3.1730 \(\int \frac{A+B x}{(a+b x) (d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{2 (B d-A e)}{e \sqrt{d+e x} (b d-a e)}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.171444, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 (B d-A e)}{e \sqrt{d+e x} (b d-a e)}-\frac{2 (A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} (b d-a e)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)*(d + e*x)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.4125, size = 76, normalized size = 0.86 \[ - \frac{2 \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} - \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\sqrt{b} \left (a e - b d\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)/(e*x+d)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.345443, size = 88, normalized size = 1. \[ \frac{2 \left (\frac{(A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{\sqrt{b} \sqrt{b d-a e}}+\frac{B d-A e}{e \sqrt{d+e x}}\right )}{a e-b d} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)*(d + e*x)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 142, normalized size = 1.6 \[ -2\,{\frac{Ab}{ \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{Ba}{ \left ( ae-bd \right ) \sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{A}{ \left ( ae-bd \right ) \sqrt{ex+d}}}+2\,{\frac{Bd}{e \left ( ae-bd \right ) \sqrt{ex+d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)/(e*x+d)^(3/2),x)

[Out]

-2/(a*e-b*d)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*A*b
+2/(a*e-b*d)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a
-2/(a*e-b*d)/(e*x+d)^(1/2)*A+2/e/(a*e-b*d)/(e*x+d)^(1/2)*B*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)/((b*x + a)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226327, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B a - A b\right )} \sqrt{e x + d} e \log \left (\frac{\sqrt{b^{2} d - a b e}{\left (b e x + 2 \, b d - a e\right )} + 2 \,{\left (b^{2} d - a b e\right )} \sqrt{e x + d}}{b x + a}\right ) - 2 \, \sqrt{b^{2} d - a b e}{\left (B d - A e\right )}}{\sqrt{b^{2} d - a b e}{\left (b d e - a e^{2}\right )} \sqrt{e x + d}}, \frac{2 \,{\left ({\left (B a - A b\right )} \sqrt{e x + d} e \arctan \left (-\frac{b d - a e}{\sqrt{-b^{2} d + a b e} \sqrt{e x + d}}\right ) - \sqrt{-b^{2} d + a b e}{\left (B d - A e\right )}\right )}}{\sqrt{-b^{2} d + a b e}{\left (b d e - a e^{2}\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)/(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)/((a + b*x)*(d + e*x)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220307, size = 126, normalized size = 1.43 \[ -\frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e}{\left (b d - a e\right )}} - \frac{2 \,{\left (B d - A e\right )}}{{\left (b d e - a e^{2}\right )} \sqrt{x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

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