Optimal. Leaf size=79 \[ \frac{b x}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.11854, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{b x}{a \sqrt{a+b x^2} (b c-a d)}-\frac{d \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(3/2)*(c + d*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.8374, size = 66, normalized size = 0.84 \[ \frac{d \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{\sqrt{c} \left (a d - b c\right )^{\frac{3}{2}}} - \frac{b x}{a \sqrt{a + b x^{2}} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.159129, size = 79, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{c} (a d-b c)^{3/2}}-\frac{b x}{a \sqrt{a+b x^2} (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^2)^(3/2)*(c + d*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.025, size = 618, normalized size = 7.8 \[{\frac{d}{2\,ad-2\,bc}{\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{bx}{ \left ( 2\,ad-2\,bc \right ) a}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{d}{2\,ad-2\,bc}\ln \left ({1 \left ( 2\,{\frac{ad-bc}{d}}+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b+2\,{\frac{b\sqrt{-cd}}{d} \left ( x-{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x-{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}}-{\frac{d}{2\,ad-2\,bc}{\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}-{\frac{bx}{ \left ( 2\,ad-2\,bc \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}}}}}+{\frac{d}{2\,ad-2\,bc}\ln \left ({1 \left ( 2\,{\frac{ad-bc}{d}}-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+2\,\sqrt{{\frac{ad-bc}{d}}}\sqrt{ \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) ^{2}b-2\,{\frac{b\sqrt{-cd}}{d} \left ( x+{\frac{\sqrt{-cd}}{d}} \right ) }+{\frac{ad-bc}{d}}} \right ) \left ( x+{\frac{1}{d}\sqrt{-cd}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}{\frac{1}{\sqrt{{\frac{ad-bc}{d}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(3/2)/(d*x^2+c),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(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.306097, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{b c^{2} - a c d} \sqrt{b x^{2} + a} b x -{\left (a b d x^{2} + a^{2} d\right )} \log \left (\frac{{\left ({\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \,{\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d} + 4 \,{\left ({\left (2 \, b^{2} c^{3} - 3 \, a b c^{2} d + a^{2} c d^{2}\right )} x^{3} +{\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt{b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \,{\left (a^{2} b c - a^{3} d +{\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt{b c^{2} - a c d}}, \frac{2 \, \sqrt{-b c^{2} + a c d} \sqrt{b x^{2} + a} b x -{\left (a b d x^{2} + a^{2} d\right )} \arctan \left (\frac{\sqrt{-b c^{2} + a c d}{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )}}{2 \,{\left (b c^{2} - a c d\right )} \sqrt{b x^{2} + a} x}\right )}{2 \,{\left (a^{2} b c - a^{3} d +{\left (a b^{2} c - a^{2} b d\right )} x^{2}\right )} \sqrt{-b c^{2} + a c d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (c + d x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.231117, size = 143, normalized size = 1.81 \[ \frac{\sqrt{b} d \arctan \left (\frac{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt{-b^{2} c^{2} + a b c d}}\right )}{\sqrt{-b^{2} c^{2} + a b c d}{\left (b c - a d\right )}} + \frac{b x}{{\left (a b c - a^{2} d\right )} \sqrt{b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/2)*(d*x^2 + c)),x, algorithm="giac")
[Out]