Optimal. Leaf size=881 \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]
[Out]
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Rubi [A] time = 1.69109, antiderivative size = 881, normalized size of antiderivative = 1., number of steps used = 26, 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} \[ -\frac{3 (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{17/4}}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 (3 b c-7 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{17/4}}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{5/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b d c+39 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{17/4} (b c-a d)^4}+\frac{3 \left (24 b^4 c^4-32 a b^3 d c^3-32 a^2 b^2 d^2 c^2+87 a^3 b d^3 c-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d \left (8 b^2 c^2+29 a b d c-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (d x^2+c\right )}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (d x^2+c\right )^2}-\frac{3 \left (24 b^3 c^3-32 a b^2 d c^2+87 a^2 b d^2 c-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
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 )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^6 \left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-9 b c+4 a d-17 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)}\\ &=\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 \left (18 b^2 c^2-16 a b c d+13 a^2 d^2\right )-52 b d (2 b c+a d) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{16 a c (b c-a d)^2}\\ &=\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-12 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )-36 b d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right ) x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{64 a c^2 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-60 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )-60 b d \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right ) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{320 a^2 c^3 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-60 \left (24 b^5 c^5-32 a b^4 c^4 d-32 a^2 b^3 c^3 d^2-32 a^3 b^2 c^2 d^3+87 a^4 b c d^4-39 a^5 d^5\right )-60 b d \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{320 a^3 c^4 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{480 b^5 c^4 (3 b c-7 a d) x^2}{(b c-a d) \left (a+b x^4\right )}+\frac{60 a^3 d^4 \left (119 b^2 c^2-126 a b c d+39 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{320 a^3 c^4 (b c-a d)^3}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\left (3 b^5 (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^3 (b c-a d)^4}+\frac{\left (3 d^4 \left (119 b^2 c^2-126 a b c d+39 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)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{\left (3 b^{9/2} (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^3 (b c-a d)^4}+\frac{\left (3 b^{9/2} (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^3 (b c-a d)^4}-\frac{\left (3 d^{7/2} \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{\left (3 d^{7/2} \left (119 b^2 c^2-126 a b c d+39 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)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{\left (3 b^4 (3 b c-7 a d)\right ) \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 )}{8 a^3 (b c-a d)^4}+\frac{\left (3 b^4 (3 b c-7 a d)\right ) \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 )}{8 a^3 (b c-a d)^4}+\frac{\left (3 b^{17/4} (3 b c-7 a d)\right ) \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 )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{\left (3 b^{17/4} (3 b c-7 a d)\right ) \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 )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{\left (3 d^3 \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{\left (3 d^3 \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}+\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{\left (3 b^{17/4} (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{\left (3 b^{17/4} (3 b c-7 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}-\frac{\left (3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}\\ &=-\frac{3 \left (24 b^3 c^3-32 a b^2 c^2 d+87 a^2 b c d^2-39 a^3 d^3\right )}{80 a^2 c^3 (b c-a d)^3 x^{5/2}}+\frac{3 \left (24 b^4 c^4-32 a b^3 c^3 d-32 a^2 b^2 c^2 d^2+87 a^3 b c d^3-39 a^4 d^4\right )}{16 a^3 c^4 (b c-a d)^3 \sqrt{x}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{5/2} \left (c+d x^2\right )^2}+\frac{b}{2 a (b c-a d) x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+29 a b c d-13 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{5/2} \left (c+d x^2\right )}-\frac{3 b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}+\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}-\frac{3 b^{17/4} (3 b c-7 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} (b c-a d)^4}+\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}-\frac{3 d^{13/4} \left (119 b^2 c^2-126 a b c d+39 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)^4}\\ \end{align*}
Mathematica [A] time = 2.20188, size = 729, normalized size = 0.83 \[ \frac{1}{640} \left (\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{15 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{17/4} (b c-a d)^4}-\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{17/4} (b c-a d)^4}+\frac{30 \sqrt{2} d^{13/4} \left (39 a^2 d^2-126 a b c d+119 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{17/4} (b c-a d)^4}-\frac{320 b^5 x^{3/2}}{a^3 \left (a+b x^2\right ) (a d-b c)^3}+\frac{120 \sqrt{2} b^{17/4} (3 b c-7 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{120 \sqrt{2} b^{17/4} (7 a d-3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (7 a d-3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{13/4} (b c-a d)^4}+\frac{240 \sqrt{2} b^{17/4} (3 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{13/4} (b c-a d)^4}+\frac{1280 (3 a d+2 b c)}{a^3 c^4 \sqrt{x}}-\frac{256}{a^2 c^3 x^{5/2}}+\frac{40 d^4 x^{3/2} (37 b c-21 a d)}{c^4 \left (c+d x^2\right ) (b c-a d)^3}+\frac{160 d^4 x^{3/2}}{c^3 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 1170, 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 = 2.71873, size = 1740, normalized size = 1.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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