Optimal. Leaf size=85 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac{35 a x}{8 b^4}-\frac{x^7}{4 b \left (a+b x^2\right )^2}+\frac{35 x^3}{24 b^3} \]
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Rubi [A] time = 0.0351929, antiderivative size = 85, 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 = {288, 302, 205} \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac{35 a x}{8 b^4}-\frac{x^7}{4 b \left (a+b x^2\right )^2}+\frac{35 x^3}{24 b^3} \]
Antiderivative was successfully verified.
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Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^7}{4 b \left (a+b x^2\right )^2}+\frac{7 \int \frac{x^6}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{x^7}{4 b \left (a+b x^2\right )^2}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac{35 \int \frac{x^4}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{x^7}{4 b \left (a+b x^2\right )^2}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac{35 \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=-\frac{35 a x}{8 b^4}+\frac{35 x^3}{24 b^3}-\frac{x^7}{4 b \left (a+b x^2\right )^2}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac{\left (35 a^2\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^4}\\ &=-\frac{35 a x}{8 b^4}+\frac{35 x^3}{24 b^3}-\frac{x^7}{4 b \left (a+b x^2\right )^2}-\frac{7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0447759, size = 77, normalized size = 0.91 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{9/2}}-\frac{175 a^2 b x^3+105 a^3 x+56 a b^2 x^5-8 b^3 x^7}{24 b^4 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 77, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{3}}}-3\,{\frac{ax}{{b}^{4}}}-{\frac{13\,{a}^{2}{x}^{3}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,{a}^{3}x}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29079, size = 493, normalized size = 5.8 \begin{align*} \left [\frac{16 \, b^{3} x^{7} - 112 \, a b^{2} x^{5} - 350 \, a^{2} b x^{3} - 210 \, a^{3} x + 105 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{48 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac{8 \, b^{3} x^{7} - 56 \, a b^{2} x^{5} - 175 \, a^{2} b x^{3} - 105 \, a^{3} x + 105 \,{\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{24 \,{\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.579039, size = 131, normalized size = 1.54 \begin{align*} - \frac{3 a x}{b^{4}} - \frac{35 \sqrt{- \frac{a^{3}}{b^{9}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac{35 \sqrt{- \frac{a^{3}}{b^{9}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{9}}}}{a} \right )}}{16} - \frac{11 a^{3} x + 13 a^{2} b x^{3}}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} + \frac{x^{3}}{3 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.53496, size = 99, normalized size = 1.16 \begin{align*} \frac{35 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{4}} - \frac{13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{4}} + \frac{b^{6} x^{3} - 9 \, a b^{5} x}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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