Optimal. Leaf size=103 \[ -\frac{(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (A b-a B)}{b (a+b x) (b d-a e)} \]
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Rubi [A] time = 0.239143, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (A b-a B)}{b (a+b x) (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 37.3715, size = 85, normalized size = 0.83 \[ \frac{\sqrt{d + e x} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} + \frac{\left (A b e + B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{3}{2}} \left (a e - b d\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.143985, size = 102, normalized size = 0.99 \[ \frac{\sqrt{d+e x} (a B-A b)}{b (a+b x) (b d-a e)}-\frac{(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)),x]
[Out]
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Maple [B] time = 0.026, size = 195, normalized size = 1.9 \[{\frac{ \left ( Ab-Ba \right ) e}{b \left ( ae-bd \right ) \left ( b \left ( ex+d \right ) +ae-bd \right ) }\sqrt{ex+d}}+{\frac{Ae}{ae-bd}\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}+{\frac{aBe}{b \left ( ae-bd \right ) }\arctan \left ({b\sqrt{ex+d}{\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}}}-2\,{\frac{Bd}{ \left ( ae-bd \right ) \sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(1/2),x)
[Out]
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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^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.297354, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} d - a b e}{\left (B a - A b\right )} \sqrt{e x + d} +{\left (2 \, B a b d -{\left (B a^{2} + A a b\right )} e +{\left (2 \, B b^{2} d -{\left (B a b + A b^{2}\right )} e\right )} x\right )} \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right )}{2 \,{\left (a b^{2} d - a^{2} b e +{\left (b^{3} d - a b^{2} e\right )} x\right )} \sqrt{b^{2} d - a b e}}, \frac{\sqrt{-b^{2} d + a b e}{\left (B a - A b\right )} \sqrt{e x + d} -{\left (2 \, B a b d -{\left (B a^{2} + A a b\right )} e +{\left (2 \, B b^{2} d -{\left (B a b + A b^{2}\right )} e\right )} x\right )} \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right )}{{\left (a b^{2} d - a^{2} b e +{\left (b^{3} d - a b^{2} e\right )} x\right )} \sqrt{-b^{2} d + a b e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right )^{2} \sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.281885, size = 182, normalized size = 1.77 \[ \frac{{\left (2 \, B b d - B a e - A b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{2} d - a b e\right )} \sqrt{-b^{2} d + a b e}} + \frac{\sqrt{x e + d} B a e - \sqrt{x e + d} A b e}{{\left (b^{2} d - a b e\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(e*x + d)),x, algorithm="giac")
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