3.1965 \(\int \frac{x^3}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]

[Out]

-((b*x)/a^2) + x^4/(4*a) - (b^(4/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1
/3))])/(Sqrt[3]*a^(7/3)) + (b^(4/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(7/3)) - (b^(
4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(7/3))

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Rubi [A]  time = 0.202148, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^3),x]

[Out]

-((b*x)/a^2) + x^4/(4*a) - (b^(4/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1
/3))])/(Sqrt[3]*a^(7/3)) + (b^(4/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(7/3)) - (b^(
4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(7/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{4}}{4 a} - \frac{\int b\, dx}{a^{2}} + \frac{b^{\frac{4}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{7}{3}}} - \frac{b^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{7}{3}}} - \frac{\sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4/(4*a) - Integral(b, x)/a**2 + b**(4/3)*log(a**(1/3)*x + b**(1/3))/(3*a**(7/
3)) - b**(4/3)*log(a**(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(6*a**(7/3))
- sqrt(3)*b**(4/3)*atan(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(3*a**(
7/3))

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Mathematica [A]  time = 0.0400798, size = 120, normalized size = 0.91 \[ \frac{-2 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+3 a^{4/3} x^4+4 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-4 \sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-12 \sqrt [3]{a} b x}{12 a^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^3),x]

[Out]

(-12*a^(1/3)*b*x + 3*a^(4/3)*x^4 - 4*Sqrt[3]*b^(4/3)*ArcTan[(1 - (2*a^(1/3)*x)/b
^(1/3))/Sqrt[3]] + 4*b^(4/3)*Log[b^(1/3) + a^(1/3)*x] - 2*b^(4/3)*Log[b^(2/3) -
a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(12*a^(7/3))

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Maple [A]  time = 0.004, size = 115, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,a}}-{\frac{bx}{{a}^{2}}}+{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242816, size = 163, normalized size = 1.23 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3} b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 12 \, b \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}}}{3 \, \left (\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (a x^{4} - 4 \, b x\right )}\right )}}{36 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(2*sqrt(3)*b*(b/a)^(1/3)*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3)) -
4*sqrt(3)*b*(b/a)^(1/3)*log(x + (b/a)^(1/3)) + 12*b*(b/a)^(1/3)*arctan(-1/3*(2*s
qrt(3)*x - sqrt(3)*(b/a)^(1/3))/(b/a)^(1/3)) - 3*sqrt(3)*(a*x^4 - 4*b*x))/a^2

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Sympy [A]  time = 1.35894, size = 37, normalized size = 0.28 \[ \operatorname{RootSum}{\left (27 t^{3} a^{7} - b^{4}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2}}{b} + x \right )} \right )\right )} + \frac{x^{4}}{4 a} - \frac{b x}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**3),x)

[Out]

RootSum(27*_t**3*a**7 - b**4, Lambda(_t, _t*log(3*_t*a**2/b + x))) + x**4/(4*a)
- b*x/a**2

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GIAC/XCAS [A]  time = 0.229689, size = 174, normalized size = 1.32 \[ -\frac{b \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} + \frac{a^{3} x^{4} - 4 \, a^{2} b x}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b*(-b/a)^(1/3)*ln(abs(x - (-b/a)^(1/3)))/a^2 + 1/3*sqrt(3)*(-a^2*b)^(1/3)*b
*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^3 + 1/6*(-a^2*b)^(1/3)*
b*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^3 + 1/4*(a^3*x^4 - 4*a^2*b*x)/a^4