Optimal. Leaf size=310 \[ -\frac{(b c-a d) (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{3/2}}{3 d^2} \]
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Rubi [A] time = 0.281799, antiderivative size = 310, 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, 459, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{(b c-a d) (a d+7 b c) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (a d+7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{x^{3/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{3/2}}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
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 )^2} \, dx &=\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{\int \frac{\sqrt{x} \left (\frac{1}{2} \left (-4 a^2 d^2+3 (b c-a d)^2\right )-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{((b c-a d) (7 b c+a d)) \int \frac{\sqrt{x}}{c+d x^2} \, dx}{4 c d^2}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{((b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c d^2}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{((b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c d^{5/2}}-\frac{((b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c d^{5/2}}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{((b c-a d) (7 b c+a d)) \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 )}{8 c d^3}-\frac{((b c-a d) (7 b c+a d)) \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 )}{8 c d^3}-\frac{((b c-a d) (7 b c+a d)) \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 )}{8 \sqrt{2} c^{5/4} d^{11/4}}-\frac{((b c-a d) (7 b c+a d)) \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 )}{8 \sqrt{2} c^{5/4} d^{11/4}}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{(b c-a d) (7 b c+a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (7 b c+a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}-\frac{((b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}+\frac{((b c-a d) (7 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}\\ &=\frac{2 b^2 x^{3/2}}{3 d^2}+\frac{(b c-a d)^2 x^{3/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{(b c-a d) (7 b c+a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (7 b c+a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} d^{11/4}}-\frac{(b c-a d) (7 b c+a d) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}+\frac{(b c-a d) (7 b c+a d) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} d^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.178779, size = 319, normalized size = 1.03 \[ \frac{-\frac{3 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{6 \sqrt{2} \left (-a^2 d^2-6 a b c d+7 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}+\frac{24 d^{3/4} x^{3/2} (b c-a d)^2}{c \left (c+d x^2\right )}+32 b^2 d^{3/4} x^{3/2}}{48 d^{11/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 499, normalized size = 1.6 \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 = 1.14783, size = 3846, normalized size = 12.41 \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 = 1.20098, size = 524, normalized size = 1.69 \begin{align*} \frac{2 \, b^{2} x^{\frac{3}{2}}}{3 \, d^{2}} + \frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} c d^{2}} - \frac{\sqrt{2}{\left (7 \, \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 - \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 )}{8 \, c^{2} d^{5}} - \frac{\sqrt{2}{\left (7 \, \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 - \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 )}{8 \, c^{2} d^{5}} + \frac{\sqrt{2}{\left (7 \, \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 - \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 )}{16 \, c^{2} d^{5}} - \frac{\sqrt{2}{\left (7 \, \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 - \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 )}{16 \, c^{2} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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