Optimal. Leaf size=88 \[ -\frac{5 b}{2 a^3 \sqrt{a+b x^2}}-\frac{5 b}{6 a^2 \left (a+b x^2\right )^{3/2}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{1}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0495339, antiderivative size = 92, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ -\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a+b x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/2}} \, dx,x,x^2\right )}{6 a}\\ &=\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^2\right )}{2 a^2}\\ &=\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^2\right )}{4 a^3}\\ &=\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^2}\right )}{2 a^3}\\ &=\frac{1}{3 a x^2 \left (a+b x^2\right )^{3/2}}+\frac{5}{3 a^2 x^2 \sqrt{a+b x^2}}-\frac{5 \sqrt{a+b x^2}}{2 a^3 x^2}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0071367, size = 37, normalized size = 0.42 \[ -\frac{b \, _2F_1\left (-\frac{3}{2},2;-\frac{1}{2};\frac{b x^2}{a}+1\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 78, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,b}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,b}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,b}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}} \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 = 1.42624, size = 527, normalized size = 5.99 \begin{align*} \left [\frac{15 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt{a} \log \left (-\frac{b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (15 \, a b^{2} x^{4} + 20 \, a^{2} b x^{2} + 3 \, a^{3}\right )} \sqrt{b x^{2} + a}}{12 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}, -\frac{15 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) +{\left (15 \, a b^{2} x^{4} + 20 \, a^{2} b x^{2} + 3 \, a^{3}\right )} \sqrt{b x^{2} + a}}{6 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.96205, size = 864, normalized size = 9.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.13894, size = 100, normalized size = 1.14 \begin{align*} -\frac{1}{6} \, b{\left (\frac{15 \, \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{2 \,{\left (6 \, b x^{2} + 7 \, a\right )}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} + \frac{3 \, \sqrt{b x^{2} + a}}{a^{3} b x^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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