3.2632 \(\int \frac{x^{-1+4 n}}{(a+b x^n)^3} \, dx\)

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

[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*Log[a + b*x^n])/(b^4*n)

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Rubi [A]  time = 0.043986, 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, Rules used = {266, 43} \[ -\frac{3 a^2}{b^4 n \left (a+b x^n\right )}+\frac{a^3}{2 b^4 n \left (a+b x^n\right )^2}-\frac{3 a \log \left (a+b x^n\right )}{b^4 n}+\frac{x^n}{b^3 n} \]

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*Log[a + b*x^n])/(b^4*n)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{-1+4 n}}{\left (a+b x^n\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^3}-\frac{a^3}{b^3 (a+b x)^3}+\frac{3 a^2}{b^3 (a+b x)^2}-\frac{3 a}{b^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{x^n}{b^3 n}+\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}\\ \end{align*}

Mathematica [A]  time = 0.130265, size = 51, normalized size = 0.73 \[ -\frac{\frac{a^2 \left (5 a+6 b x^n\right )}{\left (a+b x^n\right )^2}+6 a \log \left (a+b x^n\right )-2 b x^n}{2 b^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*b*x^n + (a^2*(5*a + 6*b*x^n))/(a + b*x^n)^2 + 6*a*Log[a + b*x^n])/(2*b^4*n)

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Maple [A]  time = 0.028, size = 75, normalized size = 1.1 \begin{align*}{\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}} \end{align*}

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 = 1.00055, size = 123, normalized size = 1.76 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)/(a+b*x^n)^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 = 1.06004, size = 216, normalized size = 3.09 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)/(a+b*x^n)^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. \begin{align*} \text{Timed out} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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