Optimal. Leaf size=402 \[ \frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}+\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\sqrt{x} \left (\frac{3 a^2 d}{c}+10 a b-\frac{45 b^2 c}{d}\right )}{16 c d^2}-\frac{x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{5/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.329627, antiderivative size = 402, 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, 457, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} d^{13/4}}+\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} d^{13/4}}-\frac{\sqrt{x} \left (\frac{3 a^2 d}{c}+10 a b-\frac{45 b^2 c}{d}\right )}{16 c d^2}-\frac{x^{5/2} (b c-a d) (3 a d+13 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{5/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 463
Rule 457
Rule 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 )^3} \, dx &=\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{x^{3/2} \left (\frac{1}{2} \left (-8 a^2 d^2+5 (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^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \int \frac{x^{3/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{32 c d^3}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c d^3}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{3/2} d^3}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{3/2} d^3}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{3/2} d^{7/2}}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{3/2} d^{7/2}}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}\\ &=\frac{\left (45 b^2 c^2-10 a b c d-3 a^2 d^2\right ) \sqrt{x}}{16 c^2 d^3}+\frac{(b c-a d)^2 x^{5/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (13 b c+3 a d) x^{5/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}+\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}-\frac{\left (45 b^2 c^2-10 a b c d-3 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^{7/4} d^{13/4}}\\ \end{align*}
Mathematica [A] time = 0.372445, size = 361, normalized size = 0.9 \[ \frac{\frac{8 \sqrt [4]{d} \sqrt{x} \left (a^2 d^2-18 a b c d+17 b^2 c^2\right )}{c \left (c+d x^2\right )}+\frac{\sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}-\frac{\sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{7/4}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2-10 a b c d+45 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{7/4}}-\frac{32 \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 \sqrt [4]{d} \sqrt{x}}{128 d^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 568, 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 = 0.964721, size = 3345, normalized size = 8.32 \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 = 1.20879, size = 575, normalized size = 1.43 \begin{align*} \frac{2 \, b^{2} \sqrt{x}}{d^{3}} - \frac{\sqrt{2}{\left (45 \, \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 - 3 \, \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^{2} d^{4}} - \frac{\sqrt{2}{\left (45 \, \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 - 3 \, \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^{2} d^{4}} - \frac{\sqrt{2}{\left (45 \, \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 - 3 \, \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^{2} d^{4}} + \frac{\sqrt{2}{\left (45 \, \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 - 3 \, \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^{2} d^{4}} + \frac{17 \, b^{2} c^{2} d x^{\frac{5}{2}} - 18 \, a b c d^{2} x^{\frac{5}{2}} + a^{2} d^{3} x^{\frac{5}{2}} + 13 \, b^{2} c^{3} \sqrt{x} - 10 \, a b c^{2} d \sqrt{x} - 3 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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