b*x^3 - a),x, algorithm="giac")

[Out]

-1/3*(a/b)^(1/3)*ln(abs(x - (a/b)^(1/3)))/b - x/b + 1/3*sqrt(3)*(a*b^2)^(1/3)*ar
ctan(1/3*sqrt(3)*(2*x + (a/b)^(1/3))/(a/b)^(1/3))/b^2 + 1/6*(a*b^2)^(1/3)*ln(x^2
+ x*(a/b)^(1/3) + (a/b)^(2/3))/b^2