3.2625 \(\int \frac{x^{-1+4 n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=70 \[ \frac{a^3}{2 b^4 n \left (a+b x^n\right )^2}-\frac{3 a^2}{b^4 n \left (a+b x^n\right )}-\frac{3 a \log \left (a+b x^n\right )}{b^4 n}+\frac{x^n}{b^3 n} \]

[Out]

x^n/(b^3*n) + a^3/(2*b^4*n*(a + b*x^n)^2) - (3*a^2)/(b^4*n*(a + b*x^n)) - (3*a*L
og[a + b*x^n])/(b^4*n)

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Rubi [A]  time = 0.101758, antiderivative size = 70, 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 \[ \frac{a^3}{2 b^4 n \left (a+b x^n\right )^2}-\frac{3 a^2}{b^4 n \left (a+b x^n\right )}-\frac{3 a \log \left (a+b x^n\right )}{b^4 n}+\frac{x^n}{b^3 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)/(a + b*x^n)^3,x]

[Out]

x^n/(b^3*n) + a^3/(2*b^4*n*(a + b*x^n)^2) - (3*a^2)/(b^4*n*(a + b*x^n)) - (3*a*L
og[a + b*x^n])/(b^4*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(a+b*x**n)**3,x)

[Out]

a**3/(2*b**4*n*(a + b*x**n)**2) - 3*a**2/(b**4*n*(a + b*x**n)) - 3*a*log(a + b*x
**n)/(b**4*n) + Integral(b**(-3), (x, x**n))/n

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Mathematica [A]  time = 0.0570648, size = 70, normalized size = 1. \[ \frac{a^3}{2 b^4 n \left (a+b x^n\right )^2}-\frac{3 a^2}{b^4 n \left (a+b x^n\right )}-\frac{3 a \log \left (a+b x^n\right )}{b^4 n}+\frac{x^n}{b^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)/(a + b*x^n)^3,x]

[Out]

x^n/(b^3*n) + a^3/(2*b^4*n*(a + b*x^n)^2) - (3*a^2)/(b^4*n*(a + b*x^n)) - (3*a*L
og[a + b*x^n])/(b^4*n)

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Maple [A]  time = 0.049, size = 75, normalized size = 1.1 \[{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{bn}}-{\frac{9\,{a}^{3}}{2\,{b}^{4}n}}-6\,{\frac{{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}} \right ) }-3\,{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)/(a+b*x^n)^3,x)

[Out]

(1/b/n*exp(n*ln(x))^3-9/2*a^3/b^4/n-6*a^2/b^3/n*exp(n*ln(x)))/(a+b*exp(n*ln(x)))
^2-3*a/b^4/n*ln(a+b*exp(n*ln(x)))

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Maxima [A]  time = 1.45841, size = 123, normalized size = 1.76 \[ \frac{2 \, b^{3} x^{3 \, n} + 4 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} - 5 \, a^{3}}{2 \,{\left (b^{6} n x^{2 \, n} + 2 \, a b^{5} n x^{n} + a^{2} b^{4} n\right )}} - \frac{3 \, a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^3,x, algorithm="maxima")

[Out]

1/2*(2*b^3*x^(3*n) + 4*a*b^2*x^(2*n) - 4*a^2*b*x^n - 5*a^3)/(b^6*n*x^(2*n) + 2*a
*b^5*n*x^n + a^2*b^4*n) - 3*a*log((b*x^n + a)/b)/(b^4*n)

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Fricas [A]  time = 0.226444, size = 138, normalized size = 1.97 \[ \frac{2 \, b^{3} x^{3 \, n} + 4 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} - 5 \, a^{3} - 6 \,{\left (a b^{2} x^{2 \, n} + 2 \, a^{2} b x^{n} + a^{3}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (b^{6} n x^{2 \, n} + 2 \, a b^{5} n x^{n} + a^{2} b^{4} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^3,x, algorithm="fricas")

[Out]

1/2*(2*b^3*x^(3*n) + 4*a*b^2*x^(2*n) - 4*a^2*b*x^n - 5*a^3 - 6*(a*b^2*x^(2*n) +
2*a^2*b*x^n + a^3)*log(b*x^n + a))/(b^6*n*x^(2*n) + 2*a*b^5*n*x^n + a^2*b^4*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+4*n)/(a+b*x**n)**3,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^3,x, algorithm="giac")

[Out]

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