3.2556 \(\int x^{-1-n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=83 \[ \frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+5 a^4 b \log (x)-\frac{a^5 x^{-n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]

[Out]

-(a^5/(n*x^n)) + (10*a^3*b^2*x^n)/n + (5*a^2*b^3*x^(2*n))/n + (5*a*b^4*x^(3*n))/(3*n) + (b^5*x^(4*n))/(4*n) +
5*a^4*b*Log[x]

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Rubi [A]  time = 0.0369853, antiderivative size = 83, 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} \[ \frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+5 a^4 b \log (x)-\frac{a^5 x^{-n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*(a + b*x^n)^5,x]

[Out]

-(a^5/(n*x^n)) + (10*a^3*b^2*x^n)/n + (5*a^2*b^3*x^(2*n))/n + (5*a*b^4*x^(3*n))/(3*n) + (b^5*x^(4*n))/(4*n) +
5*a^4*b*Log[x]

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 x^{-1-n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (10 a^3 b^2+\frac{a^5}{x^2}+\frac{5 a^4 b}{x}+10 a^2 b^3 x+5 a b^4 x^2+b^5 x^3\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-n}}{n}+\frac{10 a^3 b^2 x^n}{n}+\frac{5 a^2 b^3 x^{2 n}}{n}+\frac{5 a b^4 x^{3 n}}{3 n}+\frac{b^5 x^{4 n}}{4 n}+5 a^4 b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0374077, size = 73, normalized size = 0.88 \[ \frac{10 a^3 b^2 x^n+5 a^2 b^3 x^{2 n}+5 a^4 b n \log (x)-a^5 x^{-n}+\frac{5}{3} a b^4 x^{3 n}+\frac{1}{4} b^5 x^{4 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*(a + b*x^n)^5,x]

[Out]

(-(a^5/x^n) + 10*a^3*b^2*x^n + 5*a^2*b^3*x^(2*n) + (5*a*b^4*x^(3*n))/3 + (b^5*x^(4*n))/4 + 5*a^4*b*n*Log[x])/n

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Maple [A]  time = 0.015, size = 98, normalized size = 1.2 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 5\,{a}^{4}b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{5}}{n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{4\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{3\,n}}+5\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+10\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-n)*(a+b*x^n)^5,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.37175, size = 169, normalized size = 2.04 \begin{align*} \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(60*a^4*b*n*x^n*log(x) + 3*b^5*x^(5*n) + 20*a*b^4*x^(4*n) + 60*a^2*b^3*x^(3*n) + 120*a^3*b^2*x^(2*n) - 12
*a^5)/(n*x^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-n)*(a+b*x**n)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.22391, size = 101, normalized size = 1.22 \begin{align*} \frac{60 \, a^{4} b n x^{n} \log \left (x\right ) + 3 \, b^{5} x^{5 \, n} + 20 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} + 120 \, a^{3} b^{2} x^{2 \, n} - 12 \, a^{5}}{12 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(60*a^4*b*n*x^n*log(x) + 3*b^5*x^(5*n) + 20*a*b^4*x^(4*n) + 60*a^2*b^3*x^(3*n) + 120*a^3*b^2*x^(2*n) - 12
*a^5)/(n*x^n)