Optimal. Leaf size=171 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (A b e-4 A c d+2 b B d)}{b^3 \sqrt{d}}+\frac{\sqrt{d+e x} (b B-2 A c)}{b^2 (b+c x)}+\frac{\left (3 A b c e-4 A c^2 d+b^2 (-B) e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c} \sqrt{c d-b e}}-\frac{A \sqrt{d+e x}}{b x (b+c x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.713746, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (A b e-4 A c d+2 b B d)}{b^3 \sqrt{d}}+\frac{\sqrt{d+e x} (b B-2 A c)}{b^2 (b+c x)}-\frac{\left (-b c (3 A e+2 B d)+4 A c^2 d+b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c} \sqrt{c d-b e}}-\frac{A \sqrt{d+e x}}{b x (b+c x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 93.8459, size = 167, normalized size = 0.98 \[ \frac{\sqrt{d + e x} \left (A c - B b\right )}{b c x \left (b + c x\right )} - \frac{\sqrt{d + e x} \left (2 A c - B b\right )}{b^{2} c x} - \frac{\left (A b e - 4 A c d + 2 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3} \sqrt{d}} + \frac{\left (- 3 A b c e + 4 A c^{2} d + B b^{2} e - 2 B b c d\right ) \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \sqrt{c} \sqrt{b e - c d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.302861, size = 152, normalized size = 0.89 \[ \frac{-\frac{\left (-b c (3 A e+2 B d)+4 A c^2 d+b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{c d-b e}}+\frac{b \sqrt{d+e x} (b B x-A (b+2 c x))}{x (b+c x)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (A b e-4 A c d+2 b B d)}{\sqrt{d}}}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.03, size = 299, normalized size = 1.8 \[ -{\frac{Ace}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{Be}{b \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{Ace}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{A{c}^{2}d}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{Be}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-2\,{\frac{Bdc}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{A}{{b}^{2}x}\sqrt{ex+d}}-{\frac{Ae}{{b}^{2}}{\it Artanh} \left ({1\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{d}}}}+4\,{\frac{\sqrt{d}Ac}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{\sqrt{d}B}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^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)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.702988, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^2,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((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.28611, size = 316, normalized size = 1.85 \[ -\frac{{\left (2 \, B b c d - 4 \, A c^{2} d - B b^{2} e + 3 \, A b c e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3}} + \frac{{\left (2 \, B b d - 4 \, A c d + A b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e - \sqrt{x e + d} B b d e + 2 \, \sqrt{x e + d} A c d e - \sqrt{x e + d} A b e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]