Optimal. Leaf size=306 \[ -\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{10 a^{11/4} \sqrt{a x+b x^3}}-\frac{21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b x^3}}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{1}{a x^2 \sqrt{a x+b x^3}} \]
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Rubi [A] time = 0.322572, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {2023, 2025, 2032, 329, 305, 220, 1196} \[ -\frac{21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}-\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{11/4} \sqrt{a x+b x^3}}+\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b x^3}}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{1}{a x^2 \sqrt{a x+b x^3}} \]
Antiderivative was successfully verified.
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Rule 2023
Rule 2025
Rule 2032
Rule 329
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx &=\frac{1}{a x^2 \sqrt{a x+b x^3}}+\frac{7 \int \frac{1}{x^3 \sqrt{a x+b x^3}} \, dx}{2 a}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}-\frac{(21 b) \int \frac{1}{x \sqrt{a x+b x^3}} \, dx}{10 a^2}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{\left (21 b^2\right ) \int \frac{x}{\sqrt{a x+b x^3}} \, dx}{10 a^3}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{\left (21 b^2 \sqrt{x} \sqrt{a+b x^2}\right ) \int \frac{\sqrt{x}}{\sqrt{a+b x^2}} \, dx}{10 a^3 \sqrt{a x+b x^3}}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{\left (21 b^2 \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 a^3 \sqrt{a x+b x^3}}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}-\frac{\left (21 b^{3/2} \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 a^{5/2} \sqrt{a x+b x^3}}+\frac{\left (21 b^{3/2} \sqrt{x} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx,x,\sqrt{x}\right )}{5 a^{5/2} \sqrt{a x+b x^3}}\\ &=\frac{1}{a x^2 \sqrt{a x+b x^3}}-\frac{21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}-\frac{7 \sqrt{a x+b x^3}}{5 a^2 x^3}+\frac{21 b \sqrt{a x+b x^3}}{5 a^3 x}+\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{11/4} \sqrt{a x+b x^3}}-\frac{21 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{11/4} \sqrt{a x+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0162168, size = 56, normalized size = 0.18 \[ -\frac{2 \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{5}{4},\frac{3}{2};-\frac{1}{4};-\frac{b x^2}{a}\right )}{5 a x^2 \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 228, normalized size = 0.8 \begin{align*} -{\frac{2}{5\,{x}^{3}{a}^{2}}\sqrt{b{x}^{3}+ax}}+{\frac{ \left ( 16\,b{x}^{2}+16\,a \right ) b}{5\,{a}^{3}}{\frac{1}{\sqrt{x \left ( b{x}^{2}+a \right ) }}}}+{\frac{{b}^{2}{x}^{2}}{{a}^{3}}{\frac{1}{\sqrt{ \left ({\frac{a}{b}}+{x}^{2} \right ) bx}}}}-{\frac{21\,b}{10\,{a}^{3}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{3} + a x}}{b^{2} x^{8} + 2 \, a b x^{6} + a^{2} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (x \left (a + b x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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