3.1739 \(\int \frac{A+B x}{(a+b x)^2 (d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac{3 a B e-5 A b e+2 b B d}{\sqrt{d+e x} (b d-a e)^3}+\frac{3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2}-\frac{\sqrt{b} (3 a B e-5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}} \]

[Out]

(2*b*B*d - 5*A*b*e + 3*a*B*e)/(3*b*(b*d - a*e)^2*(d + e*x)^(3/2)) - (A*b - a*B)/
(b*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2)) + (2*b*B*d - 5*A*b*e + 3*a*B*e)/((b*d
- a*e)^3*Sqrt[d + e*x]) - (Sqrt[b]*(2*b*B*d - 5*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.385207, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac{3 a B e-5 A b e+2 b B d}{\sqrt{d+e x} (b d-a e)^3}+\frac{3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2}-\frac{\sqrt{b} (3 a B e-5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((a + b*x)^2*(d + e*x)^(5/2)),x]

[Out]

(2*b*B*d - 5*A*b*e + 3*a*B*e)/(3*b*(b*d - a*e)^2*(d + e*x)^(3/2)) - (A*b - a*B)/
(b*(b*d - a*e)*(a + b*x)*(d + e*x)^(3/2)) + (2*b*B*d - 5*A*b*e + 3*a*B*e)/((b*d
- a*e)^3*Sqrt[d + e*x]) - (Sqrt[b]*(2*b*B*d - 5*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b
]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(7/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.7624, size = 167, normalized size = 0.92 \[ \frac{\sqrt{b} \left (5 A b e - 3 B a e - 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{\left (a e - b d\right )^{\frac{7}{2}}} + \frac{5 A b e - 3 B a e - 2 B b d}{\sqrt{d + e x} \left (a e - b d\right )^{3}} - \frac{5 A b e - 3 B a e - 2 B b d}{3 b \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{b \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**2/(e*x+d)**(5/2),x)

[Out]

sqrt(b)*(5*A*b*e - 3*B*a*e - 2*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt(a*e - b*d)
)/(a*e - b*d)**(7/2) + (5*A*b*e - 3*B*a*e - 2*B*b*d)/(sqrt(d + e*x)*(a*e - b*d)*
*3) - (5*A*b*e - 3*B*a*e - 2*B*b*d)/(3*b*(d + e*x)**(3/2)*(a*e - b*d)**2) + (A*b
- B*a)/(b*(a + b*x)*(d + e*x)**(3/2)*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.874896, size = 154, normalized size = 0.85 \[ \frac{\sqrt{d+e x} \left (\frac{6 (a B e-2 A b e+b B d)}{d+e x}+\frac{2 (b d-a e) (B d-A e)}{(d+e x)^2}+\frac{3 b (a B-A b)}{a+b x}\right )}{3 (b d-a e)^3}-\frac{\sqrt{b} (3 a B e-5 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((a + b*x)^2*(d + e*x)^(5/2)),x]

[Out]

(Sqrt[d + e*x]*((3*b*(-(A*b) + a*B))/(a + b*x) + (2*(b*d - a*e)*(B*d - A*e))/(d
+ e*x)^2 + (6*(b*B*d - 2*A*b*e + a*B*e))/(d + e*x)))/(3*(b*d - a*e)^3) - (Sqrt[b
]*(2*b*B*d - 5*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]]
)/(b*d - a*e)^(7/2)

_______________________________________________________________________________________

Maple [B]  time = 0.03, size = 328, normalized size = 1.8 \[ -{\frac{2\,Ae}{3\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,Bd}{3\, \left ( ae-bd \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{Abe}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-2\,{\frac{Bae}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}-2\,{\frac{Bbd}{ \left ( ae-bd \right ) ^{3}\sqrt{ex+d}}}+{\frac{A{b}^{2}e}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}-{\frac{Babe}{ \left ( ae-bd \right ) ^{3} \left ( bxe+ae \right ) }\sqrt{ex+d}}+5\,{\frac{A{b}^{2}e}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-3\,{\frac{Babe}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{{b}^{2}Bd}{ \left ( ae-bd \right ) ^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^2/(e*x+d)^(5/2),x)

[Out]

-2/3/(a*e-b*d)^2/(e*x+d)^(3/2)*A*e+2/3/(a*e-b*d)^2/(e*x+d)^(3/2)*B*d+4/(a*e-b*d)
^3/(e*x+d)^(1/2)*A*b*e-2/(a*e-b*d)^3/(e*x+d)^(1/2)*B*a*e-2/(a*e-b*d)^3/(e*x+d)^(
1/2)*B*b*d+1/(a*e-b*d)^3*b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*A*e-1/(a*e-b*d)^3*b*(e*x+
d)^(1/2)/(b*e*x+a*e)*B*a*e+5/(a*e-b*d)^3*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^
(1/2)*b/((a*e-b*d)*b)^(1/2))*A*e-3/(a*e-b*d)^3*b/((a*e-b*d)*b)^(1/2)*arctan((e*x
+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*B*a*e-2/(a*e-b*d)^3*b^2/((a*e-b*d)*b)^(1/2)*arc
tan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(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)^2*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.229689, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^2*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/6*(4*A*a^2*e^2 + 2*(11*B*a*b - 3*A*b^2)*d^2 + 4*(2*B*a^2 - 7*A*a*b)*d*e + 6*(
2*B*b^2*d*e + (3*B*a*b - 5*A*b^2)*e^2)*x^2 + 3*(2*B*a*b*d^2 + (3*B*a^2 - 5*A*a*b
)*d*e + (2*B*b^2*d*e + (3*B*a*b - 5*A*b^2)*e^2)*x^2 + (2*B*b^2*d^2 + 5*(B*a*b -
A*b^2)*d*e + (3*B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(e*x + d)*sqrt(b/(b*d - a*e))*log((
b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)
) + 4*(4*B*b^2*d^2 + 2*(4*B*a*b - 5*A*b^2)*d*e + (3*B*a^2 - 5*A*a*b)*e^2)*x)/((a
*b^3*d^4 - 3*a^2*b^2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e^3 + (b^4*d^3*e - 3*a*b^3*
d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x^2 + (b^4*d^4 - 2*a*b^3*d^3*e + 2*a^3*b*
d*e^3 - a^4*e^4)*x)*sqrt(e*x + d)), 1/3*(2*A*a^2*e^2 + (11*B*a*b - 3*A*b^2)*d^2
+ 2*(2*B*a^2 - 7*A*a*b)*d*e + 3*(2*B*b^2*d*e + (3*B*a*b - 5*A*b^2)*e^2)*x^2 - 3*
(2*B*a*b*d^2 + (3*B*a^2 - 5*A*a*b)*d*e + (2*B*b^2*d*e + (3*B*a*b - 5*A*b^2)*e^2)
*x^2 + (2*B*b^2*d^2 + 5*(B*a*b - A*b^2)*d*e + (3*B*a^2 - 5*A*a*b)*e^2)*x)*sqrt(e
*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x
+ d)*b)) + 2*(4*B*b^2*d^2 + 2*(4*B*a*b - 5*A*b^2)*d*e + (3*B*a^2 - 5*A*a*b)*e^2)
*x)/((a*b^3*d^4 - 3*a^2*b^2*d^3*e + 3*a^3*b*d^2*e^2 - a^4*d*e^3 + (b^4*d^3*e - 3
*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 - a^3*b*e^4)*x^2 + (b^4*d^4 - 2*a*b^3*d^3*e + 2
*a^3*b*d*e^3 - a^4*e^4)*x)*sqrt(e*x + d))]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**2/(e*x+d)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232987, size = 401, normalized size = 2.22 \[ \frac{{\left (2 \, B b^{2} d + 3 \, B a b e - 5 \, A b^{2} e\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{-b^{2} d + a b e}} + \frac{\sqrt{x e + d} B a b e - \sqrt{x e + d} A b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}} + \frac{2 \,{\left (3 \,{\left (x e + d\right )} B b d + B b d^{2} + 3 \,{\left (x e + d\right )} B a e - 6 \,{\left (x e + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (x e + d\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)^2*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

(2*B*b^2*d + 3*B*a*b*e - 5*A*b^2*e)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))
/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) + (s
qrt(x*e + d)*B*a*b*e - sqrt(x*e + d)*A*b^2*e)/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*
b*d*e^2 - a^3*e^3)*((x*e + d)*b - b*d + a*e)) + 2/3*(3*(x*e + d)*B*b*d + B*b*d^2
+ 3*(x*e + d)*B*a*e - 6*(x*e + d)*A*b*e - B*a*d*e - A*b*d*e + A*a*e^2)/((b^3*d^
3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(x*e + d)^(3/2))