Optimal. Leaf size=414 \[ \frac{b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.276258, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1112, 325, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{b^{3/4} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 325
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac{\left (a b+b^2 x^2\right ) \int \frac{1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{a d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (2 b \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{a d^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \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 )}{a^{3/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (b \left (a b+b^2 x^2\right )\right ) \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 )}{a^{3/2} d^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^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 )}{2 \sqrt{2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^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 )}{2 \sqrt{2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^2\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 )}{2 a^{3/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^2\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 )}{2 a^{3/2} \sqrt{b} d^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a b+b^2 x^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 )}{\sqrt{2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a b+b^2 x^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 )}{\sqrt{2} a^{7/4} \sqrt [4]{b} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{2 \left (a+b x^2\right )}{3 a d (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b^{3/4} \left (a+b x^2\right ) \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{2 \sqrt{2} a^{7/4} d^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [C] time = 0.0140528, size = 52, normalized size = 0.13 \[ -\frac{2 x \left (a+b x^2\right ) \, _2F_1\left (-\frac{3}{4},1;\frac{1}{4};-\frac{b x^2}{a}\right )}{3 a (d x)^{5/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.227, size = 239, normalized size = 0.6 \begin{align*} -{\frac{b{x}^{2}+a}{12\,{d}^{3}{a}^{2}} \left ( 3\,b\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2} \left ( dx \right ) ^{3/2}\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 ) +6\,b\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2} \left ( dx \right ) ^{3/2}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}+\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +6\,b\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{2} \left ( dx \right ) ^{3/2}\arctan \left ({ \left ( \sqrt{2}\sqrt{dx}-\sqrt [4]{{\frac{a{d}^{2}}{b}}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \right ) +8\,a{d}^{2} \right ) \left ( dx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (b x^{2} + a\right )}^{2}} \left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65694, size = 525, normalized size = 1.27 \begin{align*} -\frac{12 \, a d^{3} x^{2} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{d x} a^{5} b d^{7} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{3}{4}} - \sqrt{a^{4} d^{6} \sqrt{-\frac{b^{3}}{a^{7} d^{10}}} + b^{2} d x} a^{5} d^{7} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{3}{4}}}{b^{3}}\right ) + 3 \, a d^{3} x^{2} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{1}{4}} \log \left (a^{2} d^{3} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{1}{4}} + \sqrt{d x} b\right ) - 3 \, a d^{3} x^{2} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{1}{4}} \log \left (-a^{2} d^{3} \left (-\frac{b^{3}}{a^{7} d^{10}}\right )^{\frac{1}{4}} + \sqrt{d x} b\right ) + 4 \, \sqrt{d x}}{6 \, a d^{3} x^{2}} \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 = 1.27133, size = 346, normalized size = 0.84 \begin{align*} -\frac{1}{12} \,{\left (\frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{2} d^{3}} + \frac{6 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{2} d^{3}} + \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{2} d^{3}} - \frac{3 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{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^{2} d^{3}} + \frac{8}{\sqrt{d x} a d^{2} x}\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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