Optimal. Leaf size=135 \[ -\frac {a^8 x^{-6 n}}{6 n}-\frac {8 a^7 b x^{-5 n}}{5 n}-\frac {7 a^6 b^2 x^{-4 n}}{n}-\frac {56 a^5 b^3 x^{-3 n}}{3 n}-\frac {35 a^4 b^4 x^{-2 n}}{n}-\frac {56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)+\frac {8 a b^7 x^n}{n}+\frac {b^8 x^{2 n}}{2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, 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} \[ -\frac {7 a^6 b^2 x^{-4 n}}{n}-\frac {56 a^5 b^3 x^{-3 n}}{3 n}-\frac {35 a^4 b^4 x^{-2 n}}{n}-\frac {56 a^3 b^5 x^{-n}}{n}+28 a^2 b^6 \log (x)-\frac {8 a^7 b x^{-5 n}}{5 n}-\frac {a^8 x^{-6 n}}{6 n}+\frac {8 a b^7 x^n}{n}+\frac {b^8 x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-6 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^7} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a b^7+\frac {a^8}{x^7}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^5}+\frac {56 a^5 b^3}{x^4}+\frac {70 a^4 b^4}{x^3}+\frac {56 a^3 b^5}{x^2}+\frac {28 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-6 n}}{6 n}-\frac {8 a^7 b x^{-5 n}}{5 n}-\frac {7 a^6 b^2 x^{-4 n}}{n}-\frac {56 a^5 b^3 x^{-3 n}}{3 n}-\frac {35 a^4 b^4 x^{-2 n}}{n}-\frac {56 a^3 b^5 x^{-n}}{n}+\frac {8 a b^7 x^n}{n}+\frac {b^8 x^{2 n}}{2 n}+28 a^2 b^6 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 116, normalized size = 0.86 \[ \frac {-\frac {1}{6} a^8 x^{-6 n}-\frac {8}{5} a^7 b x^{-5 n}-7 a^6 b^2 x^{-4 n}-\frac {56}{3} a^5 b^3 x^{-3 n}-35 a^4 b^4 x^{-2 n}-56 a^3 b^5 x^{-n}+28 a^2 b^6 n \log (x)+8 a b^7 x^n+\frac {1}{2} b^8 x^{2 n}}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{2} b^{6} n x^{6 \, n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} - 1680 \, a^{3} b^{5} x^{5 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 560 \, a^{5} b^{3} x^{3 \, n} - 210 \, a^{6} b^{2} x^{2 \, n} - 48 \, a^{7} b x^{n} - 5 \, a^{8}}{30 \, n x^{6 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{2} b^{6} n x^{6 \, n} \log \relax (x) + 15 \, b^{8} x^{8 \, n} + 240 \, a b^{7} x^{7 \, n} - 1680 \, a^{3} b^{5} x^{5 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 560 \, a^{5} b^{3} x^{3 \, n} - 210 \, a^{6} b^{2} x^{2 \, n} - 48 \, a^{7} b x^{n} - 5 \, a^{8}}{30 \, n x^{6 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 128, normalized size = 0.95 \[ 28 a^{2} b^{6} \ln \relax (x )-\frac {a^{8} x^{-6 n}}{6 n}-\frac {8 a^{7} b \,x^{-5 n}}{5 n}-\frac {7 a^{6} b^{2} x^{-4 n}}{n}-\frac {56 a^{5} b^{3} x^{-3 n}}{3 n}-\frac {35 a^{4} b^{4} x^{-2 n}}{n}-\frac {56 a^{3} b^{5} x^{-n}}{n}+\frac {8 a \,b^{7} x^{n}}{n}+\frac {b^{8} x^{2 n}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 137, normalized size = 1.01 \[ 28 \, a^{2} b^{6} \log \relax (x) + \frac {b^{8} x^{2 \, n}}{2 \, n} + \frac {8 \, a b^{7} x^{n}}{n} - \frac {a^{8}}{6 \, n x^{6 \, n}} - \frac {8 \, a^{7} b}{5 \, n x^{5 \, n}} - \frac {7 \, a^{6} b^{2}}{n x^{4 \, n}} - \frac {56 \, a^{5} b^{3}}{3 \, n x^{3 \, n}} - \frac {35 \, a^{4} b^{4}}{n x^{2 \, n}} - \frac {56 \, a^{3} b^{5}}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.50, size = 137, normalized size = 1.01 \[ \frac {b^8\,x^{2\,n}}{2\,n}-\frac {a^8}{6\,n\,x^{6\,n}}+28\,a^2\,b^6\,\ln \relax (x)-\frac {56\,a^3\,b^5}{n\,x^n}-\frac {35\,a^4\,b^4}{n\,x^{2\,n}}-\frac {56\,a^5\,b^3}{3\,n\,x^{3\,n}}-\frac {7\,a^6\,b^2}{n\,x^{4\,n}}+\frac {8\,a\,b^7\,x^n}{n}-\frac {8\,a^7\,b}{5\,n\,x^{5\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________