Optimal. Leaf size=72 \[ -\frac {a^6}{4 x^4}-\frac {3 a^5 b}{x^2}+15 a^4 b^2 \log (x)+10 a^3 b^3 x^2+\frac {15}{4} a^2 b^4 x^4+a b^5 x^6+\frac {b^6 x^8}{8} \]
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Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac {15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac {3 a^5 b}{x^2}-\frac {a^6}{4 x^4}+a b^5 x^6+\frac {b^6 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^5} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^5} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^3} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac {\operatorname {Subst}\left (\int \left (20 a^3 b^9+\frac {a^6 b^6}{x^3}+\frac {6 a^5 b^7}{x^2}+\frac {15 a^4 b^8}{x}+15 a^2 b^{10} x+6 a b^{11} x^2+b^{12} x^3\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {a^6}{4 x^4}-\frac {3 a^5 b}{x^2}+10 a^3 b^3 x^2+\frac {15}{4} a^2 b^4 x^4+a b^5 x^6+\frac {b^6 x^8}{8}+15 a^4 b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 72, normalized size = 1.00 \[ -\frac {a^6}{4 x^4}-\frac {3 a^5 b}{x^2}+15 a^4 b^2 \log (x)+10 a^3 b^3 x^2+\frac {15}{4} a^2 b^4 x^4+a b^5 x^6+\frac {b^6 x^8}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 71, normalized size = 0.99 \[ \frac {b^{6} x^{12} + 8 \, a b^{5} x^{10} + 30 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 120 \, a^{4} b^{2} x^{4} \log \relax (x) - 24 \, a^{5} b x^{2} - 2 \, a^{6}}{8 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 80, normalized size = 1.11 \[ \frac {1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac {15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac {15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac {45 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.93 \[ \frac {b^{6} x^{8}}{8}+a \,b^{5} x^{6}+\frac {15 a^{2} b^{4} x^{4}}{4}+10 a^{3} b^{3} x^{2}+15 a^{4} b^{2} \ln \relax (x )-\frac {3 a^{5} b}{x^{2}}-\frac {a^{6}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 69, normalized size = 0.96 \[ \frac {1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac {15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac {15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac {12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 69, normalized size = 0.96 \[ \frac {b^6\,x^8}{8}-\frac {\frac {a^6}{4}+3\,b\,a^5\,x^2}{x^4}+a\,b^5\,x^6+10\,a^3\,b^3\,x^2+\frac {15\,a^2\,b^4\,x^4}{4}+15\,a^4\,b^2\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 73, normalized size = 1.01 \[ 15 a^{4} b^{2} \log {\relax (x )} + 10 a^{3} b^{3} x^{2} + \frac {15 a^{2} b^{4} x^{4}}{4} + a b^{5} x^{6} + \frac {b^{6} x^{8}}{8} + \frac {- a^{6} - 12 a^{5} b x^{2}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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