3.1812 \(\int \frac{A+B x}{\sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=103 \[ -\frac{(-a B e-A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} (b d-a e)^{3/2}}-\frac{\sqrt{d+e x} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

-(((A*b - a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x))) - ((2*b*B*d - A*b*e - a
*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*(b*d - a*e)^(3/
2))

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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]

-(((A*b - a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x))) - ((2*b*B*d - A*b*e - a
*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*(b*d - a*e)^(3/
2))

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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]

sqrt(d + e*x)*(A*b - B*a)/(b*(a + b*x)*(a*e - b*d)) + (A*b*e + B*a*e - 2*B*b*d)*
atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/(b**(3/2)*(a*e - b*d)**(3/2))

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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]

((-(A*b) + a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x)) - ((2*b*B*d - A*b*e - a
*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*(b*d - a*e)^(3/
2))

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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]

(A*b-B*a)*e/b/(a*e-b*d)*(e*x+d)^(1/2)/(b*(e*x+d)+a*e-b*d)+1/(a*e-b*d)/(b*(a*e-b*
d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A*e+1/b/(a*e-b*d)/(b*(a*e-
b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B*e-2/(a*e-b*d)/(b*(a*
e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d

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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")

[Out]

Exception raised: ValueError

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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]

[1/2*(2*sqrt(b^2*d - a*b*e)*(B*a - A*b)*sqrt(e*x + d) + (2*B*a*b*d - (B*a^2 + A*
a*b)*e + (2*B*b^2*d - (B*a*b + A*b^2)*e)*x)*log((sqrt(b^2*d - a*b*e)*(b*e*x + 2*
b*d - a*e) - 2*(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)))/((a*b^2*d - a^2*b*e +
(b^3*d - a*b^2*e)*x)*sqrt(b^2*d - a*b*e)), (sqrt(-b^2*d + a*b*e)*(B*a - A*b)*sqr
t(e*x + d) - (2*B*a*b*d - (B*a^2 + A*a*b)*e + (2*B*b^2*d - (B*a*b + A*b^2)*e)*x)
*arctan(-(b*d - a*e)/(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d))))/((a*b^2*d - a^2*b*e
+ (b^3*d - a*b^2*e)*x)*sqrt(-b^2*d + a*b*e))]

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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]

Integral((A + B*x)/((a + b*x)**2*sqrt(d + e*x)), x)

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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")

[Out]

(2*B*b*d - B*a*e - A*b*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^2*d -
a*b*e)*sqrt(-b^2*d + a*b*e)) + (sqrt(x*e + d)*B*a*e - sqrt(x*e + d)*A*b*e)/((b^
2*d - a*b*e)*((x*e + d)*b - b*d + a*e))