Optimal. Leaf size=135 \[ \frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+28 a^6 b^2 \log (x)-\frac{8 a^7 b x^{-n}}{n}-\frac{a^8 x^{-2 n}}{2 n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0572692, antiderivative size = 135, 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} \[ \frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+28 a^6 b^2 \log (x)-\frac{8 a^7 b x^{-n}}{n}-\frac{a^8 x^{-2 n}}{2 n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n} \]
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 )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (56 a^5 b^3+\frac{a^8}{x^3}+\frac{8 a^7 b}{x^2}+\frac{28 a^6 b^2}{x}+70 a^4 b^4 x+56 a^3 b^5 x^2+28 a^2 b^6 x^3+8 a b^7 x^4+b^8 x^5\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-2 n}}{2 n}-\frac{8 a^7 b x^{-n}}{n}+\frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n}+28 a^6 b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0607664, size = 116, normalized size = 0.86 \[ \frac{56 a^5 b^3 x^n+35 a^4 b^4 x^{2 n}+\frac{56}{3} a^3 b^5 x^{3 n}+7 a^2 b^6 x^{4 n}+28 a^6 b^2 n \log (x)-8 a^7 b x^{-n}-\frac{1}{2} a^8 x^{-2 n}+\frac{8}{5} a b^7 x^{5 n}+\frac{1}{6} b^8 x^{6 n}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.025, size = 128, normalized size = 1. \begin{align*} 28\,{a}^{6}{b}^{2}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{6}}{6\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{3}}{3\,n}}+35\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{x}^{n}{a}^{5}{b}^{3}}{n}}-8\,{\frac{b{a}^{7}}{n{x}^{n}}}-{\frac{{a}^{8}}{2\,n \left ({x}^{n} \right ) ^{2}}} \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 = 1.3995, size = 269, normalized size = 1.99 \begin{align*} \frac{840 \, a^{6} b^{2} n x^{2 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, 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 = 1.2572, size = 157, normalized size = 1.16 \begin{align*} \frac{840 \, a^{6} b^{2} n x^{2 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]