Optimal. Leaf size=178 \[ -\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}+\frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.313232, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}+\frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(x^3*(a + b*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 27.1059, size = 165, normalized size = 0.93 \[ - \frac{2 b \left (c + d x\right )^{\frac{5}{2}}}{a x^{2} \sqrt{a + b x} \left (a d - b c\right )} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (a d - 5 b c\right )}{2 a^{2} x^{2} \left (a d - b c\right )} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (a d - 5 b c\right )}{4 a^{3} x} - \frac{3 \left (a d - 5 b c\right ) \left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{7}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/x**3/(b*x+a)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.220275, size = 165, normalized size = 0.93 \[ \frac{\frac{2 \sqrt{a} \sqrt{c+d x} \left (a^2 (-(2 c+5 d x))+a b x (5 c-13 d x)+15 b^2 c x^2\right )}{x^2 \sqrt{a+b x}}+\frac{3 \log (x) (a d-5 b c) (a d-b c)}{\sqrt{c}}-\frac{3 (a d-5 b c) (a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{c}}}{8 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(x^3*(a + b*x)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.04, size = 464, normalized size = 2.6 \[ -{\frac{1}{8\,{a}^{3}{x}^{2}}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{a}^{2}b{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}cd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{2}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{3}{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}+26\,{x}^{2}abd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-30\,{x}^{2}{b}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+10\,x{a}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xabc\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,{a}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/x^3/(b*x+a)^(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((d*x + c)^(3/2)/((b*x + a)^(3/2)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.451908, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (2 \, a^{2} c -{\left (15 \, b^{2} c - 13 \, a b d\right )} x^{2} - 5 \,{\left (a b c - a^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right )}{16 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )} \sqrt{a c}}, -\frac{2 \,{\left (2 \, a^{2} c -{\left (15 \, b^{2} c - 13 \, a b d\right )} x^{2} - 5 \,{\left (a b c - a^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right )}{8 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )} \sqrt{-a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/((b*x + a)^(3/2)*x^3),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)**(3/2)/x**3/(b*x+a)**(3/2),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((d*x + c)^(3/2)/((b*x + a)^(3/2)*x^3),x, algorithm="giac")
[Out]