Optimal. Leaf size=87 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.171883, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 \sqrt{b} (b c-a d)^{3/2}}-\frac{\sqrt{c+d x^2}}{2 \left (a+b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
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Rubi in Sympy [A] time = 21.7303, size = 70, normalized size = 0.8 \[ \frac{\sqrt{c + d x^{2}}}{2 \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 \sqrt{b} \left (a d - b c\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.148276, size = 84, normalized size = 0.97 \[ \frac{\frac{\sqrt{c+d x^2}}{a+b x^2}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{\sqrt{b} \sqrt{b c-a d}}}{2 a d-2 b c} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^2)^2*Sqrt[c + d*x^2]),x]
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Maple [B] time = 0.017, size = 513, normalized size = 5.9 \[ -{\frac{1}{4\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{4\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d+2\,{\frac{d\sqrt{-ab}}{b} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x-{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}}+{\frac{1}{4\,ab \left ( ad-bc \right ) }\sqrt{-ab}\sqrt{ \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}}-{\frac{d}{4\, \left ( ad-bc \right ) b}\ln \left ({1 \left ( -2\,{\frac{ad-bc}{b}}-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }+2\,\sqrt{-{\frac{ad-bc}{b}}}\sqrt{ \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) ^{2}d-2\,{\frac{d\sqrt{-ab}}{b} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }-{\frac{ad-bc}{b}}} \right ) \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{ad-bc}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)^2/(d*x^2+c)^(1/2),x)
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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/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="maxima")
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Fricas [A] time = 0.257694, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b d x^{2} + a d\right )} \log \left (\frac{{\left (b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b^{2} c - a b d} - 4 \,{\left (2 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2} +{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, \sqrt{b^{2} c - a b d} \sqrt{d x^{2} + c}}{8 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{b^{2} c - a b d}}, -\frac{{\left (b d x^{2} + a d\right )} \arctan \left (-\frac{{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt{-b^{2} c + a b d}}{2 \,{\left (b^{2} c - a b d\right )} \sqrt{d x^{2} + c}}\right ) + 2 \, \sqrt{-b^{2} c + a b d} \sqrt{d x^{2} + c}}{4 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-b^{2} c + a b d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a)**2/(d*x**2+c)**(1/2),x)
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GIAC/XCAS [A] time = 0.238996, size = 124, normalized size = 1.43 \[ -\frac{1}{2} \, d{\left (\frac{\arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d}{\left (b c - a d\right )}} + \frac{\sqrt{d x^{2} + c}}{{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )}{\left (b c - a d\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*sqrt(d*x^2 + c)),x, algorithm="giac")
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