Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac{\left (a+b x^2\right )^6}{14 a x^{14}} \]
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Rubi [A] time = 0.0176447, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac{\left (a+b x^2\right )^6}{14 a x^{14}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5}{x^{15}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^8} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^6}{14 a x^{14}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^7} \, dx,x,x^2\right )}{14 a}\\ &=-\frac{\left (a+b x^2\right )^6}{14 a x^{14}}+\frac{b \left (a+b x^2\right )^6}{84 a^2 x^{12}}\\ \end{align*}
Mathematica [A] time = 0.005927, size = 67, normalized size = 1.68 \[ -\frac{a^3 b^2}{x^{10}}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^5}{14 x^{14}}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 58, normalized size = 1.5 \begin{align*} -{\frac{5\,{a}^{2}{b}^{3}}{4\,{x}^{8}}}-{\frac{5\,{a}^{4}b}{12\,{x}^{12}}}-{\frac{{a}^{3}{b}^{2}}{{x}^{10}}}-{\frac{{b}^{5}}{4\,{x}^{4}}}-{\frac{5\,a{b}^{4}}{6\,{x}^{6}}}-{\frac{{a}^{5}}{14\,{x}^{14}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.96214, size = 80, normalized size = 2. \begin{align*} -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23035, size = 134, normalized size = 3.35 \begin{align*} -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.644176, size = 63, normalized size = 1.58 \begin{align*} - \frac{6 a^{5} + 35 a^{4} b x^{2} + 84 a^{3} b^{2} x^{4} + 105 a^{2} b^{3} x^{6} + 70 a b^{4} x^{8} + 21 b^{5} x^{10}}{84 x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.45779, size = 80, normalized size = 2. \begin{align*} -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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