3.93 \(\int \frac{A+B x+C x^2}{(a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=424 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{4 b (a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]

[Out]

-((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(b*e - a
*f)*(a + b*x)^2) + ((2*a^3*C*d*f + a*b^2*(8*c*C*e + B*d*e + B*c*f - 6*A*d*f) - b
^3*(4*B*c*e - 3*A*(d*e + c*f)) + a^2*b*(2*B*d*f - 5*C*(d*e + c*f)))*Sqrt[c + d*x
]*Sqrt[e + f*x])/(4*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)) - ((b^2*(3*A*d^2*e^
2 - 2*c*d*e*(2*B*e - A*f) + c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)) + a*b*(d^2*e*(B*e
- 8*A*f) - c^2*f*(8*C*e - B*f) - 2*c*d*(4*C*e^2 - 7*B*e*f + 4*A*f^2)) + a^2*(C*
(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))))*ArcTanh[
(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*(b*c - a*d)
^(5/2)*(b*e - a*f)^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 2.35524, antiderivative size = 424, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{4 b (a+b x) (b c-a d)^2 (b e-a f)^2}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(2*b*(b*c - a*d)*(b*e - a
*f)*(a + b*x)^2) + ((2*a^3*C*d*f + a*b^2*(8*c*C*e + B*d*e + B*c*f - 6*A*d*f) - b
^3*(4*B*c*e - 3*A*(d*e + c*f)) + a^2*b*(2*B*d*f - 5*C*(d*e + c*f)))*Sqrt[c + d*x
]*Sqrt[e + f*x])/(4*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)) - ((b^2*(3*A*d^2*e^
2 - 2*c*d*e*(2*B*e - A*f) + c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)) + a*b*(d^2*e*(B*e
- 8*A*f) - c^2*f*(8*C*e - B*f) - 2*c*d*(4*C*e^2 - 7*B*e*f + 4*A*f^2)) + a^2*(C*
(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))))*ArcTanh[
(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(4*(b*c - a*d)
^(5/2)*(b*e - a*f)^(5/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((C*x**2+B*x+A)/(b*x+a)**3/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 2.70396, size = 604, normalized size = 1.42 \[ \frac{b (a+b x)^2 \log (a+b x) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 f (B f-8 C e)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )+2 c d e (A f-2 B e)+3 A d^2 e^2\right )\right )-b (a+b x)^2 \log \left (2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d} \sqrt{b e-a f}-a (c f+d e+2 d f x)+b (2 c e+c f x+d e x)\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 f (B f-8 C e)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )+2 c d e (A f-2 B e)+3 A d^2 e^2\right )\right )-2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d} \sqrt{b e-a f} \left ((a+b x) \left (-2 a^3 C d f+a^2 b (5 C (c f+d e)-2 B d f)-a b^2 (-6 A d f+B c f+B d e+8 c C e)+b^3 (4 B c e-3 A (c f+d e))\right )+2 (b c-a d) (b e-a f) \left (a (a C-b B)+A b^2\right )\right )}{8 b (a+b x)^2 (b c-a d)^{5/2} (b e-a f)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*Sqrt[c + d*x]*Sqrt[e + f*x]*(2*(A*b^2 + a*(-
(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-2*a^3*C*d*f - a*b^2*(8*c*C*e + B*d*e +
B*c*f - 6*A*d*f) + b^3*(4*B*c*e - 3*A*(d*e + c*f)) + a^2*b*(-2*B*d*f + 5*C*(d*e
+ c*f)))*(a + b*x)) + b*(b^2*(3*A*d^2*e^2 + 2*c*d*e*(-2*B*e + A*f) + c^2*(8*C*e
^2 - 4*B*e*f + 3*A*f^2)) + a*b*(d^2*e*(B*e - 8*A*f) + c^2*f*(-8*C*e + B*f) - 2*c
*d*(4*C*e^2 - 7*B*e*f + 4*A*f^2)) + a^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) +
4*d*f*(2*A*d*f - B*(d*e + c*f))))*(a + b*x)^2*Log[a + b*x] - b*(b^2*(3*A*d^2*e^
2 + 2*c*d*e*(-2*B*e + A*f) + c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)) + a*b*(d^2*e*(B*
e - 8*A*f) + c^2*f*(-8*C*e + B*f) - 2*c*d*(4*C*e^2 - 7*B*e*f + 4*A*f^2)) + a^2*(
C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))))*(a + b
*x)^2*Log[2*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*Sqrt[c + d*x]*Sqrt[e + f*x] + b*(2*c
*e + d*e*x + c*f*x) - a*(d*e + c*f + 2*d*f*x)])/(8*b*(b*c - a*d)^(5/2)*(b*e - a*
f)^(5/2)*(a + b*x)^2)

_______________________________________________________________________________________

Maple [B]  time = 0.165, size = 7119, normalized size = 16.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

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((C*x^2 + B*x + A)/((b*x + a)^3*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 32.6876, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*((2*(3*C*a^2*b - B*a*b^2 -
A*b^3)*c - (3*C*a^3 + B*a^2*b - 5*A*a*b^2)*d)*e - ((3*C*a^3 + B*a^2*b - 5*A*a*b^
2)*c - 4*(B*a^3 - 2*A*a^2*b)*d)*f + ((4*(2*C*a*b^2 - B*b^3)*c - (5*C*a^2*b - B*a
*b^2 - 3*A*b^3)*d)*e - ((5*C*a^2*b - B*a*b^2 - 3*A*b^3)*c - 2*(C*a^3 + B*a^2*b -
3*A*a*b^2)*d)*f)*x)*sqrt(d*x + c)*sqrt(f*x + e) + ((8*C*a^2*b^2*c^2 - 4*(2*C*a^
3*b + B*a^2*b^2)*c*d + (3*C*a^4 + B*a^3*b + 3*A*a^2*b^2)*d^2)*e^2 - 2*(2*(2*C*a^
3*b + B*a^2*b^2)*c^2 - (C*a^4 + 7*B*a^3*b + A*a^2*b^2)*c*d + 2*(B*a^4 + 2*A*a^3*
b)*d^2)*e*f + (8*A*a^4*d^2 + (3*C*a^4 + B*a^3*b + 3*A*a^2*b^2)*c^2 - 4*(B*a^4 +
2*A*a^3*b)*c*d)*f^2 + ((8*C*b^4*c^2 - 4*(2*C*a*b^3 + B*b^4)*c*d + (3*C*a^2*b^2 +
B*a*b^3 + 3*A*b^4)*d^2)*e^2 - 2*(2*(2*C*a*b^3 + B*b^4)*c^2 - (C*a^2*b^2 + 7*B*a
*b^3 + A*b^4)*c*d + 2*(B*a^2*b^2 + 2*A*a*b^3)*d^2)*e*f + (8*A*a^2*b^2*d^2 + (3*C
*a^2*b^2 + B*a*b^3 + 3*A*b^4)*c^2 - 4*(B*a^2*b^2 + 2*A*a*b^3)*c*d)*f^2)*x^2 + 2*
((8*C*a*b^3*c^2 - 4*(2*C*a^2*b^2 + B*a*b^3)*c*d + (3*C*a^3*b + B*a^2*b^2 + 3*A*a
*b^3)*d^2)*e^2 - 2*(2*(2*C*a^2*b^2 + B*a*b^3)*c^2 - (C*a^3*b + 7*B*a^2*b^2 + A*a
*b^3)*c*d + 2*(B*a^3*b + 2*A*a^2*b^2)*d^2)*e*f + (8*A*a^3*b*d^2 + (3*C*a^3*b + B
*a^2*b^2 + 3*A*a*b^3)*c^2 - 4*(B*a^3*b + 2*A*a^2*b^2)*c*d)*f^2)*x)*log(-(4*((2*b
^3*c^2 - 3*a*b^2*c*d + a^2*b*d^2)*e^2 - (3*a*b^2*c^2 - 4*a^2*b*c*d + a^3*d^2)*e*
f + (a^2*b*c^2 - a^3*c*d)*f^2 + ((b^3*c*d - a*b^2*d^2)*e^2 + (b^3*c^2 - 4*a*b^2*
c*d + 3*a^2*b*d^2)*e*f - (a*b^2*c^2 - 3*a^2*b*c*d + 2*a^3*d^2)*f^2)*x)*sqrt(d*x
+ c)*sqrt(f*x + e) - (a^2*c^2*f^2 + (8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*e^2 - 2*(4
*a*b*c^2 - 3*a^2*c*d)*e*f + (b^2*d^2*e^2 + 2*(3*b^2*c*d - 4*a*b*d^2)*e*f + (b^2*
c^2 - 8*a*b*c*d + 8*a^2*d^2)*f^2)*x^2 + 2*((4*b^2*c*d - 3*a*b*d^2)*e^2 + 2*(2*b^
2*c^2 - 5*a*b*c*d + 2*a^2*d^2)*e*f - (3*a*b*c^2 - 4*a^2*c*d)*f^2)*x)*sqrt((b^2*c
- a*b*d)*e - (a*b*c - a^2*d)*f))/(b^2*x^2 + 2*a*b*x + a^2)))/(((a^2*b^4*c^2 - 2
*a^3*b^3*c*d + a^4*b^2*d^2)*e^2 - 2*(a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*e*
f + (a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*f^2 + ((b^6*c^2 - 2*a*b^5*c*d + a^2*b^
4*d^2)*e^2 - 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*e*f + (a^2*b^4*c^2 - 2*
a^3*b^3*c*d + a^4*b^2*d^2)*f^2)*x^2 + 2*((a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^
2)*e^2 - 2*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*e*f + (a^3*b^3*c^2 - 2*a^
4*b^2*c*d + a^5*b*d^2)*f^2)*x)*sqrt((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)), 1/8
*(2*sqrt(-(b^2*c - a*b*d)*e + (a*b*c - a^2*d)*f)*((2*(3*C*a^2*b - B*a*b^2 - A*b^
3)*c - (3*C*a^3 + B*a^2*b - 5*A*a*b^2)*d)*e - ((3*C*a^3 + B*a^2*b - 5*A*a*b^2)*c
- 4*(B*a^3 - 2*A*a^2*b)*d)*f + ((4*(2*C*a*b^2 - B*b^3)*c - (5*C*a^2*b - B*a*b^2
- 3*A*b^3)*d)*e - ((5*C*a^2*b - B*a*b^2 - 3*A*b^3)*c - 2*(C*a^3 + B*a^2*b - 3*A
*a*b^2)*d)*f)*x)*sqrt(d*x + c)*sqrt(f*x + e) + ((8*C*a^2*b^2*c^2 - 4*(2*C*a^3*b
+ B*a^2*b^2)*c*d + (3*C*a^4 + B*a^3*b + 3*A*a^2*b^2)*d^2)*e^2 - 2*(2*(2*C*a^3*b
+ B*a^2*b^2)*c^2 - (C*a^4 + 7*B*a^3*b + A*a^2*b^2)*c*d + 2*(B*a^4 + 2*A*a^3*b)*d
^2)*e*f + (8*A*a^4*d^2 + (3*C*a^4 + B*a^3*b + 3*A*a^2*b^2)*c^2 - 4*(B*a^4 + 2*A*
a^3*b)*c*d)*f^2 + ((8*C*b^4*c^2 - 4*(2*C*a*b^3 + B*b^4)*c*d + (3*C*a^2*b^2 + B*a
*b^3 + 3*A*b^4)*d^2)*e^2 - 2*(2*(2*C*a*b^3 + B*b^4)*c^2 - (C*a^2*b^2 + 7*B*a*b^3
+ A*b^4)*c*d + 2*(B*a^2*b^2 + 2*A*a*b^3)*d^2)*e*f + (8*A*a^2*b^2*d^2 + (3*C*a^2
*b^2 + B*a*b^3 + 3*A*b^4)*c^2 - 4*(B*a^2*b^2 + 2*A*a*b^3)*c*d)*f^2)*x^2 + 2*((8*
C*a*b^3*c^2 - 4*(2*C*a^2*b^2 + B*a*b^3)*c*d + (3*C*a^3*b + B*a^2*b^2 + 3*A*a*b^3
)*d^2)*e^2 - 2*(2*(2*C*a^2*b^2 + B*a*b^3)*c^2 - (C*a^3*b + 7*B*a^2*b^2 + A*a*b^3
)*c*d + 2*(B*a^3*b + 2*A*a^2*b^2)*d^2)*e*f + (8*A*a^3*b*d^2 + (3*C*a^3*b + B*a^2
*b^2 + 3*A*a*b^3)*c^2 - 4*(B*a^3*b + 2*A*a^2*b^2)*c*d)*f^2)*x)*arctan(1/2*(a*c*f
- (2*b*c - a*d)*e - (b*d*e + (b*c - 2*a*d)*f)*x)*sqrt(-(b^2*c - a*b*d)*e + (a*b
*c - a^2*d)*f)/(((b^2*c - a*b*d)*e - (a*b*c - a^2*d)*f)*sqrt(d*x + c)*sqrt(f*x +
e))))/(((a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*e^2 - 2*(a^3*b^3*c^2 - 2*a^
4*b^2*c*d + a^5*b*d^2)*e*f + (a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)*f^2 + ((b^6*c
^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*e^2 - 2*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2
)*e*f + (a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*f^2)*x^2 + 2*((a*b^5*c^2 - 2
*a^2*b^4*c*d + a^3*b^3*d^2)*e^2 - 2*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*
e*f + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*f^2)*x)*sqrt(-(b^2*c - a*b*d)*e
+ (a*b*c - a^2*d)*f))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x**2+B*x+A)/(b*x+a)**3/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out