Optimal. Leaf size=133 \[ -\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+56 a^5 b^3 \log (x)-\frac{4 a^7 b x^{-2 n}}{n}-\frac{a^8 x^{-3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0584728, antiderivative size = 133, 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} \[ -\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+56 a^5 b^3 \log (x)-\frac{4 a^7 b x^{-2 n}}{n}-\frac{a^8 x^{-3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-3 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^4} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (70 a^4 b^4+\frac{a^8}{x^4}+\frac{8 a^7 b}{x^3}+\frac{28 a^6 b^2}{x^2}+\frac{56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-3 n}}{3 n}-\frac{4 a^7 b x^{-2 n}}{n}-\frac{28 a^6 b^2 x^{-n}}{n}+\frac{70 a^4 b^4 x^n}{n}+\frac{28 a^3 b^5 x^{2 n}}{n}+\frac{28 a^2 b^6 x^{3 n}}{3 n}+\frac{2 a b^7 x^{4 n}}{n}+\frac{b^8 x^{5 n}}{5 n}+56 a^5 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0607483, size = 114, normalized size = 0.86 \[ \frac{-28 a^6 b^2 x^{-n}+70 a^4 b^4 x^n+28 a^3 b^5 x^{2 n}+\frac{28}{3} a^2 b^6 x^{3 n}+56 a^5 b^3 n \log (x)-4 a^7 b x^{-2 n}-\frac{1}{3} a^8 x^{-3 n}+2 a b^7 x^{4 n}+\frac{1}{5} b^8 x^{5 n}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 128, normalized size = 1. \begin{align*} 56\,{a}^{5}{b}^{3}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{5}}{5\,n}}+2\,{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{3}}{3\,n}}+28\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{2}}{n}}+70\,{\frac{{x}^{n}{a}^{4}{b}^{4}}{n}}-28\,{\frac{{a}^{6}{b}^{2}}{n{x}^{n}}}-4\,{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{{a}^{8}}{3\,n \left ({x}^{n} \right ) ^{3}}} \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 = 1.36124, size = 265, normalized size = 1.99 \begin{align*} \frac{840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, 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 = 1.26142, size = 157, normalized size = 1.18 \begin{align*} \frac{840 \, a^{5} b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]