Optimal. Leaf size=364 \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\sqrt{x} (7 a d+9 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\sqrt{x} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.297039, 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, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} d^{9/4}}+\frac{\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} d^{9/4}}-\frac{\sqrt{x} (7 a d+9 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\sqrt{x} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 463
Rule 457
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{\sqrt{x} \left (c+d x^2\right )^3} \, dx &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (-8 a^2 d^2+(b c-a d)^2\right )-4 b^2 c d x^2}{\sqrt{x} \left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{32 c^2 d^2}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^2 d^2}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{5/2} d^2}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{5/2} d^2}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{5/2} d^{5/2}}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{5/2} d^{5/2}}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}\\ &=\frac{(b c-a d)^2 \sqrt{x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (9 b c+7 a d) \sqrt{x}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}-\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac{\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}\\ \end{align*}
Mathematica [A] time = 0.195867, size = 339, normalized size = 0.93 \[ \frac{-\frac{8 c^{3/4} \sqrt [4]{d} \sqrt{x} \left (-7 a^2 d^2-2 a b c d+9 b^2 c^2\right )}{c+d x^2}-\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+\sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+2 \sqrt{2} \left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+\frac{32 c^{7/4} \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 c^{11/4} d^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 514, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{ \left ( d{x}^{2}+c \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 7\,{a}^{2}{d}^{2}+2\,cabd-9\,{b}^{2}{c}^{2} \right ){x}^{5/2}}{{c}^{2}d}}+1/32\,{\frac{ \left ( 11\,{a}^{2}{d}^{2}-6\,cabd-5\,{b}^{2}{c}^{2} \right ) \sqrt{x}}{{d}^{2}c}} \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{64\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}ab}{32\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{64\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{64\,{c}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}ab}{32\,{c}^{2}d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}{b}^{2}}{64\,{d}^{2}c}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}{a}^{2}}{128\,{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{3\,\sqrt{2}ab}{64\,{c}^{2}d}\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{5\,\sqrt{2}{b}^{2}}{128\,{d}^{2}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 ) } \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 = 1.33032, size = 3345, normalized size = 9.19 \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 = 1.20145, size = 562, normalized size = 1.54 \begin{align*} \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \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 )}{64 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \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 )}{64 \, c^{3} d^{3}} + \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \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 )}{128 \, c^{3} d^{3}} - \frac{\sqrt{2}{\left (5 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 21 \, \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 )}{128 \, c^{3} d^{3}} - \frac{9 \, b^{2} c^{2} d x^{\frac{5}{2}} - 2 \, a b c d^{2} x^{\frac{5}{2}} - 7 \, a^{2} d^{3} x^{\frac{5}{2}} + 5 \, b^{2} c^{3} \sqrt{x} + 6 \, a b c^{2} d \sqrt{x} - 11 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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