3.1728 \(\int \frac{(A+B x) \sqrt{d+e x}}{a+b x} \, dx\)

Optimal. Leaf size=98 \[ -\frac{2 (A b-a B) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 \sqrt{d+e x} (A b-a B)}{b^2}+\frac{2 B (d+e x)^{3/2}}{3 b e} \]

[Out]

(2*(A*b - a*B)*Sqrt[d + e*x])/b^2 + (2*B*(d + e*x)^(3/2))/(3*b*e) - (2*(A*b - a*
B)*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(5/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.163181, antiderivative size = 98, 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 \[ -\frac{2 (A b-a B) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 \sqrt{d+e x} (A b-a B)}{b^2}+\frac{2 B (d+e x)^{3/2}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*Sqrt[d + e*x])/b^2 + (2*B*(d + e*x)^(3/2))/(3*b*e) - (2*(A*b - a*
B)*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(5/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.1014, size = 85, normalized size = 0.87 \[ \frac{2 B \left (d + e x\right )^{\frac{3}{2}}}{3 b e} + \frac{2 \sqrt{d + e x} \left (A b - B a\right )}{b^{2}} - \frac{2 \left (A b - B a\right ) \sqrt{a e - b d} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(d + e*x)**(3/2)/(3*b*e) + 2*sqrt(d + e*x)*(A*b - B*a)/b**2 - 2*(A*b - B*a)*
sqrt(a*e - b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d))/b**(5/2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.217269, size = 94, normalized size = 0.96 \[ \frac{2 (a B-A b) \sqrt{b d-a e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{5/2}}+\frac{2 \sqrt{d+e x} (-3 a B e+3 A b e+b B (d+e x))}{3 b^2 e} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(3*A*b*e - 3*a*B*e + b*B*(d + e*x)))/(3*b^2*e) + (2*(-(A*b) + a
*B)*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(5/2)

_______________________________________________________________________________________

Maple [B]  time = 0.013, size = 211, normalized size = 2.2 \[{\frac{2\,B}{3\,be} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{A\sqrt{ex+d}}{b}}-2\,{\frac{Ba\sqrt{ex+d}}{{b}^{2}}}-2\,{\frac{Aae}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{Ad}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{B{a}^{2}e}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Bad}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*(e*x+d)^(3/2)/b/e+2/b*A*(e*x+d)^(1/2)-2/b^2*B*a*(e*x+d)^(1/2)-2*e/b/((a*e-
b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*a+2/((a*e-b*d)*b)^(1
/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*d+2*e/b^2/((a*e-b*d)*b)^(1/2)*
arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a^2-2/b/((a*e-b*d)*b)^(1/2)*arctan
((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*d

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.22322, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} e \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) - 2 \,{\left (B b e x + B b d - 3 \,{\left (B a - A b\right )} e\right )} \sqrt{e x + d}}{3 \, b^{2} e}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} e \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (B b e x + B b d - 3 \,{\left (B a - A b\right )} e\right )} \sqrt{e x + d}\right )}}{3 \, b^{2} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 15.7031, size = 212, normalized size = 2.16 \[ \frac{2 \left (\frac{B \left (d + e x\right )^{\frac{3}{2}}}{3 b} + \frac{e \left (- A b + B a\right ) \left (a e - b d\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{2}} + \frac{\sqrt{d + e x} \left (A b e - B a e\right )}{b^{2}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*(d + e*x)**(3/2)/(3*b) + e*(-A*b + B*a)*(a*e - b*d)*Piecewise((atan(sqrt(d
+ e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e - b*d)/b)), (a*e - b*d)/b > 0), (-acoth
(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b <
0) & (d + e*x > (-a*e + b*d)/b)), (-atanh(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b
*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x < (-a*e + b*d)/b)))/b**2
+ sqrt(d + e*x)*(A*b*e - B*a*e)/b**2)/e

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21087, size = 170, normalized size = 1.73 \[ -\frac{2 \,{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{2}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{2} e^{2} - 3 \, \sqrt{x e + d} B a b e^{3} + 3 \, \sqrt{x e + d} A b^{2} e^{3}\right )} e^{\left (-3\right )}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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