Optimal. Leaf size=84 \[ \frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{\left (a+b x^2\right )^9}{24 a x^{24}} \]
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Rubi [A] time = 0.0414562, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{\left (a+b x^2\right )^9}{24 a x^{24}} \]
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 )^8}{x^{25}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{13}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{12}} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{11}} \, dx,x,x^2\right )}{44 a^2}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{10}} \, dx,x,x^2\right )}{440 a^3}\\ &=-\frac{\left (a+b x^2\right )^9}{24 a x^{24}}+\frac{b \left (a+b x^2\right )^9}{88 a^2 x^{22}}-\frac{b^2 \left (a+b x^2\right )^9}{440 a^3 x^{20}}+\frac{b^3 \left (a+b x^2\right )^9}{3960 a^4 x^{18}}\\ \end{align*}
Mathematica [A] time = 0.0040744, size = 106, normalized size = 1.26 \[ -\frac{7 a^6 b^2}{5 x^{20}}-\frac{28 a^5 b^3}{9 x^{18}}-\frac{35 a^4 b^4}{8 x^{16}}-\frac{4 a^3 b^5}{x^{14}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{4 a^7 b}{11 x^{22}}-\frac{a^8}{24 x^{24}}-\frac{4 a b^7}{5 x^{10}}-\frac{b^8}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 91, normalized size = 1.1 \begin{align*} -{\frac{{b}^{8}}{8\,{x}^{8}}}-{\frac{{a}^{8}}{24\,{x}^{24}}}-{\frac{35\,{a}^{4}{b}^{4}}{8\,{x}^{16}}}-{\frac{4\,a{b}^{7}}{5\,{x}^{10}}}-4\,{\frac{{a}^{3}{b}^{5}}{{x}^{14}}}-{\frac{7\,{a}^{6}{b}^{2}}{5\,{x}^{20}}}-{\frac{4\,{a}^{7}b}{11\,{x}^{22}}}-{\frac{7\,{a}^{2}{b}^{6}}{3\,{x}^{12}}}-{\frac{28\,{a}^{5}{b}^{3}}{9\,{x}^{18}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.7703, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23595, size = 235, normalized size = 2.8 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.32326, size = 99, normalized size = 1.18 \begin{align*} - \frac{165 a^{8} + 1440 a^{7} b x^{2} + 5544 a^{6} b^{2} x^{4} + 12320 a^{5} b^{3} x^{6} + 17325 a^{4} b^{4} x^{8} + 15840 a^{3} b^{5} x^{10} + 9240 a^{2} b^{6} x^{12} + 3168 a b^{7} x^{14} + 495 b^{8} x^{16}}{3960 x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.86287, size = 124, normalized size = 1.48 \begin{align*} -\frac{495 \, b^{8} x^{16} + 3168 \, a b^{7} x^{14} + 9240 \, a^{2} b^{6} x^{12} + 15840 \, a^{3} b^{5} x^{10} + 17325 \, a^{4} b^{4} x^{8} + 12320 \, a^{5} b^{3} x^{6} + 5544 \, a^{6} b^{2} x^{4} + 1440 \, a^{7} b x^{2} + 165 \, a^{8}}{3960 \, x^{24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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