Optimal. Leaf size=311 \[ -\frac{c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{17/4}}-\frac{2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac{2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 c \sqrt{x} (b c-a d)^2}{d^4}+\frac{2 b^2 x^{13/2}}{13 d} \]
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Rubi [A] time = 0.312552, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {461, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{17/4}}-\frac{2 b x^{9/2} (b c-2 a d)}{9 d^2}+\frac{2 x^{5/2} (b c-a d)^2}{5 d^3}-\frac{2 c \sqrt{x} (b c-a d)^2}{d^4}+\frac{2 b^2 x^{13/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 461
Rule 321
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^{7/2} \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac{b (b c-2 a d) x^{7/2}}{d^2}+\frac{b^2 x^{11/2}}{d}+\frac{\left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^{7/2}}{d^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}+\frac{(b c-a d)^2 \int \frac{x^{7/2}}{c+d x^2} \, dx}{d^2}\\ &=\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}-\frac{\left (c (b c-a d)^2\right ) \int \frac{x^{3/2}}{c+d x^2} \, dx}{d^3}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}+\frac{\left (c^2 (b c-a d)^2\right ) \int \frac{1}{\sqrt{x} \left (c+d x^2\right )} \, dx}{d^4}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}+\frac{\left (2 c^2 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{d^4}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}+\frac{\left (c^{3/2} (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{d^4}+\frac{\left (c^{3/2} (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{d^4}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}+\frac{\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}+\frac{\left (c^{3/2} (b c-a 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 )}{2 d^{9/2}}-\frac{\left (c^{5/4} (b c-a 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 )}{2 \sqrt{2} d^{17/4}}-\frac{\left (c^{5/4} (b c-a 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 )}{2 \sqrt{2} d^{17/4}}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}-\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{\left (c^{5/4} (b c-a 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 )}{\sqrt{2} d^{17/4}}-\frac{\left (c^{5/4} (b c-a 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 )}{\sqrt{2} d^{17/4}}\\ &=-\frac{2 c (b c-a d)^2 \sqrt{x}}{d^4}+\frac{2 (b c-a d)^2 x^{5/2}}{5 d^3}-\frac{2 b (b c-2 a d) x^{9/2}}{9 d^2}+\frac{2 b^2 x^{13/2}}{13 d}-\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{17/4}}-\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}+\frac{c^{5/4} (b c-a d)^2 \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{2 \sqrt{2} d^{17/4}}\\ \end{align*}
Mathematica [A] time = 0.134275, size = 299, normalized size = 0.96 \[ \frac{-585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+585 \sqrt{2} c^{5/4} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )+1170 \sqrt{2} c^{5/4} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-520 b d^{9/4} x^{9/2} (b c-2 a d)+936 d^{5/4} x^{5/2} (b c-a d)^2-4680 c \sqrt [4]{d} \sqrt{x} (b c-a d)^2+360 b^2 d^{13/4} x^{13/2}}{2340 d^{17/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 545, normalized size = 1.8 \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 = 1.11715, size = 2813, normalized size = 9.05 \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 = 1.40537, size = 589, normalized size = 1.89 \begin{align*} \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c 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 )}{2 \, d^{5}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c 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 )}{2 \, d^{5}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{5}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{3} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c^{2} d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} c d^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{5}} + \frac{2 \,{\left (45 \, b^{2} d^{12} x^{\frac{13}{2}} - 65 \, b^{2} c d^{11} x^{\frac{9}{2}} + 130 \, a b d^{12} x^{\frac{9}{2}} + 117 \, b^{2} c^{2} d^{10} x^{\frac{5}{2}} - 234 \, a b c d^{11} x^{\frac{5}{2}} + 117 \, a^{2} d^{12} x^{\frac{5}{2}} - 585 \, b^{2} c^{3} d^{9} \sqrt{x} + 1170 \, a b c^{2} d^{10} \sqrt{x} - 585 \, a^{2} c d^{11} \sqrt{x}\right )}}{585 \, d^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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