Optimal. Leaf size=364 \[ \frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.292597, antiderivative size = 364, 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 = {463, 457, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{x^{3/2} (5 a d+11 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{3/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 463
Rule 457
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{\sqrt{x} \left (\frac{1}{2} \left (-8 a^2 d^2+3 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \int \frac{\sqrt{x}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^2 d^2}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^2 d^{5/2}}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^2 d^{5/2}}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{64 c^2 d^3}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{64 c^2 d^3}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{64 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \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 )}{64 \sqrt{2} c^{9/4} d^{11/4}}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}\\ &=\frac{(b c-a d)^2 x^{3/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (11 b c+5 a d) x^{3/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{9/4} d^{11/4}}+\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}-\frac{\left (21 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{9/4} d^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.203085, size = 339, normalized size = 0.93 \[ \frac{-\frac{8 \sqrt [4]{c} d^{3/4} x^{3/2} \left (-5 a^2 d^2-6 a b c d+11 b^2 c^2\right )}{c+d x^2}+\sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-\sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (5 a^2 d^2+6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+\frac{32 c^{5/4} d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 c^{9/4} d^{11/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 514, normalized size = 1.4 \begin{align*} \text{result too large to display} \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 = 1.29552, size = 4366, normalized size = 11.99 \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 = 1.29434, size = 562, normalized size = 1.54 \begin{align*} -\frac{11 \, b^{2} c^{2} d x^{\frac{7}{2}} - 6 \, a b c d^{2} x^{\frac{7}{2}} - 5 \, a^{2} d^{3} x^{\frac{7}{2}} + 7 \, b^{2} c^{3} x^{\frac{3}{2}} + 2 \, a b c^{2} d x^{\frac{3}{2}} - 9 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{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 )}{64 \, c^{3} d^{5}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{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 )}{64 \, c^{3} d^{5}} - \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{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 )}{128 \, c^{3} d^{5}} + \frac{\sqrt{2}{\left (21 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{3}{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 )}{128 \, c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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