Optimal. Leaf size=242 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.163816, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 290
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (a+b x^2\right )^3} \, dx &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{1}{\sqrt{x} \left (a+b x^2\right )^2} \, dx}{8 b}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}+\frac{3 \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{32 a b}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{16 a b}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^{3/2} b}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{32 a^{3/2} b}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}+\frac{3 \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^{3/2} b^{3/2}}+\frac{3 \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^{3/2} b^{3/2}}-\frac{3 \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^{7/4} b^{5/4}}-\frac{3 \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^{7/4} b^{5/4}}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}-\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \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^{7/4} b^{5/4}}-\frac{3 \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^{7/4} b^{5/4}}\\ &=-\frac{\sqrt{x}}{4 b \left (a+b x^2\right )^2}+\frac{\sqrt{x}}{16 a b \left (a+b x^2\right )}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{5/4}}-\frac{3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}+\frac{3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.10785, size = 223, normalized size = 0.92 \[ \frac{\frac{8 \sqrt [4]{b} \sqrt{x}}{a^2+a b x^2}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{32 \sqrt [4]{b} \sqrt{x}}{\left (a+b x^2\right )^2}}{128 b^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 169, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{{x}^{5/2}}{a}}-{\frac{3\,\sqrt{x}}{32\,b}} \right ) }+{\frac{3\,\sqrt{2}}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\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{3\,\sqrt{2}}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \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 = 1.44062, size = 608, normalized size = 2.51 \begin{align*} \frac{12 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\sqrt{a^{4} b^{2} \sqrt{-\frac{1}{a^{7} b^{5}}} + x} a^{5} b^{4} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{3}{4}} - a^{5} b^{4} \sqrt{x} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{3}{4}}\right ) + 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (a^{2} b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) - 3 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{4}} \log \left (-a^{2} b \left (-\frac{1}{a^{7} b^{5}}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 4 \,{\left (b x^{2} - 3 \, a\right )} \sqrt{x}}{64 \,{\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\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 = 2.3881, size = 285, normalized size = 1.18 \begin{align*} \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{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^{2} b^{2}} + \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{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^{2} b^{2}} + \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{2}} - \frac{3 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{2}} + \frac{b x^{\frac{5}{2}} - 3 \, a \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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