3.323 \(\int \frac{x^4}{a+b x^3} \, dx\)

Optimal. Leaf size=124 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}+\frac{x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.174805, antiderivative size = 124, 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 \[ -\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.0274, size = 116, normalized size = 0.94 \[ \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}}} - \frac{a^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a),x)

[Out]

a**(2/3)*log(a**(1/3) + b**(1/3)*x)/(3*b**(5/3)) - a**(2/3)*log(a**(2/3) - a**(1
/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*b**(5/3)) + sqrt(3)*a**(2/3)*atan(sqrt(3)*(a*
*(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(5/3)) + x**2/(2*b)

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Mathematica [A]  time = 0.0614524, size = 111, normalized size = 0.9 \[ \frac{-a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+3 b^{2/3} x^2}{6 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.009, size = 102, normalized size = 0.8 \[{\frac{{x}^{2}}{2\,b}}+{\frac{a}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a),x)

[Out]

1/2*x^2/b+1/3*a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-1/6*a/b^2/(a/b)^(1/3)*ln(x^2-x
*(a/b)^(1/3)+(a/b)^(2/3))-1/3*a/b^2*3^(1/2)/(a/b)^(1/3)*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^4/(b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220925, size = 190, normalized size = 1.53 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3} x^{2} - \sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) + 6 \, \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.25563, size = 32, normalized size = 0.26 \[ \operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a),x)

[Out]

RootSum(27*_t**3*b**5 - a**2, Lambda(_t, _t*log(9*_t**2*b**3/a + x))) + x**2/(2*
b)

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GIAC/XCAS [A]  time = 0.222057, size = 154, normalized size = 1.24 \[ \frac{x^{2}}{2 \, b} + \frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b} + \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b^{3}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a),x, algorithm="giac")

[Out]

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