Optimal. Leaf size=56 \[ -\frac{3 a^2 b^2}{5 x^{10}}-\frac{a^3 b}{3 x^{12}}-\frac{a^4}{14 x^{14}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]
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Rubi [A] time = 0.0364443, antiderivative size = 56, normalized size of antiderivative = 1., 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} \[ -\frac{3 a^2 b^2}{5 x^{10}}-\frac{a^3 b}{3 x^{12}}-\frac{a^4}{14 x^{14}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{15}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^{15}} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^8} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^4 b^4}{x^8}+\frac{4 a^3 b^5}{x^7}+\frac{6 a^2 b^6}{x^6}+\frac{4 a b^7}{x^5}+\frac{b^8}{x^4}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{14 x^{14}}-\frac{a^3 b}{3 x^{12}}-\frac{3 a^2 b^2}{5 x^{10}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.0040352, size = 56, normalized size = 1. \[ -\frac{3 a^2 b^2}{5 x^{10}}-\frac{a^3 b}{3 x^{12}}-\frac{a^4}{14 x^{14}}-\frac{a b^3}{2 x^8}-\frac{b^4}{6 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 47, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}}{14\,{x}^{14}}}-{\frac{{a}^{3}b}{3\,{x}^{12}}}-{\frac{3\,{b}^{2}{a}^{2}}{5\,{x}^{10}}}-{\frac{a{b}^{3}}{2\,{x}^{8}}}-{\frac{{b}^{4}}{6\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992685, size = 65, normalized size = 1.16 \begin{align*} -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67089, size = 113, normalized size = 2.02 \begin{align*} -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.591961, size = 51, normalized size = 0.91 \begin{align*} - \frac{15 a^{4} + 70 a^{3} b x^{2} + 126 a^{2} b^{2} x^{4} + 105 a b^{3} x^{6} + 35 b^{4} x^{8}}{210 x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22416, size = 65, normalized size = 1.16 \begin{align*} -\frac{35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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