Optimal. Leaf size=346 \[ -\frac{(b c-a d) (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{x^{5/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac{\sqrt{x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2} \]
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Rubi [A] time = 0.328888, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {463, 459, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{(b c-a d) (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{x^{5/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac{\sqrt{x} (b c-a d) (9 b c-a d)}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
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Rule 463
Rule 459
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{\int \frac{x^{3/2} \left (\frac{1}{2} \left (-4 a^2 d^2+5 (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^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{((b c-a d) (9 b c-a d)) \int \frac{x^{3/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{((b c-a d) (9 b c-a d)) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{4 d^3}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{((b c-a d) (9 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 d^3}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{((b c-a d) (9 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 \sqrt{c} d^3}+\frac{((b c-a d) (9 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 \sqrt{c} d^3}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}+\frac{((b c-a d) (9 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 \sqrt{c} d^{7/2}}+\frac{((b c-a d) (9 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 \sqrt{c} d^{7/2}}-\frac{((b c-a d) (9 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^{3/4} d^{13/4}}-\frac{((b c-a d) (9 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^{3/4} d^{13/4}}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{(b c-a d) (9 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^{3/4} d^{13/4}}+\frac{(b c-a d) (9 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^{3/4} d^{13/4}}+\frac{((b c-a d) (9 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^{3/4} d^{13/4}}-\frac{((b c-a d) (9 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^{3/4} d^{13/4}}\\ &=-\frac{(b c-a d) (9 b c-a d) \sqrt{x}}{2 c d^3}+\frac{2 b^2 x^{5/2}}{5 d^2}+\frac{(b c-a d)^2 x^{5/2}}{2 c d^2 \left (c+d x^2\right )}-\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}+\frac{(b c-a d) (9 b c-a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{3/4} d^{13/4}}-\frac{(b c-a d) (9 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^{3/4} d^{13/4}}+\frac{(b c-a d) (9 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^{3/4} d^{13/4}}\\ \end{align*}
Mathematica [A] time = 0.177896, size = 333, normalized size = 0.96 \[ \frac{-\frac{5 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}+\frac{5 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{3/4}}-\frac{10 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{3/4}}+\frac{10 \sqrt{2} \left (a^2 d^2-10 a b c d+9 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{3/4}}-\frac{40 \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{c+d x^2}-320 b \sqrt [4]{d} \sqrt{x} (b c-a d)+32 b^2 d^{5/4} x^{5/2}}{80 d^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 523, normalized size = 1.5 \begin{align*}{\frac{2\,{b}^{2}}{5\,{d}^{2}}{x}^{{\frac{5}{2}}}}+4\,{\frac{ab\sqrt{x}}{{d}^{2}}}-4\,{\frac{{b}^{2}\sqrt{x}c}{{d}^{3}}}-{\frac{{a}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }\sqrt{x}}+{\frac{abc}{{d}^{2} \left ( d{x}^{2}+c \right ) }\sqrt{x}}-{\frac{{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }\sqrt{x}}+{\frac{\sqrt{2}{a}^{2}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}ab}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{9\,c\sqrt{2}{b}^{2}}{8\,{d}^{3}}\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}}{8\,cd}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}ab}{4\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{9\,c\sqrt{2}{b}^{2}}{8\,{d}^{3}}\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}}{16\,cd}\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}ab}{8\,{d}^{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{9\,c\sqrt{2}{b}^{2}}{16\,{d}^{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 ) } \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 = 1.08931, size = 3036, normalized size = 8.77 \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 = 1.21174, size = 551, normalized size = 1.59 \begin{align*} \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \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 )}{8 \, c d^{4}} + \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \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 )}{8 \, c d^{4}} + \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \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 )}{16 \, c d^{4}} - \frac{\sqrt{2}{\left (9 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 10 \, \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 )}{16 \, c d^{4}} - \frac{b^{2} c^{2} \sqrt{x} - 2 \, a b c d \sqrt{x} + a^{2} d^{2} \sqrt{x}}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{2 \,{\left (b^{2} d^{8} x^{\frac{5}{2}} - 10 \, b^{2} c d^{7} \sqrt{x} + 10 \, a b d^{8} \sqrt{x}\right )}}{5 \, d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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