Optimal. Leaf size=85 \[ \frac{10 a^2 b^3 x^n}{n}+10 a^3 b^2 \log (x)-\frac{5 a^4 b x^{-n}}{n}-\frac{a^5 x^{-2 n}}{2 n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0370283, antiderivative size = 85, 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} \[ \frac{10 a^2 b^3 x^n}{n}+10 a^3 b^2 \log (x)-\frac{5 a^4 b x^{-n}}{n}-\frac{a^5 x^{-2 n}}{2 n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-2 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (10 a^2 b^3+\frac{a^5}{x^3}+\frac{5 a^4 b}{x^2}+\frac{10 a^3 b^2}{x}+5 a b^4 x+b^5 x^2\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-2 n}}{2 n}-\frac{5 a^4 b x^{-n}}{n}+\frac{10 a^2 b^3 x^n}{n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n}+10 a^3 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0510785, size = 72, normalized size = 0.85 \[ \frac{x^{-2 n} \left (60 a^2 b^3 x^{3 n}-30 a^4 b x^n-3 a^5+15 a b^4 x^{4 n}+2 b^5 x^{5 n}\right )}{6 n}+10 a^3 b^2 \log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 98, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( 10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{5}}{2\,n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{3\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{2\,n}}+10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.29222, size = 170, normalized size = 2. \begin{align*} \frac{60 \, a^{3} b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} - 30 \, a^{4} b x^{n} - 3 \, a^{5}}{6 \, n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18681, size = 104, normalized size = 1.22 \begin{align*} \frac{60 \, a^{3} b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} - 30 \, a^{4} b x^{n} - 3 \, a^{5}}{6 \, n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]