Optimal. Leaf size=191 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{12 a^2 c x^2}+\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.507498, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-3 a d) (a d+3 b c)}{24 a^3 c^2 x}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d)}{12 a^2 c x^2}+\frac{(b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{7/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 56.4023, size = 175, normalized size = 0.92 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{3 a x^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - 5 b c\right )}{12 a^{2} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d + 3 b c\right ) \left (3 a d - 5 b c\right )}{24 a^{3} c^{2} x} - \frac{\left (a d - b c\right ) \left (a^{2} d^{2} + 2 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{7}{2}} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/x**4/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.188111, size = 213, normalized size = 1.12 \[ \frac{-3 x^3 \log (x) (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right )+3 x^3 (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+2 c d x-3 d^2 x^2\right )-2 a b c x (5 c+2 d x)+15 b^2 c^2 x^2\right )}{48 a^{7/2} c^{5/2} x^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(x^4*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.036, size = 408, normalized size = 2.1 \[ -{\frac{1}{48\,{a}^{3}{c}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}{a}^{2}{x}^{2}\sqrt{ac}-8\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dbca{x}^{2}\sqrt{ac}+30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{2}{x}^{2}\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dc{a}^{2}x\sqrt{ac}-20\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{2}ax\sqrt{ac}+16\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{2}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/x^4/(b*x+a)^(1/2),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(sqrt(d*x + c)/(sqrt(b*x + a)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.435119, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 4 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{96 \, \sqrt{a c} a^{3} c^{2} x^{3}}, \frac{3 \,{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (8 \, a^{2} c^{2} +{\left (15 \, b^{2} c^{2} - 4 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} - 2 \,{\left (5 \, a b c^{2} - a^{2} c d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{48 \, \sqrt{-a c} a^{3} c^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(1/2)/x**4/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*x^4),x, algorithm="giac")
[Out]