3.1978 \(\int \frac{x^3}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=155 \[ -\frac{7 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{10/3}}+\frac{7 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{10/3}}-\frac{7 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{10/3}}-\frac{7 b x}{3 a^3}+\frac{7 x^4}{12 a^2}-\frac{x^7}{3 a \left (a x^3+b\right )} \]

[Out]

(-7*b*x)/(3*a^3) + (7*x^4)/(12*a^2) - x^7/(3*a*(b + a*x^3)) - (7*b^(4/3)*ArcTan[
(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(10/3)) + (7*b^(4/3)*Lo
g[b^(1/3) + a^(1/3)*x])/(9*a^(10/3)) - (7*b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*
x + a^(2/3)*x^2])/(18*a^(10/3))

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Rubi [A]  time = 0.225125, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ -\frac{7 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{10/3}}+\frac{7 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{10/3}}-\frac{7 b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{10/3}}-\frac{7 b x}{3 a^3}+\frac{7 x^4}{12 a^2}-\frac{x^7}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b/x^3)^2,x]

[Out]

(-7*b*x)/(3*a^3) + (7*x^4)/(12*a^2) - x^7/(3*a*(b + a*x^3)) - (7*b^(4/3)*ArcTan[
(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(10/3)) + (7*b^(4/3)*Lo
g[b^(1/3) + a^(1/3)*x])/(9*a^(10/3)) - (7*b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*
x + a^(2/3)*x^2])/(18*a^(10/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x^{7}}{3 a \left (a x^{3} + b\right )} + \frac{7 x^{4}}{12 a^{2}} - \frac{7 \int b\, dx}{3 a^{3}} + \frac{7 b^{\frac{4}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 a^{\frac{10}{3}}} - \frac{7 b^{\frac{4}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{18 a^{\frac{10}{3}}} - \frac{7 \sqrt{3} b^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 a^{\frac{10}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**7/(3*a*(a*x**3 + b)) + 7*x**4/(12*a**2) - 7*Integral(b, x)/(3*a**3) + 7*b**(
4/3)*log(a**(1/3)*x + b**(1/3))/(9*a**(10/3)) - 7*b**(4/3)*log(a**(2/3)*x**2 - a
**(1/3)*b**(1/3)*x + b**(2/3))/(18*a**(10/3)) - 7*sqrt(3)*b**(4/3)*atan(sqrt(3)*
(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(9*a**(10/3))

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Mathematica [A]  time = 0.170798, size = 140, normalized size = 0.9 \[ \frac{-14 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+9 a^{4/3} x^4+28 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-28 \sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-\frac{12 \sqrt [3]{a} b^2 x}{a x^3+b}-72 \sqrt [3]{a} b x}{36 a^{10/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b/x^3)^2,x]

[Out]

(-72*a^(1/3)*b*x + 9*a^(4/3)*x^4 - (12*a^(1/3)*b^2*x)/(b + a*x^3) - 28*Sqrt[3]*b
^(4/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 28*b^(4/3)*Log[b^(1/3) + a^
(1/3)*x] - 14*b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(36*a^(10/
3))

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Maple [A]  time = 0.011, size = 133, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,{a}^{2}}}-2\,{\frac{bx}{{a}^{3}}}-{\frac{{b}^{2}x}{3\,{a}^{3} \left ( a{x}^{3}+b \right ) }}+{\frac{7\,{b}^{2}}{9\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{7\,{b}^{2}}{18\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,{b}^{2}\sqrt{3}}{9\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^3)^2,x)

[Out]

1/4*x^4/a^2-2*b*x/a^3-1/3/a^3*b^2*x/(a*x^3+b)+7/9/a^4*b^2/(b/a)^(2/3)*ln(x+(b/a)
^(1/3))-7/18/a^4*b^2/(b/a)^(2/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))+7/9/a^4*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225632, size = 232, normalized size = 1.5 \[ -\frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (a b x^{3} + b^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3}{\left (a b x^{3} + b^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (a b x^{3} + b^{2}\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}}}{3 \, \left (\frac{b}{a}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, a^{2} x^{7} - 21 \, a b x^{4} - 28 \, b^{2} x\right )}\right )}}{108 \,{\left (a^{4} x^{3} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(a + b/x^3)^2,x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*(14*sqrt(3)*(a*b*x^3 + b^2)*(b/a)^(1/3)*log(x^2 - x*(b/a)^(1/3) +
(b/a)^(2/3)) - 28*sqrt(3)*(a*b*x^3 + b^2)*(b/a)^(1/3)*log(x + (b/a)^(1/3)) + 84
*(a*b*x^3 + b^2)*(b/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*x - sqrt(3)*(b/a)^(1/3))/(b/
a)^(1/3)) - 3*sqrt(3)*(3*a^2*x^7 - 21*a*b*x^4 - 28*b^2*x))/(a^4*x^3 + a^3*b)

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Sympy [A]  time = 1.92968, size = 65, normalized size = 0.42 \[ - \frac{b^{2} x}{3 a^{4} x^{3} + 3 a^{3} b} + \operatorname{RootSum}{\left (729 t^{3} a^{10} - 343 b^{4}, \left ( t \mapsto t \log{\left (\frac{9 t a^{3}}{7 b} + x \right )} \right )\right )} + \frac{x^{4}}{4 a^{2}} - \frac{2 b x}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b/x**3)**2,x)

[Out]

-b**2*x/(3*a**4*x**3 + 3*a**3*b) + RootSum(729*_t**3*a**10 - 343*b**4, Lambda(_t
, _t*log(9*_t*a**3/(7*b) + x))) + x**4/(4*a**2) - 2*b*x/a**3

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GIAC/XCAS [A]  time = 0.228954, size = 198, normalized size = 1.28 \[ -\frac{7 \, b \left (-\frac{b}{a}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} - \frac{b^{2} x}{3 \,{\left (a x^{3} + b\right )} a^{3}} + \frac{7 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4}} + \frac{7 \, \left (-a^{2} b\right )^{\frac{1}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{18 \, a^{4}} + \frac{a^{6} x^{4} - 8 \, a^{5} b x}{4 \, a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/9*b*(-b/a)^(1/3)*ln(abs(x - (-b/a)^(1/3)))/a^3 - 1/3*b^2*x/((a*x^3 + b)*a^3)
+ 7/9*sqrt(3)*(-a^2*b)^(1/3)*b*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1
/3))/a^4 + 7/18*(-a^2*b)^(1/3)*b*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^4 + 1
/4*(a^6*x^4 - 8*a^5*b*x)/a^8