3.2626 \(\int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac{2 a}{b^3 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^3 n} \]

[Out]

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

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Rubi [A]  time = 0.0892145, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^2}{2 b^3 n \left (a+b x^n\right )^2}+\frac{2 a}{b^3 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 3*n)/(a + b*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0085, size = 46, normalized size = 0.82 \[ - \frac{a^{2}}{2 b^{3} n \left (a + b x^{n}\right )^{2}} + \frac{2 a}{b^{3} n \left (a + b x^{n}\right )} + \frac{\log{\left (a + b x^{n} \right )}}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3*n)/(a+b*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0546697, size = 42, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 b x^n\right )}{\left (a+b x^n\right )^2}+2 \log \left (a+b x^n\right )}{2 b^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 3*n)/(a + b*x^n)^3,x]

[Out]

((a*(3*a + 4*b*x^n))/(a + b*x^n)^2 + 2*Log[a + b*x^n])/(2*b^3*n)

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Maple [A]  time = 0.039, size = 57, normalized size = 1. \[{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{3\,{a}^{2}}{2\,{b}^{3}n}}+2\,{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}} \right ) }+{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(a+b*x^n)^3,x)

[Out]

(3/2*a^2/b^3/n+2*a/b^2/n*exp(n*ln(x)))/(a+b*exp(n*ln(x)))^2+1/b^3/n*ln(a+b*exp(n
*ln(x)))

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Maxima [A]  time = 1.45562, size = 89, normalized size = 1.59 \[ \frac{4 \, a b x^{n} + 3 \, a^{2}}{2 \,{\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{3} n\right )}} + \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(b*x^n + a)^3,x, algorithm="maxima")

[Out]

1/2*(4*a*b*x^n + 3*a^2)/(b^5*n*x^(2*n) + 2*a*b^4*n*x^n + a^2*b^3*n) + log((b*x^n
+ a)/b)/(b^3*n)

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Fricas [A]  time = 0.221159, size = 103, normalized size = 1.84 \[ \frac{4 \, a b x^{n} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (b^{5} n x^{2 \, n} + 2 \, a b^{4} n x^{n} + a^{2} b^{3} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(b*x^n + a)^3,x, algorithm="fricas")

[Out]

1/2*(4*a*b*x^n + 3*a^2 + 2*(b^2*x^(2*n) + 2*a*b*x^n + a^2)*log(b*x^n + a))/(b^5*
n*x^(2*n) + 2*a*b^4*n*x^n + a^2*b^3*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+3*n)/(a+b*x**n)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3*n - 1)/(b*x^n + a)^3,x, algorithm="giac")

[Out]

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