3.2541 \(\int x^{-1-5 n} \left (a+b x^n\right )^3 \, dx\)

Optimal. Leaf size=50 \[ \frac{b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]

[Out]

-(a + b*x^n)^4/(5*a*n*x^(5*n)) + (b*(a + b*x^n)^4)/(20*a^2*n*x^(4*n))

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Rubi [A]  time = 0.0558028, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 5*n)*(a + b*x^n)^3,x]

[Out]

-(a + b*x^n)^4/(5*a*n*x^(5*n)) + (b*(a + b*x^n)^4)/(20*a^2*n*x^(4*n))

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Rubi in Sympy [A]  time = 10.9464, size = 51, normalized size = 1.02 \[ - \frac{a^{3} x^{- 5 n}}{5 n} - \frac{3 a^{2} b x^{- 4 n}}{4 n} - \frac{a b^{2} x^{- 3 n}}{n} - \frac{b^{3} x^{- 2 n}}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-5*n)*(a+b*x**n)**3,x)

[Out]

-a**3*x**(-5*n)/(5*n) - 3*a**2*b*x**(-4*n)/(4*n) - a*b**2*x**(-3*n)/n - b**3*x**
(-2*n)/(2*n)

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Mathematica [A]  time = 0.0267365, size = 48, normalized size = 0.96 \[ -\frac{x^{-5 n} \left (4 a^3+15 a^2 b x^n+20 a b^2 x^{2 n}+10 b^3 x^{3 n}\right )}{20 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 5*n)*(a + b*x^n)^3,x]

[Out]

-(4*a^3 + 15*a^2*b*x^n + 20*a*b^2*x^(2*n) + 10*b^3*x^(3*n))/(20*n*x^(5*n))

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Maple [A]  time = 0.029, size = 63, normalized size = 1.3 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ( -{\frac{{a}^{3}}{5\,n}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,n}}-{\frac{a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{3\,{a}^{2}b{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-5*n)*(a+b*x^n)^3,x)

[Out]

(-1/5*a^3/n-1/2*b^3/n*exp(n*ln(x))^3-a*b^2/n*exp(n*ln(x))^2-3/4*a^2*b/n*exp(n*ln
(x)))/exp(n*ln(x))^5

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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((b*x^n + a)^3*x^(-5*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223662, size = 65, normalized size = 1.3 \[ -\frac{10 \, b^{3} x^{3 \, n} + 20 \, a b^{2} x^{2 \, n} + 15 \, a^{2} b x^{n} + 4 \, a^{3}}{20 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-5*n - 1),x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^(3*n) + 20*a*b^2*x^(2*n) + 15*a^2*b*x^n + 4*a^3)/(n*x^(5*n))

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Sympy [A]  time = 118.366, size = 60, normalized size = 1.2 \[ \begin{cases} - \frac{a^{3} x^{- 5 n}}{5 n} - \frac{3 a^{2} b x^{- 4 n}}{4 n} - \frac{a b^{2} x^{- 3 n}}{n} - \frac{b^{3} x^{- 2 n}}{2 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-5*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((-a**3*x**(-5*n)/(5*n) - 3*a**2*b*x**(-4*n)/(4*n) - a*b**2*x**(-3*n)/n
- b**3*x**(-2*n)/(2*n), Ne(n, 0)), ((a + b)**3*log(x), True))

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GIAC/XCAS [A]  time = 0.222688, size = 69, normalized size = 1.38 \[ -\frac{{\left (10 \, b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 20 \, a b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 15 \, a^{2} b e^{\left (n{\rm ln}\left (x\right )\right )} + 4 \, a^{3}\right )} e^{\left (-5 \, n{\rm ln}\left (x\right )\right )}}{20 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^3*x^(-5*n - 1),x, algorithm="giac")

[Out]

-1/20*(10*b^3*e^(3*n*ln(x)) + 20*a*b^2*e^(2*n*ln(x)) + 15*a^2*b*e^(n*ln(x)) + 4*
a^3)*e^(-5*n*ln(x))/n