Optimal. Leaf size=103 \[ \frac{-\frac{3 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{2 c \sqrt{c+d x^2}}-\frac{a^2}{2 c x^2 \sqrt{c+d x^2}}-\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{5/2}} \]
[Out]
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Rubi [A] time = 0.283707, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{-\frac{3 a^2 d}{c}+4 a b-\frac{2 b^2 c}{d}}{2 c \sqrt{c+d x^2}}-\frac{a^2}{2 c x^2 \sqrt{c+d x^2}}-\frac{a (4 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 24.6581, size = 92, normalized size = 0.89 \[ - \frac{a^{2}}{2 c x^{2} \sqrt{c + d x^{2}}} + \frac{a \left (3 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 c^{\frac{5}{2}}} - \frac{\frac{a d \left (3 a d - 4 b c\right )}{2} + b^{2} c^{2}}{c^{2} d \sqrt{c + d x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(3/2),x)
[Out]
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Mathematica [A] time = 0.250312, size = 105, normalized size = 1.02 \[ \frac{-\sqrt{c} \sqrt{c+d x^2} \left (\frac{a^2}{x^2}+\frac{2 (b c-a d)^2}{d \left (c+d x^2\right )}\right )+a (3 a d-4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )-a \log (x) (3 a d-4 b c)}{2 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^2/(x^3*(c + d*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.016, size = 135, normalized size = 1.3 \[ -{\frac{{b}^{2}}{d}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{a}^{2}d}{2\,{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{a}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{ab}{c\sqrt{d{x}^{2}+c}}}-2\,{\frac{ab}{{c}^{3/2}}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2/x^3/(d*x^2+c)^(3/2),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((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243675, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (a^{2} c d +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} +{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} +{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{4 \,{\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )} \sqrt{c}}, -\frac{{\left (a^{2} c d +{\left (2 \, b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} +{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} +{\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{2 \,{\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )} \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2/x**3/(d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.258964, size = 189, normalized size = 1.83 \[ \frac{{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c} c^{2}} - \frac{2 \,{\left (d x^{2} + c\right )} b^{2} c^{2} - 2 \, b^{2} c^{3} - 4 \,{\left (d x^{2} + c\right )} a b c d + 4 \, a b c^{2} d + 3 \,{\left (d x^{2} + c\right )} a^{2} d^{2} - 2 \, a^{2} c d^{2}}{2 \,{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} - \sqrt{d x^{2} + c} c\right )} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^3),x, algorithm="giac")
[Out]