Optimal. Leaf size=230 \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]
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Rubi [A] time = 0.166363, 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 = {288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{7/2}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \int \frac{x^{3/2}}{a+b x^2} \, dx}{4 b}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}-\frac{(5 a) \int \frac{1}{\sqrt{x} \left (a+b x^2\right )} \, dx}{4 b^2}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b^2}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (5 \sqrt{a}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^2}-\frac{\left (5 \sqrt{a}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^2}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}-\frac{\left (5 \sqrt{a}\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 b^{5/2}}-\frac{\left (5 \sqrt{a}\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 b^{5/2}}+\frac{\left (5 \sqrt [4]{a}\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} b^{9/4}}+\frac{\left (5 \sqrt [4]{a}\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} b^{9/4}}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt [4]{a} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{\left (5 \sqrt [4]{a}\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} b^{9/4}}+\frac{\left (5 \sqrt [4]{a}\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} b^{9/4}}\\ &=\frac{5 \sqrt{x}}{2 b^2}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.10183, size = 221, normalized size = 0.96 \[ \frac{\frac{32 b^{5/4} x^{5/2}}{a+b x^2}+\frac{40 a \sqrt [4]{b} \sqrt{x}}{a+b x^2}+5 \sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-5 \sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{16 b^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 158, normalized size = 0.7 \begin{align*} 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}}{16\,{b}^{2}}\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{5\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\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 = 1.34265, size = 443, normalized size = 1.93 \begin{align*} -\frac{20 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{b^{4} \sqrt{-\frac{a}{b^{9}}} + x} b^{7} \left (-\frac{a}{b^{9}}\right )^{\frac{3}{4}} - b^{7} \sqrt{x} \left (-\frac{a}{b^{9}}\right )^{\frac{3}{4}}}{a}\right ) + 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) - 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (-5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) - 4 \,{\left (4 \, b x^{2} + 5 \, a\right )} \sqrt{x}}{8 \,{\left (b^{3} x^{2} + a b^{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 = 2.55286, size = 265, normalized size = 1.15 \begin{align*} -\frac{5 \, \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 \, b^{3}} - \frac{5 \, \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 \, b^{3}} - \frac{5 \, \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 \, b^{3}} + \frac{5 \, \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 \, b^{3}} + \frac{a \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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