Optimal. Leaf size=281 \[ \frac{3 d^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.279858, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {28, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{3 d^{5/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(d x)^{5/2}}{a^2+2 a b x^2+b^2 x^4} \, dx &=b^2 \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac{1}{4} \left (3 d^2\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac{1}{2} (3 d) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}-\frac{(3 d) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4 \sqrt{b}}+\frac{(3 d) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4 \sqrt{b}}\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac{\left (3 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{\left (3 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{8 b^2}+\frac{\left (3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{8 b^2}\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}+\frac{3 d^{5/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{\left (3 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{\left (3 d^{5/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}\\ &=-\frac{d (d x)^{3/2}}{2 b \left (a+b x^2\right )}-\frac{3 d^{5/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{7/4}}+\frac{3 d^{5/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}-\frac{3 d^{5/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.0152097, size = 54, normalized size = 0.19 \[ \frac{2 d (d x)^{3/2} \left (\left (a+b x^2\right ) \, _2F_1\left (\frac{3}{4},2;\frac{7}{4};-\frac{b x^2}{a}\right )-a\right )}{a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 209, normalized size = 0.7 \begin{align*} -{\frac{{d}^{3}}{2\,b \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{16\,{b}^{2}}\ln \left ({ \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{3\,{d}^{3}\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \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 = 1.37561, size = 552, normalized size = 1.96 \begin{align*} -\frac{4 \, \sqrt{d x} d^{2} x + 12 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \sqrt{d x} b^{2} d^{7} - \sqrt{d^{15} x - \sqrt{-\frac{d^{10}}{a b^{7}}} a b^{3} d^{10}} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} b^{2}}{d^{10}}\right ) - 3 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (27 \, \sqrt{d x} d^{7} + 27 \, \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{3}{4}} a b^{5}\right ) + 3 \,{\left (b^{2} x^{2} + a b\right )} \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{1}{4}} \log \left (27 \, \sqrt{d x} d^{7} - 27 \, \left (-\frac{d^{10}}{a b^{7}}\right )^{\frac{3}{4}} a b^{5}\right )}{8 \,{\left (b^{2} x^{2} + a b\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{\left (d x\right )^{\frac{5}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20044, size = 355, normalized size = 1.26 \begin{align*} -\frac{1}{16} \,{\left (\frac{8 \, \sqrt{d x} d^{3} x}{{\left (b d^{2} x^{2} + a d^{2}\right )} b} - \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} - \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{4}} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \log \left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \log \left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{4}}\right )} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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