Optimal. Leaf size=94 \[ -\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10} \]
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Rubi [A] time = 0.06, antiderivative size = 94, 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} \begin {gather*} \frac {14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac {2 a^7 b}{x^4}-\frac {a^8}{6 x^6}+a b^7 x^8+\frac {b^8 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^8}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \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^2\right )\\ &=-\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10}+56 a^5 b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 94, normalized size = 1.00 \begin {gather*} -\frac {a^8}{6 x^6}-\frac {2 a^7 b}{x^4}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac {14}{3} a^2 b^6 x^6+a b^7 x^8+\frac {b^8 x^{10}}{10} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^8}{x^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.29, size = 94, normalized size = 1.00 \begin {gather*} \frac {3 \, b^{8} x^{16} + 30 \, a b^{7} x^{14} + 140 \, a^{2} b^{6} x^{12} + 420 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} \log \relax (x) - 420 \, a^{6} b^{2} x^{4} - 60 \, a^{7} b x^{2} - 5 \, a^{8}}{30 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 102, normalized size = 1.09 \begin {gather*} \frac {1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac {14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac {308 \, a^{5} b^{3} x^{6} + 84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 0.95 \begin {gather*} \frac {b^{8} x^{10}}{10}+a \,b^{7} x^{8}+\frac {14 a^{2} b^{6} x^{6}}{3}+14 a^{3} b^{5} x^{4}+35 a^{4} b^{4} x^{2}+56 a^{5} b^{3} \ln \relax (x )-\frac {14 a^{6} b^{2}}{x^{2}}-\frac {2 a^{7} b}{x^{4}}-\frac {a^{8}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 91, normalized size = 0.97 \begin {gather*} \frac {1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac {14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac {84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 91, normalized size = 0.97 \begin {gather*} \frac {b^8\,x^{10}}{10}-\frac {\frac {a^8}{6}+2\,a^7\,b\,x^2+14\,a^6\,b^2\,x^4}{x^6}+a\,b^7\,x^8+35\,a^4\,b^4\,x^2+14\,a^3\,b^5\,x^4+\frac {14\,a^2\,b^6\,x^6}{3}+56\,a^5\,b^3\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 97, normalized size = 1.03 \begin {gather*} 56 a^{5} b^{3} \log {\relax (x )} + 35 a^{4} b^{4} x^{2} + 14 a^{3} b^{5} x^{4} + \frac {14 a^{2} b^{6} x^{6}}{3} + a b^{7} x^{8} + \frac {b^{8} x^{10}}{10} + \frac {- a^{8} - 12 a^{7} b x^{2} - 84 a^{6} b^{2} x^{4}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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