Optimal. Leaf size=627 \[ -\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]
[Out]
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Rubi [A] time = 1.49367, antiderivative size = 627, 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 \[ -\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}-\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{7/4} \sqrt [4]{d} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} (b c-a d)^3}+\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} (b c-a d)^3}-\frac{\sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} (b c-a d)^3}+\frac{\sqrt{x} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^2}+\frac{\sqrt{x}}{4 \left (c+d x^2\right )^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/((a + b*x^2)*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(b*x**2+a)/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 1.66039, size = 543, normalized size = 0.87 \[ \frac{-\frac{\sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4} \sqrt [4]{d}}+\frac{\sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{7/4} \sqrt [4]{d}}-\frac{2 \sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{7/4} \sqrt [4]{d}}+\frac{2 \sqrt{2} \left (-3 a^2 d^2+14 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{7/4} \sqrt [4]{d}}+32 \sqrt{2} \sqrt [4]{a} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-32 \sqrt{2} \sqrt [4]{a} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+64 \sqrt{2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-64 \sqrt{2} \sqrt [4]{a} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\frac{32 \sqrt{x} (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{8 \sqrt{x} (a d+7 b c) (b c-a d)}{c \left (c+d x^2\right )}}{128 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/((a + b*x^2)*(c + d*x^2)^3),x]
[Out]
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Maple [A] time = 0.027, size = 848, normalized size = 1.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(b*x^2+a)/(d*x^2+c)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(b*x**2+a)/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.398723, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="giac")
[Out]