Optimal. Leaf size=40 \[ \frac {b \left (a+b x^2\right )^5}{60 a^2 x^{10}}-\frac {\left (a+b x^2\right )^5}{12 a x^{12}} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \[ \frac {b \left (a+b x^2\right )^5}{60 a^2 x^{10}}-\frac {\left (a+b x^2\right )^5}{12 a x^{12}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{13}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^{13}} \, dx}{b^4}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^7} \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {\left (a+b x^2\right )^5}{12 a x^{12}}-\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^6} \, dx,x,x^2\right )}{12 a b^3}\\ &=-\frac {\left (a+b x^2\right )^5}{12 a x^{12}}+\frac {b \left (a+b x^2\right )^5}{60 a^2 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 56, normalized size = 1.40 \[ -\frac {a^4}{12 x^{12}}-\frac {2 a^3 b}{5 x^{10}}-\frac {3 a^2 b^2}{4 x^8}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 48, normalized size = 1.20 \[ -\frac {15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.20 \[ -\frac {15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 1.18 \[ -\frac {b^{4}}{4 x^{4}}-\frac {2 a \,b^{3}}{3 x^{6}}-\frac {3 a^{2} b^{2}}{4 x^{8}}-\frac {2 a^{3} b}{5 x^{10}}-\frac {a^{4}}{12 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 48, normalized size = 1.20 \[ -\frac {15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 48, normalized size = 1.20 \[ -\frac {\frac {a^4}{12}+\frac {2\,a^3\,b\,x^2}{5}+\frac {3\,a^2\,b^2\,x^4}{4}+\frac {2\,a\,b^3\,x^6}{3}+\frac {b^4\,x^8}{4}}{x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 51, normalized size = 1.28 \[ \frac {- 5 a^{4} - 24 a^{3} b x^{2} - 45 a^{2} b^{2} x^{4} - 40 a b^{3} x^{6} - 15 b^{4} x^{8}}{60 x^{12}} \]
Verification of antiderivative is not currently implemented for this CAS.
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