Optimal. Leaf size=681 \[ -\frac{77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^3}-\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{b^{15/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{15/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d (19 b c-11 a d)}{16 c^2 x^{3/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.927946, antiderivative size = 681, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {466, 472, 579, 583, 522, 211, 1165, 628, 1162, 617, 204} \[ -\frac{77 a^2 d^2-133 a b c d+32 b^2 c^2}{48 a c^3 x^{3/2} (b c-a d)^2}-\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^3}-\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^3}+\frac{b^{15/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}+\frac{b^{15/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d (19 b c-11 a d)}{16 c^2 x^{3/2} \left (c+d x^2\right ) (b c-a d)^2}-\frac{d}{4 c x^{3/2} \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 472
Rule 579
Rule 583
Rule 522
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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^{3/2} \left (c+d x^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{8 b c-11 a d-11 b d x^4}{x^4 \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^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{32 b^2 c^2-133 a b c d+77 a^2 d^2-7 b d (19 b c-11 a d) x^4}{x^4 \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}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{3 \left (32 b^3 c^3+32 a b^2 c^2 d-133 a^2 b c d^2+77 a^3 d^3\right )+3 b d \left (32 b^2 c^2-133 a b c d+77 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{48 a c^3 (b c-a d)^2}\\ &=-\frac{\frac{32 b^2 c}{a}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a (b c-a d)^3}+\frac{\left (d^2 \left (165 b^2 c^2-210 a b c d+77 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^3 (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac{b^4 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^{3/2} (b c-a d)^3}-\frac{b^4 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{a^{3/2} (b c-a d)^3}+\frac{\left (d^2 \left (165 b^2 c^2-210 a b c d+77 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^{7/2} (b c-a d)^3}+\frac{\left (d^2 \left (165 b^2 c^2-210 a b c d+77 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^{7/2} (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}-\frac{b^{7/2} \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^{3/2} (b c-a d)^3}-\frac{b^{7/2} \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^{3/2} (b c-a d)^3}+\frac{b^{15/4} \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^{7/4} (b c-a d)^3}+\frac{b^{15/4} \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^{7/4} (b c-a d)^3}+\frac{\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 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^{7/2} (b c-a d)^3}+\frac{\left (d^{3/2} \left (165 b^2 c^2-210 a b c d+77 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^{7/2} (b c-a d)^3}-\frac{\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}-\frac{\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac{b^{15/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}+\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}-\frac{b^{15/4} \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^{7/4} (b c-a d)^3}+\frac{b^{15/4} \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^{7/4} (b c-a d)^3}+\frac{\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}-\frac{\left (d^{7/4} \left (165 b^2 c^2-210 a b c d+77 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^{15/4} (b c-a d)^3}\\ &=-\frac{\frac{32 b^2 c}{a}-133 b d+\frac{77 a d^2}{c}}{48 c^2 (b c-a d)^2 x^{3/2}}-\frac{d}{4 c (b c-a d) x^{3/2} \left (c+d x^2\right )^2}-\frac{d (19 b c-11 a d)}{16 c^2 (b c-a d)^2 x^{3/2} \left (c+d x^2\right )}+\frac{b^{15/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}+\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}+\frac{b^{15/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{b^{15/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{7/4} (b c-a d)^3}-\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}+\frac{d^{7/4} \left (165 b^2 c^2-210 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} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 1.01227, size = 639, normalized size = 0.94 \[ \frac{1}{384} \left (\frac{3 \sqrt{2} d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (a d-b c)^3}+\frac{3 \sqrt{2} d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (b c-a d)^3}-\frac{6 \sqrt{2} d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{15/4} (b c-a d)^3}+\frac{6 \sqrt{2} d^{7/4} \left (77 a^2 d^2-210 a b c d+165 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{15/4} (b c-a d)^3}+\frac{96 \sqrt{2} b^{15/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^3}+\frac{96 \sqrt{2} b^{15/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (a d-b c)^3}-\frac{192 \sqrt{2} b^{15/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4} (a d-b c)^3}+\frac{192 \sqrt{2} b^{15/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4} (a d-b c)^3}+\frac{24 d^2 \sqrt{x} (23 b c-15 a d)}{c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{96 d^2 \sqrt{x}}{c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac{256}{a c^3 x^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 906, 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 = 2.05851, size = 1343, normalized size = 1.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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