Optimal. Leaf size=624 \[ \frac{b^{5/4} (b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.858001, antiderivative size = 624, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {466, 472, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac{b^{5/4} (b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d x^{3/2} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2}+\frac{b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 472
Rule 579
Rule 584
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-b c+4 a d-5 b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (-4 \left (b^2 c^2-8 a b c d+a^2 d^2\right )-4 b d (b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{8 a c (b c-a d)^2}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{4 b^2 c (b c-9 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{4 a d^2 (-9 b c+a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{8 a c (b c-a d)^2}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{\left (b^2 (b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)^3}+\frac{\left (d^2 (9 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 c (b c-a d)^3}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{\left (b^{3/2} (b c-9 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 (b c-a d)^3}+\frac{\left (b^{3/2} (b c-9 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 (b c-a d)^3}-\frac{\left (d^{3/2} (9 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c (b c-a d)^3}+\frac{\left (d^{3/2} (9 b c-a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 c (b c-a d)^3}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{(b (b c-9 a d)) \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 (b c-a d)^3}+\frac{(b (b c-9 a d)) \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 (b c-a d)^3}+\frac{\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac{\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac{(d (9 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 (b c-a d)^3}+\frac{(d (9 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 (b c-a d)^3}+\frac{\left (d^{5/4} (9 b c-a d)\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 )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{\left (d^{5/4} (9 b c-a d)\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 )}{8 \sqrt{2} c^{5/4} (b c-a d)^3}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 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^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 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^{5/4} (b c-a d)^3}+\frac{\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}-\frac{\left (b^{5/4} (b c-9 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^{5/4} (b c-a d)^3}+\frac{\left (d^{5/4} (9 b c-a d)\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 )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}-\frac{\left (d^{5/4} (9 b c-a d)\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 )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}\\ &=\frac{d (b c+a d) x^{3/2}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac{b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 b c-a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{5/4} (b c-a d)^3}+\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}-\frac{b^{5/4} (b c-9 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} (b c-a d)^3}+\frac{d^{5/4} (9 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^{5/4} (b c-a d)^3}-\frac{d^{5/4} (9 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^{5/4} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 1.14288, size = 589, normalized size = 0.94 \[ \frac{1}{16} \left (\frac{\sqrt{2} b^{5/4} (9 a d-b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (a d-b c)^3}+\frac{\sqrt{2} b^{5/4} (b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4} (a d-b c)^3}+\frac{2 \sqrt{2} b^{5/4} (9 a d-b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4} (a d-b c)^3}+\frac{8 b^2 x^{3/2}}{a \left (a+b x^2\right ) (b c-a d)^2}+\frac{\sqrt{2} d^{5/4} (9 b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} (b c-a d)^3}+\frac{\sqrt{2} d^{5/4} (a d-9 b c) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (a d-9 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4} (b c-a d)^3}+\frac{2 \sqrt{2} d^{5/4} (9 b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4} (b c-a d)^3}+\frac{8 d^2 x^{3/2}}{c \left (c+d x^2\right ) (b c-a d)^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 778, 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 [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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