Optimal. Leaf size=140 \[ -\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}}+\frac{\sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.264834, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}}+\frac{\sqrt{d+e x} (-3 a B e+A b e+2 b B d)}{b^2 (b d-a e)}-\frac{(d+e x)^{3/2} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.7772, size = 124, normalized size = 0.89 \[ \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\sqrt{d + e x} \left (A b e - 3 B a e + 2 B b d\right )}{b^{2} \left (a e - b d\right )} + \frac{\left (A b e - 3 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{5}{2}} \sqrt{a e - b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.127544, size = 99, normalized size = 0.71 \[ \sqrt{d+e x} \left (\frac{a B-A b}{b^2 (a+b x)}+\frac{2 B}{b^2}\right )-\frac{(-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2} \sqrt{b d-a e}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/(a + b*x)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.022, size = 186, normalized size = 1.3 \[ 2\,{\frac{B\sqrt{ex+d}}{{b}^{2}}}-{\frac{Ae}{b \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{Bae}{{b}^{2} \left ( bxe+ae \right ) }\sqrt{ex+d}}+{\frac{Ae}{b}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( ae-bd \right ) b}}}}-3\,{\frac{Bae}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{Bd}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^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)*sqrt(e*x + d)/(b*x + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.225052, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} d - a b e}{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{e x + d} +{\left (2 \, B a b d -{\left (3 \, B a^{2} - A a b\right )} e +{\left (2 \, B b^{2} d -{\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{2 \,{\left (b^{3} x + a b^{2}\right )} \sqrt{b^{2} d - a b e}}, \frac{\sqrt{-b^{2} d + a b e}{\left (2 \, B b x + 3 \, B a - A b\right )} \sqrt{e x + d} -{\left (2 \, B a b d -{\left (3 \, B a^{2} - A a b\right )} e +{\left (2 \, B b^{2} d -{\left (3 \, B a b - A b^{2}\right )} e\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{{\left (b^{3} x + a b^{2}\right )} \sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(b*x + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 39.9498, size = 1559, normalized size = 11.14 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.218078, size = 170, normalized size = 1.21 \[ \frac{2 \, \sqrt{x e + d} B}{b^{2}} + \frac{{\left (2 \, B b d - 3 \, B a e + A b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{\sqrt{x e + d} B a e - \sqrt{x e + d} A b e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(b*x + a)^2,x, algorithm="giac")
[Out]