3.2607 \(\int \frac{x^{3+4 (-1+n)}}{a+b x^n} \, dx\)

Optimal. Leaf size=64 \[ \frac{a^2 x^n}{b^3 n}-\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]

[Out]

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

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Rubi [A]  time = 0.0310129, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac{a^2 x^n}{b^3 n}-\frac{a^3 \log \left (a+b x^n\right )}{b^4 n}-\frac{a x^{2 n}}{2 b^2 n}+\frac{x^{3 n}}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(3 + 4*(-1 + n))/(a + b*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0213367, size = 52, normalized size = 0.81 \[ \frac{b x^n \left (6 a^2-3 a b x^n+2 b^2 x^{2 n}\right )-6 a^3 \log \left (a+b x^n\right )}{6 b^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^n*(6*a^2 - 3*a*b*x^n + 2*b^2*x^(2*n)) - 6*a^3*Log[a + b*x^n])/(6*b^4*n)

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Maple [A]  time = 0., size = 69, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{3}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,bn}}-{\frac{a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{2}n}}-{\frac{{a}^{3}\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),x)

[Out]

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

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Maxima [A]  time = 0.985651, size = 81, normalized size = 1.27 \begin{align*} -\frac{a^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} + \frac{2 \, b^{2} x^{3 \, n} - 3 \, a b x^{2 \, n} + 6 \, a^{2} x^{n}}{6 \, b^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.07423, size = 113, normalized size = 1.77 \begin{align*} \frac{2 \, b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} + 6 \, a^{2} b x^{n} - 6 \, a^{3} \log \left (b x^{n} + a\right )}{6 \, 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),x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^(3*n) - 3*a*b^2*x^(2*n) + 6*a^2*b*x^n - 6*a^3*log(b*x^n + a))/(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),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}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(4*n - 1)/(b*x^n + a), x)