Optimal. Leaf size=121 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.30431, antiderivative size = 121, 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 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 34.4658, size = 107, normalized size = 0.88 \[ - \frac{2 b}{a \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} + \frac{2 d \sqrt{a + b x} \left (a d + b c\right )}{a c \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.209966, size = 125, normalized size = 1.03 \[ \frac{\frac{2 \sqrt{a} \sqrt{c} \left (a^2 d^2+a b d^2 x+b^2 c (c+d x)\right )}{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )+\log (x)}{a^{3/2} c^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*x)^(3/2)*(c + d*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.05, size = 638, normalized size = 5.3 \[{\frac{1}{ac \left ( ad-bc \right ) ^{2}} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{a}^{2}b{d}^{3}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}c{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{3}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{3}{d}^{3}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{2}bc{d}^{2}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xa{b}^{2}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{3}{c}^{3}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){a}^{3}c{d}^{2}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}b{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) a{b}^{2}{c}^{3}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xab{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}x{b}^{2}cd+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{a}^{2}{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b*x+a)^(3/2)/(d*x+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(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.356424, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} +{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{2 \,{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \sqrt{a c}}, \frac{2 \,{\left (b^{2} c^{2} + a^{2} d^{2} +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} -{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{{\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} +{\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{2} +{\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x\right )} \sqrt{-a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.986374, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*(d*x + c)^(3/2)*x),x, algorithm="giac")
[Out]