Optimal. Leaf size=269 \[ -\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}} \]
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Rubi [A] time = 0.275532, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {462, 453, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac{(b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 453
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^{9/2} \left (c+d x^2\right )} \, dx &=-\frac{2 a^2}{7 c x^{7/2}}+\frac{2 \int \frac{\frac{7}{2} a (2 b c-a d)+\frac{7}{2} b^2 c x^2}{x^{5/2} \left (c+d x^2\right )} \, dx}{7 c}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac{(b c-a d)^2 \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{c^2}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac{\left (2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c^2}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c^{5/2}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{c^{5/2}}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{2 c^{5/2} \sqrt{d}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{2 c^{5/2} \sqrt{d}}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac{(b c-a d)^2 \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}\\ &=-\frac{2 a^2}{7 c x^{7/2}}-\frac{2 a (2 b c-a d)}{3 c^2 x^{3/2}}-\frac{(b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{d}}-\frac{(b c-a d)^2 \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}+\frac{(b c-a d)^2 \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{d}}\\ \end{align*}
Mathematica [A] time = 0.115048, size = 254, normalized size = 0.94 \[ \frac{-\frac{24 a^2 c^{7/4}}{x^{7/2}}+\frac{56 a c^{3/4} (a d-2 b c)}{x^{3/2}}-\frac{21 \sqrt{2} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d}}+\frac{21 \sqrt{2} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{d}}-\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{d}}+\frac{42 \sqrt{2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{d}}}{84 c^{11/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 461, normalized size = 1.7 \begin{align*}{\frac{\sqrt{2}{a}^{2}{d}^{2}}{2\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}abd}{{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{2\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}{a}^{2}{d}^{2}}{4\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}abd}{2\,{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}{b}^{2}}{4\,c}\sqrt [4]{{\frac{c}{d}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}{a}^{2}{d}^{2}}{2\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}abd}{{c}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}{b}^{2}}{2\,c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{2\,{a}^{2}}{7\,c}{x}^{-{\frac{7}{2}}}}+{\frac{2\,{a}^{2}d}{3\,{c}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{4\,ab}{3\,c}{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 [B] time = 1.07121, size = 2587, normalized size = 9.62 \begin{align*} \text{result too large to display} \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 = 1.17359, size = 478, normalized size = 1.78 \begin{align*} \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3} d} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3} d} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, c^{3} d} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, c^{3} d} - \frac{2 \,{\left (14 \, a b c x^{2} - 7 \, a^{2} d x^{2} + 3 \, a^{2} c\right )}}{21 \, c^{2} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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