Optimal. Leaf size=251 \[ \frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4}}-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.190952, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{13/4}}-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \left (a+b x^2\right )^3} \, dx &=\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9 \int \frac{1}{x^{3/2} \left (a+b x^2\right )^2} \, dx}{8 a}\\ &=\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}+\frac{45 \int \frac{1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^2}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{(45 b) \int \frac{\sqrt{x}}{a+b x^2} \, dx}{32 a^3}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{(45 b) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{16 a^3}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}+\frac{\left (45 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}-\frac{\left (45 \sqrt{b}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^3}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{45 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^3}-\frac{45 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{64 a^3}-\frac{\left (45 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{13/4}}-\frac{\left (45 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{13/4}}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}-\frac{\left (45 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}+\frac{\left (45 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}\\ &=-\frac{45}{16 a^3 \sqrt{x}}+\frac{1}{4 a \sqrt{x} \left (a+b x^2\right )^2}+\frac{9}{16 a^2 \sqrt{x} \left (a+b x^2\right )}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}+\frac{45 \sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{13/4}}\\ \end{align*}
Mathematica [C] time = 0.0059544, size = 27, normalized size = 0.11 \[ -\frac{2 \, _2F_1\left (-\frac{1}{4},3;\frac{3}{4};-\frac{b x^2}{a}\right )}{a^3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 178, normalized size = 0.7 \begin{align*} -{\frac{13\,{b}^{2}}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{17\,b}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{45\,\sqrt{2}}{128\,{a}^{3}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{45\,\sqrt{2}}{64\,{a}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-2\,{\frac{1}{{a}^{3}\sqrt{x}}} \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 = 1.41627, size = 672, normalized size = 2.68 \begin{align*} \frac{180 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{91125 \, a^{3} b \sqrt{x} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{4}} - \sqrt{-8303765625 \, a^{7} b \sqrt{-\frac{b}{a^{13}}} + 8303765625 \, b^{2} x} a^{3} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{4}}}{91125 \, b}\right ) - 45 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} \left (-\frac{b}{a^{13}}\right )^{\frac{3}{4}} + 91125 \, b \sqrt{x}\right ) + 45 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac{b}{a^{13}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} \left (-\frac{b}{a^{13}}\right )^{\frac{3}{4}} + 91125 \, b \sqrt{x}\right ) - 4 \,{\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt{x}}{64 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.14213, size = 297, normalized size = 1.18 \begin{align*} -\frac{2}{a^{3} \sqrt{x}} - \frac{45 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{2}} - \frac{45 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b^{2}} + \frac{45 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac{45 \, \sqrt{2} \left (a b^{3}\right )^{\frac{3}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac{13 \, b^{2} x^{\frac{7}{2}} + 17 \, a b x^{\frac{3}{2}}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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