Optimal. Leaf size=743 \[ -\frac{117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac{-189 a^2 b c d^2+117 a^3 d^3+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt{x} (b c-a d)^2}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 1.23311, antiderivative size = 743, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {466, 472, 579, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac{117 a^2 d^2-189 a b c d+32 b^2 c^2}{80 a c^3 x^{5/2} (b c-a d)^2}+\frac{-189 a^2 b c d^2+117 a^3 d^3+32 a b^2 c^2 d+32 b^3 c^3}{16 a^2 c^4 \sqrt{x} (b c-a d)^2}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^3}+\frac{b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d (21 b c-13 a d)}{16 c^2 x^{5/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{5/2} \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 472
Rule 579
Rule 583
Rule 584
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{8 b c-13 a d-13 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{4 c (b c-a d)}\\ &=-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{32 b^2 c^2-189 a b c d+117 a^2 d^2-9 b d (21 b c-13 a d) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{5 \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right )+5 b d \left (32 b^2 c^2-189 a b c d+117 a^2 d^2\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{80 a c^3 (b c-a d)^2}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (5 \left (32 b^4 c^4+32 a b^3 c^3 d+32 a^2 b^2 c^2 d^2-189 a^3 b c d^3+117 a^4 d^4\right )+5 b d \left (32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{160 b^5 c^4 x^2}{(b c-a d) \left (a+b x^4\right )}+\frac{5 a^2 d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{80 a^2 c^4 (b c-a d)^2}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{\left (2 b^5\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^3}-\frac{\left (d^3 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^4 (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac{b^{9/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^3}+\frac{b^{9/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^2 (b c-a d)^3}+\frac{\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^4 (b c-a d)^3}-\frac{\left (d^{5/2} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^4 (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 a^2 (b c-a d)^3}+\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{2 a^2 (b c-a d)^3}+\frac{b^{17/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^4 (b c-a d)^3}-\frac{\left (d^2 \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^4 (b c-a d)^3}-\frac{\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^{17/4} (b c-a d)^3}-\frac{\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^{17/4} (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}+\frac{b^{17/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}+\frac{b^{17/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}-\frac{\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^{17/4} (b c-a d)^3}+\frac{\left (d^{9/4} \left (221 b^2 c^2-306 a b c d+117 a^2 d^2\right )\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^{17/4} (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-189 b d+\frac{117 a d^2}{c}}{80 c^2 (b c-a d)^2 x^{5/2}}+\frac{32 b^3 c^3+32 a b^2 c^2 d-189 a^2 b c d^2+117 a^3 d^3}{16 a^2 c^4 (b c-a d)^2 \sqrt{x}}-\frac{d}{4 c (b c-a d) x^{5/2} \left (c+d x^2\right )^2}-\frac{d (21 b c-13 a d)}{16 c^2 (b c-a d)^2 x^{5/2} \left (c+d x^2\right )}-\frac{b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{b^{17/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4} (b c-a d)^3}+\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}-\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}+\frac{b^{17/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{b^{17/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4} (b c-a d)^3}-\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}+\frac{d^{9/4} \left (221 b^2 c^2-306 a b c d+117 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^{17/4} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 1.28456, size = 660, normalized size = 0.89 \[ \frac{1}{640} \left (\frac{5 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (a d-b c)^3}+\frac{5 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^3}+\frac{10 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^3}-\frac{10 \sqrt{2} d^{9/4} \left (117 a^2 d^2-306 a b c d+221 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^3}+\frac{160 \sqrt{2} b^{17/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (b c-a d)^3}+\frac{160 \sqrt{2} b^{17/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} (a d-b c)^3}+\frac{320 \sqrt{2} b^{17/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} (a d-b c)^3}-\frac{320 \sqrt{2} b^{17/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} (a d-b c)^3}+\frac{1280 (3 a d+b c)}{a^2 c^4 \sqrt{x}}+\frac{40 d^3 x^{3/2} (21 a d-29 b c)}{c^4 \left (c+d x^2\right ) (b c-a d)^2}-\frac{160 d^3 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)}-\frac{256}{a c^3 x^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 933, normalized size = 1.3 \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 [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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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 = 1.87921, size = 1350, normalized size = 1.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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