Optimal. Leaf size=412 \[ -\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.286151, antiderivative size = 412, 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, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{2 \sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{\sqrt{2} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 321
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{(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{(d x)^{5/2}}{a b+b^2 x^2} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a d^2 \left (a b+b^2 x^2\right )\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (2 a d \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a d \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 )}{b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a d \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 )}{b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{3/4} d^{5/2} \left (a b+b^2 x^2\right )\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} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{3/4} d^{5/2} \left (a b+b^2 x^2\right )\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} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a d^3 \left (a b+b^2 x^2\right )\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 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a d^3 \left (a b+b^2 x^2\right )\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 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a^{3/4} d^{5/2} \left (a b+b^2 x^2\right )\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} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a^{3/4} d^{5/2} \left (a b+b^2 x^2\right )\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} b^{11/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 d (d x)^{3/2} \left (a+b x^2\right )}{3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{a^{3/4} d^{5/2} \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} b^{7/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0538894, size = 110, normalized size = 0.27 \[ \frac{(d x)^{5/2} \left (a+b x^2\right ) \left (3 (-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )-3 (-a)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{-a}}\right )+2 b^{3/4} x^{3/2}\right )}{3 b^{7/4} x^{5/2} \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.223, size = 221, normalized size = 0.5 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) d}{12\,{b}^{2}} \left ( 8\, \left ( dx \right ) ^{3/2}b\sqrt [4]{{\frac{a{d}^{2}}{b}}}-3\,a{d}^{2}\sqrt{2}\ln \left ( -{ \left ( \sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}-dx-\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\,a{d}^{2}\sqrt{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\,a{d}^{2}\sqrt{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 ) \right ){\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{5}{2}}}{\sqrt{{\left (b x^{2} + a\right )}^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63825, size = 486, normalized size = 1.18 \begin{align*} \frac{4 \, \sqrt{d x} d^{2} x + 12 \, \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{2} d^{7} - \sqrt{a^{4} d^{15} x - \sqrt{-\frac{a^{3} d^{10}}{b^{7}}} a^{3} b^{3} d^{10}} \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{1}{4}} b^{2}}{a^{3} d^{10}}\right ) - 3 \, \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\sqrt{d x} a^{2} d^{7} + \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{3}{4}} b^{5}\right ) + 3 \, \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{1}{4}} b \log \left (\sqrt{d x} a^{2} d^{7} - \left (-\frac{a^{3} d^{10}}{b^{7}}\right )^{\frac{3}{4}} b^{5}\right )}{6 \, b} \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 = 1.29573, size = 325, normalized size = 0.79 \begin{align*} \frac{1}{12} \,{\left (\frac{8 \, \sqrt{d x} d x}{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 )}{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 )}{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 )}{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 )}{b^{4}}\right )} d \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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