Optimal. Leaf size=363 \[ -\frac{9 a^2 d^2-10 a b c d+5 b^2 c^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac{(b c-9 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}} \]
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Rubi [A] time = 0.377162, antiderivative size = 363, 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, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{9 a^2 d^2-10 a b c d+5 b^2 c^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac{(b c-9 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 462
Rule 457
Rule 325
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^{7/2} \left (c+d x^2\right )^2} \, dx &=-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac{2 \int \frac{\frac{1}{2} a (10 b c-9 a d)+\frac{5}{2} b^2 c x^2}{x^{3/2} \left (c+d x^2\right )^2} \, dx}{5 c}\\ &=-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{((b c-9 a d) (b c-a d)) \int \frac{1}{x^{3/2} \left (c+d x^2\right )} \, dx}{4 c^2 d}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}+\frac{((b c-9 a d) (b c-a d)) \int \frac{\sqrt{x}}{c+d x^2} \, dx}{4 c^3}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}+\frac{((b c-9 a d) (b c-a d)) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c^3}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{((b c-9 a d) (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 c^3 \sqrt{d}}+\frac{((b c-9 a d) (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 c^3 \sqrt{d}}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}+\frac{((b c-9 a d) (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 c^3 d}+\frac{((b c-9 a d) (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 c^3 d}+\frac{((b c-9 a d) (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^{13/4} d^{3/4}}+\frac{((b c-9 a d) (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^{13/4} d^{3/4}}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}+\frac{(b c-9 a d) (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^{13/4} d^{3/4}}-\frac{(b c-9 a d) (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^{13/4} d^{3/4}}+\frac{((b c-9 a d) (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^{13/4} d^{3/4}}-\frac{((b c-9 a d) (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^{13/4} d^{3/4}}\\ &=\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}-\frac{5 b^2 c^2-10 a b c d+9 a^2 d^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (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^{13/4} d^{3/4}}-\frac{(b c-9 a d) (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^{13/4} d^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.198704, size = 333, normalized size = 0.92 \[ \frac{\frac{5 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{5 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{10 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{3/4}}+\frac{10 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{3/4}}-\frac{32 a^2 c^{5/4}}{x^{5/2}}+\frac{40 \sqrt [4]{c} x^{3/2} (b c-a d)^2}{c+d x^2}+\frac{320 a \sqrt [4]{c} (a d-b c)}{\sqrt{x}}}{80 c^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 524, 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 = 1.20001, size = 3997, normalized size = 11.01 \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 = 1.21053, size = 541, normalized size = 1.49 \begin{align*} \frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} c^{3}} - \frac{2 \,{\left (10 \, a b c x^{2} - 10 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{3} x^{\frac{5}{2}}} + \frac{\sqrt{2}{\left (\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 + 9 \, \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 )}{8 \, c^{4} d^{3}} + \frac{\sqrt{2}{\left (\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 + 9 \, \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 )}{8 \, c^{4} d^{3}} - \frac{\sqrt{2}{\left (\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 + 9 \, \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 )}{16 \, c^{4} d^{3}} + \frac{\sqrt{2}{\left (\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 + 9 \, \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 )}{16 \, c^{4} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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