Optimal. Leaf size=113 \[ -\frac{3 d}{2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{1}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.223024, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 d}{2 \sqrt{c+d x^2} (b c-a d)^2}-\frac{1}{2 \left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 (b c-a d)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.2105, size = 97, normalized size = 0.86 \[ - \frac{3 \sqrt{b} d \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{2 \left (a d - b c\right )^{\frac{5}{2}}} - \frac{3 d}{2 \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}} + \frac{1}{2 \left (a + b x^{2}\right ) \sqrt{c + d x^{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.320424, size = 102, normalized size = 0.9 \[ \frac{1}{2} \left (\frac{-2 a d-b \left (c+3 d x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2} (b c-a d)^2}+\frac{3 \sqrt{b} d \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{5/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^2)^2*(c + d*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.019, size = 989, normalized size = 8.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a)^2/(d*x^2+c)^(3/2),x)
[Out]
_______________________________________________________________________________________
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*(d*x^2 + c)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.281814, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b d^{2} x^{4} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{b c - a d}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \,{\left (3 \, b d x^{2} + b c + 2 \, a d\right )} \sqrt{d x^{2} + c}}{8 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}, -\frac{3 \,{\left (b d^{2} x^{4} + a c d +{\left (b c d + a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{b c - a d}} \arctan \left (-\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c}{\left (b c - a d\right )} \sqrt{-\frac{b}{b c - a d}}}\right ) + 2 \,{\left (3 \, b d x^{2} + b c + 2 \, a d\right )} \sqrt{d x^{2} + c}}{4 \,{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.222084, size = 203, normalized size = 1.8 \[ -\frac{1}{2} \, d{\left (\frac{3 \, b \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-b^{2} c + a b d}} + \frac{3 \,{\left (d x^{2} + c\right )} b - 2 \, b c + 2 \, a d}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left ({\left (d x^{2} + c\right )}^{\frac{3}{2}} b - \sqrt{d x^{2} + c} b c + \sqrt{d x^{2} + c} a d\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]