Optimal. Leaf size=99 \[ -\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}-\frac {2 a^7 b}{3 x^{12}}-\frac {a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^8} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^8+\frac {a^8}{x^8}+\frac {8 a^7 b}{x^7}+\frac {28 a^6 b^2}{x^6}+\frac {56 a^5 b^3}{x^5}+\frac {70 a^4 b^4}{x^4}+\frac {56 a^3 b^5}{x^3}+\frac {28 a^2 b^6}{x^2}+\frac {8 a b^7}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+\frac {b^8 x^2}{2}+8 a b^7 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 99, normalized size = 1.00 \[ -\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+8 a b^7 \log (x)+\frac {b^8 x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 94, normalized size = 0.95 \[ \frac {105 \, b^{8} x^{16} + 1680 \, a b^{7} x^{14} \log \relax (x) - 2940 \, a^{2} b^{6} x^{12} - 2940 \, a^{3} b^{5} x^{10} - 2450 \, a^{4} b^{4} x^{8} - 1470 \, a^{5} b^{3} x^{6} - 588 \, a^{6} b^{2} x^{4} - 140 \, a^{7} b x^{2} - 15 \, a^{8}}{210 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.08, size = 103, normalized size = 1.04 \[ \frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2178 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 90, normalized size = 0.91 \[ \frac {b^{8} x^{2}}{2}+8 a \,b^{7} \ln \relax (x )-\frac {14 a^{2} b^{6}}{x^{2}}-\frac {14 a^{3} b^{5}}{x^{4}}-\frac {35 a^{4} b^{4}}{3 x^{6}}-\frac {7 a^{5} b^{3}}{x^{8}}-\frac {14 a^{6} b^{2}}{5 x^{10}}-\frac {2 a^{7} b}{3 x^{12}}-\frac {a^{8}}{14 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 94, normalized size = 0.95 \[ \frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.15, size = 94, normalized size = 0.95 \[ -\frac {\frac {a^8}{14}-\frac {b^8\,x^{16}}{2}+\frac {2\,a^7\,b\,x^2}{3}+\frac {14\,a^6\,b^2\,x^4}{5}+7\,a^5\,b^3\,x^6+\frac {35\,a^4\,b^4\,x^8}{3}+14\,a^3\,b^5\,x^{10}+14\,a^2\,b^6\,x^{12}-8\,a\,b^7\,x^{14}\,\ln \relax (x)}{x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.73, size = 99, normalized size = 1.00 \[ 8 a b^{7} \log {\relax (x )} + \frac {b^{8} x^{2}}{2} + \frac {- 15 a^{8} - 140 a^{7} b x^{2} - 588 a^{6} b^{2} x^{4} - 1470 a^{5} b^{3} x^{6} - 2450 a^{4} b^{4} x^{8} - 2940 a^{3} b^{5} x^{10} - 2940 a^{2} b^{6} x^{12}}{210 x^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________