Optimal. Leaf size=402 \[ -\frac{\sqrt{x} \left (11 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} d^{5/4}}-\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\sqrt{x} \left (7 a d (6 b c-11 a d)+3 b^2 c^2\right )}{48 c^3 d \left (c+d x^2\right )} \]
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Rubi [A] time = 0.405564, antiderivative size = 401, 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, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt{x} \left (11 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^2 d \left (c+d x^2\right )^2}-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} d^{5/4}}-\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\left (7 a d (6 b c-11 a d)+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} d^{5/4}}+\frac{\sqrt{x} \left (\frac{7 a (6 b c-11 a d)}{c^2}+\frac{3 b^2}{d}\right )}{48 c \left (c+d x^2\right )} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
Rule 290
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}{x^{5/2} \left (c+d x^2\right )^3} \, dx &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}+\frac{2 \int \frac{\frac{1}{2} a (6 b c-11 a d)+\frac{3}{2} b^2 c x^2}{\sqrt{x} \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{1}{24} \left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )^2} \, dx\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{32 c}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^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}}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^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}}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{7/2} d^{3/2}}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{7/2} d^{3/2}}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ &=-\frac{2 a^2}{3 c x^{3/2} \left (c+d x^2\right )^2}-\frac{\left (3 b^2 c^2-6 a b c d+11 a^2 d^2\right ) \sqrt{x}}{12 c^2 d \left (c+d x^2\right )^2}+\frac{\left (\frac{3 b^2}{d}+\frac{7 a (6 b c-11 a d)}{c^2}\right ) \sqrt{x}}{48 c \left (c+d x^2\right )}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}-\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}+\frac{\left (3 b^2 c^2+42 a b c d-77 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^{15/4} d^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.259722, size = 365, normalized size = 0.91 \[ \frac{\frac{24 c^{3/4} \sqrt{x} \left (-15 a^2 d^2+14 a b c d+b^2 c^2\right )}{d \left (c+d x^2\right )}+\frac{3 \sqrt{2} \left (77 a^2 d^2-42 a b c d-3 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{5/4}}+\frac{3 \sqrt{2} \left (-77 a^2 d^2+42 a b c d+3 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{5/4}}+\frac{6 \sqrt{2} \left (77 a^2 d^2-42 a b c d-3 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{5/4}}+\frac{6 \sqrt{2} \left (-77 a^2 d^2+42 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{5/4}}-\frac{256 a^2 c^{3/4}}{x^{3/2}}-\frac{96 c^{7/4} \sqrt{x} (b c-a d)^2}{d \left (c+d x^2\right )^2}}{384 c^{15/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 562, 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 = 1.36903, size = 3515, normalized size = 8.74 \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 = 1.22447, size = 575, normalized size = 1.43 \begin{align*} -\frac{2 \, a^{2}}{3 \, c^{3} x^{\frac{3}{2}}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 77 \, \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^{4} d^{2}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 77 \, \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^{4} d^{2}} + \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 77 \, \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^{4} d^{2}} - \frac{\sqrt{2}{\left (3 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} + 42 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d - 77 \, \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^{4} d^{2}} + \frac{b^{2} c^{2} d x^{\frac{5}{2}} + 14 \, a b c d^{2} x^{\frac{5}{2}} - 15 \, a^{2} d^{3} x^{\frac{5}{2}} - 3 \, b^{2} c^{3} \sqrt{x} + 22 \, a b c^{2} d \sqrt{x} - 19 \, a^{2} c d^{2} \sqrt{x}}{16 \,{\left (d x^{2} + c\right )}^{2} c^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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