Optimal. Leaf size=399 \[ -\frac{x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{x^{3/2} \left (5 a d (2 b c-9 a d)+3 b^2 c^2\right )}{16 c^3 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.39452, antiderivative size = 398, 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 = {462, 457, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{x^{3/2} \left (9 a^2 d^2-2 a b c d+b^2 c^2\right )}{4 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{13/4} d^{7/4}}-\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{\left (5 a d (2 b c-9 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{13/4} d^{7/4}}+\frac{x^{3/2} \left (\frac{5 a (2 b c-9 a d)}{c^2}+\frac{3 b^2}{d}\right )}{16 c \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
Rule 290
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^{3/2} \left (c+d x^2\right )^3} \, dx &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}+\frac{2 \int \frac{\sqrt{x} \left (\frac{1}{2} a (2 b c-9 a d)+\frac{1}{2} b^2 c x^2\right )}{\left (c+d x^2\right )^3} \, dx}{c}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{1}{8} \left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) \int \frac{\sqrt{x}}{\left (c+d x^2\right )^2} \, dx\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) \int \frac{\sqrt{x}}{c+d x^2} \, dx}{32 c}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2+10 a b c d-45 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 d^{3/2}}+\frac{\left (3 b^2 c^2+10 a b c d-45 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 d^{3/2}}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2+10 a b c d-45 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 d^2}+\frac{\left (3 b^2 c^2+10 a b c d-45 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 d^2}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}-\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}-\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}\\ &=-\frac{2 a^2}{c \sqrt{x} \left (c+d x^2\right )^2}-\frac{\left (b^2 c^2-2 a b c d+9 a^2 d^2\right ) x^{3/2}}{4 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{5 a (2 b c-9 a d)}{c^2}\right ) x^{3/2}}{16 c \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}+\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}-\frac{\left (3 b^2 c^2+10 a b c d-45 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^{13/4} d^{7/4}}\\ \end{align*}
Mathematica [A] time = 0.40807, size = 364, normalized size = 0.91 \[ \frac{\frac{8 \sqrt [4]{c} x^{3/2} \left (-13 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{d \left (c+d x^2\right )}+\frac{\sqrt{2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{\sqrt{2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (45 a^2 d^2-10 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{7/4}}+\frac{2 \sqrt{2} \left (-45 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{7/4}}-\frac{256 a^2 \sqrt [4]{c}}{\sqrt{x}}-\frac{32 c^{5/4} x^{3/2} (b c-a d)^2}{d \left (c+d x^2\right )^2}}{128 c^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, 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 = 1.51384, size = 4365, normalized size = 10.94 \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 = 1.20441, size = 576, normalized size = 1.44 \begin{align*} -\frac{2 \, a^{2}}{c^{3} \sqrt{x}} + \frac{3 \, b^{2} c^{2} d x^{\frac{7}{2}} + 10 \, a b c d^{2} x^{\frac{7}{2}} - 13 \, a^{2} d^{3} x^{\frac{7}{2}} - b^{2} c^{3} x^{\frac{3}{2}} + 18 \, a b c^{2} d x^{\frac{3}{2}} - 17 \, a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{3} d} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \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^{4} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \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^{4} d^{4}} - \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \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^{4} d^{4}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} + 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 45 \, \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^{4} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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