Optimal. Leaf size=633 \[ \frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}-\frac{d \sqrt{x} (15 b c-7 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d \sqrt{x}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
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Rubi [A] time = 0.831815, antiderivative size = 633, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {466, 414, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^3}+\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^3}-\frac{b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}-\frac{b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}-\frac{d \sqrt{x} (15 b c-7 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{d \sqrt{x}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 414
Rule 527
Rule 522
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{8 b c-7 a d-7 b d 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 \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{32 b^2 c^2-45 a b c d+21 a^2 d^2-3 b d (15 b c-7 a d) 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{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{(b c-a d)^3}-\frac{\left (d \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^2 (b c-a d)^3}\\ &=-\frac{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{a} (b c-a d)^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{\sqrt{a} (b c-a d)^3}-\frac{\left (d \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}-\frac{\left (d \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}\\ &=-\frac{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac{b^{5/2} \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 \sqrt{a} (b c-a d)^3}+\frac{b^{5/2} \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 \sqrt{a} (b c-a d)^3}-\frac{b^{11/4} \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^{3/4} (b c-a d)^3}-\frac{b^{11/4} \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^{3/4} (b c-a d)^3}-\frac{\left (\sqrt{d} \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}-\frac{\left (\sqrt{d} \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}+\frac{\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac{\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ &=-\frac{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac{b^{11/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac{b^{11/4} \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^{3/4} (b c-a d)^3}-\frac{b^{11/4} \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^{3/4} (b c-a d)^3}-\frac{\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac{\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ &=-\frac{d \sqrt{x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac{d (15 b c-7 a d) \sqrt{x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac{b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac{b^{11/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{b^{11/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} (b c-a d)^3}+\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac{d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.908967, size = 620, normalized size = 0.98 \[ \frac{1}{128} \left (\frac{\sqrt{2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^3}+\frac{\sqrt{2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (a d-b c)^3}+\frac{2 \sqrt{2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^3}-\frac{2 \sqrt{2} d^{3/4} \left (21 a^2 d^2-66 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^3}+\frac{32 \sqrt{2} b^{11/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} (a d-b c)^3}+\frac{32 \sqrt{2} b^{11/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4} (b c-a d)^3}+\frac{64 \sqrt{2} b^{11/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4} (a d-b c)^3}-\frac{64 \sqrt{2} b^{11/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} (a d-b c)^3}+\frac{8 d \sqrt{x} (7 a d-15 b c)}{c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{32 d \sqrt{x}}{c \left (c+d x^2\right )^2 (b c-a d)}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 882, normalized size = 1.4 \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 = 1.76236, size = 1296, normalized size = 2.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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