Optimal. Leaf size=193 \[ -\frac{(b d-a e)^2 (a B e-6 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{3/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (a B e-6 A b e+5 b B d)}{8 b e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (a B e-6 A b e+5 b B d)}{12 b e^2}+\frac{B (a+b x)^{5/2} \sqrt{d+e x}}{3 b e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.385366, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b d-a e)^2 (a B e-6 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{3/2} e^{7/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (a B e-6 A b e+5 b B d)}{8 b e^3}-\frac{(a+b x)^{3/2} \sqrt{d+e x} (a B e-6 A b e+5 b B d)}{12 b e^2}+\frac{B (a+b x)^{5/2} \sqrt{d+e x}}{3 b e} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.2749, size = 182, normalized size = 0.94 \[ \frac{B \left (a + b x\right )^{\frac{5}{2}} \sqrt{d + e x}}{3 b e} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (6 A b e - B a e - 5 B b d\right )}{12 b e^{2}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (6 A b e - B a e - 5 B b d\right )}{8 b e^{3}} + \frac{\left (a e - b d\right )^{2} \left (6 A b e - B a e - 5 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{8 b^{\frac{3}{2}} e^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.232473, size = 178, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (3 a^2 B e^2+2 a b e (15 A e-11 B d+7 B e x)+b^2 \left (6 A e (2 e x-3 d)+B \left (15 d^2-10 d e x+8 e^2 x^2\right )\right )\right )}{24 b e^3}-\frac{(b d-a e)^2 (a B e-6 A b e+5 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{3/2} e^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/Sqrt[d + e*x],x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.033, size = 636, normalized size = 3.3 \[{\frac{1}{48\,{e}^{3}b}\sqrt{bx+a}\sqrt{ex+d} \left ( 16\,B{x}^{2}{b}^{2}{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+18\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}A{e}^{3}b-36\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aA{b}^{2}d{e}^{2}+18\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{2}Ae+24\,A\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }x{b}^{2}{e}^{2}\sqrt{be}-3\,B{e}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}-9\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}Bd{e}^{2}b+27\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aB{b}^{2}{d}^{2}e-15\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{3}B+28\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xab{e}^{2}\sqrt{be}-20\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xd{b}^{2}e\sqrt{be}+60\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Aa{e}^{2}\sqrt{be}b-36\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }A{b}^{2}de\sqrt{be}+6\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{2}{e}^{2}\sqrt{be}-44\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Bade\sqrt{be}b+30\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }B{b}^{2}{d}^{2}\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(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((B*x + A)*(b*x + a)^(3/2)/sqrt(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.530251, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, B b^{2} e^{2} x^{2} + 15 \, B b^{2} d^{2} - 2 \,{\left (11 \, B a b + 9 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} + 10 \, A a b\right )} e^{2} - 2 \,{\left (5 \, B b^{2} d e -{\left (7 \, B a b + 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} - 3 \,{\left (5 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e + 3 \,{\left (B a^{2} b + 4 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{96 \, \sqrt{b e} b e^{3}}, \frac{2 \,{\left (8 \, B b^{2} e^{2} x^{2} + 15 \, B b^{2} d^{2} - 2 \,{\left (11 \, B a b + 9 \, A b^{2}\right )} d e + 3 \,{\left (B a^{2} + 10 \, A a b\right )} e^{2} - 2 \,{\left (5 \, B b^{2} d e -{\left (7 \, B a b + 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} - 3 \,{\left (5 \, B b^{3} d^{3} - 3 \,{\left (3 \, B a b^{2} + 2 \, A b^{3}\right )} d^{2} e + 3 \,{\left (B a^{2} b + 4 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{48 \, \sqrt{-b e} b e^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/sqrt(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{\frac{3}{2}}}{\sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.236843, size = 362, normalized size = 1.88 \[ \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} B e^{\left (-1\right )}}{b^{2}} - \frac{{\left (5 \, B b^{3} d e^{3} + B a b^{2} e^{4} - 6 \, A b^{3} e^{4}\right )} e^{\left (-5\right )}}{b^{4}}\right )} + \frac{3 \,{\left (5 \, B b^{4} d^{2} e^{2} - 4 \, B a b^{3} d e^{3} - 6 \, A b^{4} d e^{3} - B a^{2} b^{2} e^{4} + 6 \, A a b^{3} e^{4}\right )} e^{\left (-5\right )}}{b^{4}}\right )} + \frac{3 \,{\left (5 \, B b^{3} d^{3} - 9 \, B a b^{2} d^{2} e - 6 \, A b^{3} d^{2} e + 3 \, B a^{2} b d e^{2} + 12 \, A a b^{2} d e^{2} + B a^{3} e^{3} - 6 \, A a^{2} b e^{3}\right )} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{3}{2}}}\right )} b}{24 \,{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/sqrt(e*x + d),x, algorithm="giac")
[Out]