Optimal. Leaf size=101 \[ -\frac {a^8}{4 x^4}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 \log (x)+28 a^5 b^3 x^2+\frac {35}{2} a^4 b^4 x^4+\frac {28}{3} a^3 b^5 x^6+\frac {7}{2} a^2 b^6 x^8+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12} \]
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Rubi [A] time = 0.06, antiderivative size = 101, 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 {7}{2} a^2 b^6 x^8+\frac {28}{3} a^3 b^5 x^6+\frac {35}{2} a^4 b^4 x^4+28 a^5 b^3 x^2+28 a^6 b^2 \log (x)-\frac {4 a^7 b}{x^2}-\frac {a^8}{4 x^4}+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12} \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^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \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^2\right )\\ &=-\frac {a^8}{4 x^4}-\frac {4 a^7 b}{x^2}+28 a^5 b^3 x^2+\frac {35}{2} a^4 b^4 x^4+\frac {28}{3} a^3 b^5 x^6+\frac {7}{2} a^2 b^6 x^8+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12}+28 a^6 b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 101, normalized size = 1.00 \begin {gather*} -\frac {a^8}{4 x^4}-\frac {4 a^7 b}{x^2}+28 a^6 b^2 \log (x)+28 a^5 b^3 x^2+\frac {35}{2} a^4 b^4 x^4+\frac {28}{3} a^3 b^5 x^6+\frac {7}{2} a^2 b^6 x^8+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12} \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^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.39, size = 94, normalized size = 0.93 \begin {gather*} \frac {5 \, b^{8} x^{16} + 48 \, a b^{7} x^{14} + 210 \, a^{2} b^{6} x^{12} + 560 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} + 1680 \, a^{6} b^{2} x^{4} \log \relax (x) - 240 \, a^{7} b x^{2} - 15 \, a^{8}}{60 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 103, normalized size = 1.02 \begin {gather*} \frac {1}{12} \, b^{8} x^{12} + \frac {4}{5} \, a b^{7} x^{10} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac {84 \, a^{6} b^{2} x^{4} + 16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.89 \begin {gather*} \frac {b^{8} x^{12}}{12}+\frac {4 a \,b^{7} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{8}}{2}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{4}}{2}+28 a^{5} b^{3} x^{2}+28 a^{6} b^{2} \ln \relax (x )-\frac {4 a^{7} b}{x^{2}}-\frac {a^{8}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 92, normalized size = 0.91 \begin {gather*} \frac {1}{12} \, b^{8} x^{12} + \frac {4}{5} \, a b^{7} x^{10} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac {16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 92, normalized size = 0.91 \begin {gather*} \frac {b^8\,x^{12}}{12}-\frac {\frac {a^8}{4}+4\,b\,a^7\,x^2}{x^4}+\frac {4\,a\,b^7\,x^{10}}{5}+28\,a^5\,b^3\,x^2+\frac {35\,a^4\,b^4\,x^4}{2}+\frac {28\,a^3\,b^5\,x^6}{3}+\frac {7\,a^2\,b^6\,x^8}{2}+28\,a^6\,b^2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 104, normalized size = 1.03 \begin {gather*} 28 a^{6} b^{2} \log {\relax (x )} + 28 a^{5} b^{3} x^{2} + \frac {35 a^{4} b^{4} x^{4}}{2} + \frac {28 a^{3} b^{5} x^{6}}{3} + \frac {7 a^{2} b^{6} x^{8}}{2} + \frac {4 a b^{7} x^{10}}{5} + \frac {b^{8} x^{12}}{12} + \frac {- a^{8} - 16 a^{7} b x^{2}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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