Optimal. Leaf size=230 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.176691, antiderivative size = 230, 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 = {290, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} \left (a+b x^2\right )^2} \, dx &=\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}+\frac{7 \int \frac{1}{x^{5/2} \left (a+b x^2\right )} \, dx}{4 a}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}-\frac{(7 b) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{4 a^2}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^2}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2}}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{5/2}}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}-\frac{\left (7 \sqrt{b}\right ) \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 )}{8 a^{5/2}}-\frac{\left (7 \sqrt{b}\right ) \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 )}{8 a^{5/2}}+\frac{\left (7 b^{3/4}\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 )}{8 \sqrt{2} a^{11/4}}+\frac{\left (7 b^{3/4}\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 )}{8 \sqrt{2} a^{11/4}}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}+\frac{7 b^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{\left (7 b^{3/4}\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 )}{4 \sqrt{2} a^{11/4}}+\frac{\left (7 b^{3/4}\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 )}{4 \sqrt{2} a^{11/4}}\\ &=-\frac{7}{6 a^2 x^{3/2}}+\frac{1}{2 a x^{3/2} \left (a+b x^2\right )}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}-\frac{7 b^{3/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.0060163, size = 29, normalized size = 0.13 \[ -\frac{2 \, _2F_1\left (-\frac{3}{4},2;\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 161, normalized size = 0.7 \begin{align*} -{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,b\sqrt{2}}{16\,{a}^{3}}\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{7\,b\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{7\,b\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{2}{3\,{a}^{2}}{x}^{-{\frac{3}{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 = 1.45044, size = 518, normalized size = 2.25 \begin{align*} -\frac{84 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{8} b \sqrt{x} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{4}} - \sqrt{a^{6} \sqrt{-\frac{b^{3}}{a^{11}}} + b^{2} x} a^{8} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{3}{4}}}{b^{3}}\right ) + 21 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, b \sqrt{x}\right ) - 21 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} \left (-\frac{b^{3}}{a^{11}}\right )^{\frac{1}{4}} + 7 \, b \sqrt{x}\right ) + 4 \,{\left (7 \, b x^{2} + 4 \, a\right )} \sqrt{x}}{24 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\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 = 2.93922, size = 265, normalized size = 1.15 \begin{align*} -\frac{7 \, \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 )}{8 \, a^{3}} - \frac{7 \, \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 )}{8 \, a^{3}} - \frac{7 \, \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 )}{16 \, a^{3}} + \frac{7 \, \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 )}{16 \, a^{3}} - \frac{b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{2}} - \frac{2}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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