3.1862 \(\int \frac{A+B x}{\sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=124 \[ \frac{2 B (a+b x) \sqrt{d+e x}}{b e \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]

[Out]

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

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Rubi [A]  time = 0.272011, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.114 \[ \frac{2 B (a+b x) \sqrt{d+e x}}{b e \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.137069, size = 106, normalized size = 0.85 \[ \frac{2 B (a+b x) \sqrt{d+e x}}{b e \sqrt{(a+b x)^2}}-\frac{2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{3/2} \sqrt{(a+b x)^2} \sqrt{b d-a e}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.015, size = 110, normalized size = 0.9 \[ 2\,{\frac{bx+a}{\sqrt{ \left ( bx+a \right ) ^{2}}eb\sqrt{b \left ( ae-bd \right ) }} \left ( A\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) be-B\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) ae+B\sqrt{ex+d}\sqrt{b \left ( ae-bd \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2),x)

[Out]

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

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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)/(sqrt((b*x + a)^2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286365, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a - A b\right )} e \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} - 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) - 2 \, \sqrt{b^{2} d - a b e} \sqrt{e x + d} B}{\sqrt{b^{2} d - a b e} b e}, \frac{2 \,{\left ({\left (B a - A b\right )} e \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) + \sqrt{-b^{2} d + a b e} \sqrt{e x + d} B\right )}}{\sqrt{-b^{2} d + a b e} b e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{d + e x} \sqrt{\left (a + b x\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(e*x+d)**(1/2)/((b*x+a)**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.287185, size = 117, normalized size = 0.94 \[ \frac{2 \, \sqrt{x e + d} B e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{b} - \frac{2 \,{\left (B a{\rm sign}\left (b x + a\right ) - A b{\rm sign}\left (b x + a\right )\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt((b*x + a)^2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

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