3.324 \(\int \frac{x^3}{a+b x^3} \, dx\)

Optimal. Leaf size=119 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3}}+\frac{x}{b} \]

[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 = 0.146383, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3}}+\frac{x}{b} \]

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 = 31.1975, size = 112, normalized size = 0.94 \[ - \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{4}{3}}} + \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{4}{3}}} + \frac{x}{b} \]

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 = 0.0288017, size = 108, normalized size = 0.91 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{b} x}{6 b^{4/3}} \]

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 = 0.003, size = 99, 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 = 0.222932, size = 157, normalized size = 1.32 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, \sqrt{3} x + 6 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, b} \]

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 = 1.20084, size = 22, normalized size = 0.18 \[ \operatorname{RootSum}{\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (- 3 t b + x \right )} \right )\right )} + \frac{x}{b} \]

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 = 0.217019, size = 150, normalized size = 1.26 \[ \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} + \frac{x}{b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{2}} \]

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)*
arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^2 - 1/6*(-a*b^2)^(1/3)*l
n(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^2