3.2475 \(\int \frac{x}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=33 \[ \frac{x^2 \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3} \]

[Out]

(x^2*Hypergeometric2F1[3, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^3)

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Rubi [A]  time = 0.0267032, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2 \, _2F_1\left (3,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^n)^3,x]

[Out]

(x^2*Hypergeometric2F1[3, 2/n, (2 + n)/n, -((b*x^n)/a)])/(2*a^3)

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Rubi in Sympy [A]  time = 3.43929, size = 22, normalized size = 0.67 \[ \frac{x^{2}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*hyper((3, 2/n), ((n + 2)/n,), -b*x**n/a)/(2*a**3)

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Mathematica [B]  time = 0.0981823, size = 74, normalized size = 2.24 \[ \frac{x^2 \left (\left (n^2-3 n+2\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+\frac{a \left (a (3 n-2)+2 b (n-1) x^n\right )}{\left (a+b x^n\right )^2}\right )}{2 a^3 n^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b*x^n)^3,x]

[Out]

(x^2*((a*(a*(-2 + 3*n) + 2*b*(-1 + n)*x^n))/(a + b*x^n)^2 + (2 - 3*n + n^2)*Hype
rgeometric2F1[1, 2/n, (2 + n)/n, -((b*x^n)/a)]))/(2*a^3*n^2)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)^3,x)

[Out]

int(x/(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (n^{2} - 3 \, n + 2\right )} \int \frac{x}{a^{2} b n^{2} x^{n} + a^{3} n^{2}}\,{d x} + \frac{2 \, b{\left (n - 1\right )} x^{2} x^{n} + a{\left (3 \, n - 2\right )} x^{2}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(n^2 - 3*n + 2)*integrate(x/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(2*b*(n - 1)*x^2
*x^n + a*(3*n - 2)*x^2)/(a^2*b^2*n^2*x^(2*n) + 2*a^3*b*n^2*x^n + a^4*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b*x**n)**3,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^n + a)^3, x)