Optimal. Leaf size=123 \[ -\frac{3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{7/2}}+\frac{3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac{5 b c-4 a d}{4 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b}{4 d x^5 \sqrt{c+\frac{d}{x^2}}} \]
[Out]
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Rubi [A] time = 0.224599, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{7/2}}+\frac{3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac{5 b c-4 a d}{4 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b}{4 d x^5 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x]
[Out]
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Rubi in Sympy [A] time = 18.7608, size = 114, normalized size = 0.93 \[ - \frac{b}{4 d x^{5} \sqrt{c + \frac{d}{x^{2}}}} + \frac{3 c \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{7}{2}}} + \frac{4 a d - 5 b c}{4 d^{2} x^{3} \sqrt{c + \frac{d}{x^{2}}}} - \frac{3 \sqrt{c + \frac{d}{x^{2}}} \left (4 a d - 5 b c\right )}{8 d^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**6,x)
[Out]
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Mathematica [A] time = 0.216183, size = 143, normalized size = 1.16 \[ \frac{\sqrt{d} \left (b \left (15 c^2 x^4+5 c d x^2-2 d^2\right )-4 a d x^2 \left (3 c x^2+d\right )\right )+3 c x^4 \log (x) \sqrt{c x^2+d} (5 b c-4 a d)-3 c x^4 \sqrt{c x^2+d} (5 b c-4 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{8 d^{7/2} x^5 \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x]
[Out]
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Maple [A] time = 0.019, size = 159, normalized size = 1.3 \[ -{\frac{c{x}^{2}+d}{8\,{x}^{7}} \left ( 12\,ac{d}^{9/2}{x}^{4}-15\,b{c}^{2}{x}^{4}{d}^{7/2}-12\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{4}{x}^{4}\sqrt{c{x}^{2}+d}+4\,a{d}^{11/2}{x}^{2}-5\,bc{d}^{9/2}{x}^{2}+15\,b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{3}{x}^{4}\sqrt{c{x}^{2}+d}+2\,b{d}^{11/2} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{13}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/(c+d/x^2)^(3/2)/x^6,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((a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254471, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} +{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt{d} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} +{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \,{\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}, \frac{3 \,{\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} +{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \,{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} +{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \,{\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 53.3011, size = 180, normalized size = 1.46 \[ a \left (- \frac{3 \sqrt{c}}{2 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{5}{2}}} - \frac{1}{2 \sqrt{c} d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}}\right ) + b \left (\frac{15 c^{\frac{3}{2}}}{8 d^{3} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{5 \sqrt{c}}{8 d^{2} x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{15 c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{7}{2}}} - \frac{1}{4 \sqrt{c} d x^{5} \sqrt{1 + \frac{d}{c x^{2}}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**6,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x, algorithm="giac")
[Out]