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

Optimal. Leaf size=718 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{64 d^{9/2} f^{9/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)+(4 a d f-5 b (c f+d e)) (2 a C d f-b (8 B d f-7 C (c f+d e))))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{192 b d^4 f^4}-\frac{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} (2 a C d f-b (8 B d f-7 C (c f+d e)))}{24 b d^2 f^2}+\frac{C (a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}}{4 b d f} \]

[Out]

-((2*a*C*d*f - b*(8*B*d*f - 7*C*(d*e + c*f)))*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e +
f*x])/(24*b*d^2*f^2) + (C*(a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b*d*f) -
(Sqrt[c + d*x]*Sqrt[e + f*x]*(32*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(16*B*d*f - 11*
C*(d*e + c*f)) - 16*a*b^2*d*f*(C*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2) + 6*d*f*
(4*A*d*f - 3*B*(d*e + c*f))) + b^3*(5*C*(21*d^3*e^3 + 19*c*d^2*e^2*f + 19*c^2*d*
e*f^2 + 21*c^3*f^3) + 8*d*f*(18*A*d*f*(d*e + c*f) - B*(15*d^2*e^2 + 14*c*d*e*f +
15*c^2*f^2))) + 2*b*d*f*(6*b*d*f*(6*b*c*C*e + a*C*d*e + a*c*C*f - 8*A*b*d*f) +
(4*a*d*f - 5*b*(d*e + c*f))*(2*a*C*d*f - b*(8*B*d*f - 7*C*(d*e + c*f))))*x))/(19
2*b*d^4*f^4) + ((16*a^2*d^2*f^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))) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e
*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^
2*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e
*f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) - B*(5*d
^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c +
d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(9/2)*f^(9/2))

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Rubi [A]  time = 3.39722, antiderivative size = 715, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{64 d^{9/2} f^{9/2}}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{192 b d^4 f^4}+\frac{(a+b x)^2 \sqrt{c+d x} \sqrt{e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{24 b d^2 f^2}+\frac{C (a+b x)^3 \sqrt{c+d x} \sqrt{e+f x}}{4 b d f} \]

Antiderivative was successfully verified.

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

[Out]

((8*b*B*d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f))*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e +
f*x])/(24*b*d^2*f^2) + (C*(a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b*d*f) - (
Sqrt[c + d*x]*Sqrt[e + f*x]*(32*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2*(16*B*d*f - 11*C
*(d*e + c*f)) - 16*a*b^2*d*f*(C*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2) + 6*d*f*(
4*A*d*f - 3*B*(d*e + c*f))) + b^3*(5*C*(21*d^3*e^3 + 19*c*d^2*e^2*f + 19*c^2*d*e
*f^2 + 21*c^3*f^3) + 8*d*f*(18*A*d*f*(d*e + c*f) - B*(15*d^2*e^2 + 14*c*d*e*f +
15*c^2*f^2))) + 2*b*d*f*(6*b*d*f*(6*b*c*C*e + a*C*d*e + a*c*C*f - 8*A*b*d*f) - (
4*a*d*f - 5*b*(d*e + c*f))*(8*b*B*d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f)))*x))/(192
*b*d^4*f^4) + ((16*a^2*d^2*f^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))) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*
f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2
*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e*
f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) - B*(5*d^
3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*Sqrt[c +
d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(9/2)*f^(9/2))

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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((b*x+a)**2*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.61591, size = 645, normalized size = 0.9 \[ \frac{3 \log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )-2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x} \left (-48 a^2 d^2 f^2 (4 B d f+C (-3 c f-3 d e+2 d f x))-16 a b d f \left (6 d f (4 A d f+B (-3 c f-3 d e+2 d f x))+C \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b^2 \left (C \left (105 c^3 f^3+5 c^2 d f^2 (19 e-14 f x)+c d^2 f \left (95 e^2-68 e f x+56 f^2 x^2\right )+d^3 \left (105 e^3-70 e^2 f x+56 e f^2 x^2-48 f^3 x^3\right )\right )-8 d f \left (6 A d f (-3 c f-3 d e+2 d f x)+B \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )\right )}{384 d^{9/2} f^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]*(-48*a^2*d^2*f^2*(4*B*d*f + C*(-
3*d*e - 3*c*f + 2*d*f*x)) - 16*a*b*d*f*(6*d*f*(4*A*d*f + B*(-3*d*e - 3*c*f + 2*d
*f*x)) + C*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*
x^2))) + b^2*(C*(105*c^3*f^3 + 5*c^2*d*f^2*(19*e - 14*f*x) + c*d^2*f*(95*e^2 - 6
8*e*f*x + 56*f^2*x^2) + d^3*(105*e^3 - 70*e^2*f*x + 56*e*f^2*x^2 - 48*f^3*x^3))
- 8*d*f*(6*A*d*f*(-3*d*e - 3*c*f + 2*d*f*x) + B*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f
*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))))) + 3*(16*a^2*d^2*f^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))) - 16*a*b*d*f*(C*(5*d
^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f)
- B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f
+ 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e*f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2
+ 2*c*d*e*f + 3*c^2*f^2) - B*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3
*f^3))))*Log[d*e + c*f + 2*d*f*x + 2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]
])/(384*d^(9/2)*f^(9/2))

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Maple [B]  time = 0.059, size = 2528, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(384*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d
)^(1/2))*a^2*d^4*f^4+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)
+c*f+d*e)/(f*d)^(1/2))*b^2*c^4*f^4+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/
2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b^2*d^4*e^4+144*C*ln(1/2*(2*d*f*x+2*((d*x+c
)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a^2*c^2*d^2*f^4-576*B*((d*x+c
)*(f*x+e))^(1/2)*(f*d)^(1/2)*a*b*c*d^2*f^3+448*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(
1/2)*a*b*c*d^2*e*f^2-320*((d*x+c)*(f*x+e))^(1/2)*x*C*a*b*c*f^3*d^2*(f*d)^(1/2)-3
20*((d*x+c)*(f*x+e))^(1/2)*x*C*a*b*e*f^2*d^3*(f*d)^(1/2)+136*((d*x+c)*(f*x+e))^(
1/2)*x*C*b^2*c*e*f^2*d^2*(f*d)^(1/2)+140*((d*x+c)*(f*x+e))^(1/2)*x*C*b^2*c^2*f^3
*d*(f*d)^(1/2)+140*((d*x+c)*(f*x+e))^(1/2)*x*C*b^2*e^2*f*d^3*(f*d)^(1/2)-576*B*(
(d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a*b*d^3*e*f^2+224*B*((d*x+c)*(f*x+e))^(1/2)*(
f*d)^(1/2)*b^2*c*d^2*e*f^2+480*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a*b*c^2*d*f
^3+480*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a*b*d^3*e^2*f-190*C*((d*x+c)*(f*x+e
))^(1/2)*(f*d)^(1/2)*b^2*c^2*d*e*f^2-190*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*b
^2*c*d^2*e^2*f+256*C*x^2*a*b*d^3*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-112*C*x
^2*b^2*c*d^2*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-112*C*x^2*b^2*d^3*e*f^2*((d
*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+192*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(
f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*e*B*a*b*f^3*d^3+384*((d*x+c)*(f*x+e))^(1/2)*x
*B*a*b*f^3*d^3*(f*d)^(1/2)-160*((d*x+c)*(f*x+e))^(1/2)*x*B*b^2*c*f^3*d^2*(f*d)^(
1/2)-160*((d*x+c)*(f*x+e))^(1/2)*x*B*b^2*e*f^2*d^3*(f*d)^(1/2)-144*ln(1/2*(2*d*f
*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^2*C*a*b*e*f^3*d
^2-144*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2
))*c*e^2*C*a*b*f^2*d^3+96*C*x^3*b^2*d^3*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+
128*B*x^2*b^2*d^3*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-384*A*ln(1/2*(2*d*f*x+
2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b*c*d^3*f^4-384*A*
ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b*
d^4*e*f^3+288*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(
f*d)^(1/2))*a*b*c^2*d^2*f^4+288*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d
)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b*d^4*e^2*f^2+96*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x
+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*e*A*b^2*f^3*d^3+192*((d*x+c)*(f*x
+e))^(1/2)*x*A*b^2*f^3*d^3*(f*d)^(1/2)-72*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1
/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^2*B*b^2*e*f^3*d^2-240*C*ln(1/2*(2*d*f*x+
2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b*c^3*d*f^4-240*C*
ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*b*
d^4*e^3*f+768*A*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a*b*d^3*f^3-288*A*((d*x+c)*(
f*x+e))^(1/2)*(f*d)^(1/2)*b^2*c*d^2*f^3-288*A*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2
)*b^2*d^3*e*f^2+240*B*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*b^2*c^2*d*f^3+240*B*((
d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*b^2*d^3*e^2*f-288*C*((d*x+c)*(f*x+e))^(1/2)*(f
*d)^(1/2)*a^2*c*d^2*f^3-288*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a^2*d^3*e*f^2-
72*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c
*e^2*B*b^2*f^2*d^3+96*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+
d*e)/(f*d)^(1/2))*c*e*a^2*C*f^3*d^3+60*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)
*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^3*C*b^2*e*f^3*d+54*C*b^2*ln(1/2*(2*d*f*x+2*
((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^2*e^2*f^2*d^2+60*ln(
1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*e^3*C
*b^2*f*d^3+192*((d*x+c)*(f*x+e))^(1/2)*x*a^2*C*f^3*d^3*(f*d)^(1/2)+144*C*ln(1/2*
(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a^2*d^4*e^2
*f^2+384*B*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*a^2*d^3*f^3-210*C*((d*x+c)*(f*x+e
))^(1/2)*(f*d)^(1/2)*b^2*c^3*f^3-210*C*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)*b^2*d
^3*e^3+144*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d
)^(1/2))*b^2*c^2*d^2*f^4+144*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(
1/2)+c*f+d*e)/(f*d)^(1/2))*b^2*d^4*e^2*f^2-192*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x
+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a^2*c*d^3*f^4-192*B*ln(1/2*(2*d*f*x
+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a^2*d^4*e*f^3-120*B
*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*b^2
*c^3*d*f^4-120*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+c*f+d*e)/
(f*d)^(1/2))*b^2*d^4*e^3*f)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(f*d)^(1/2)/d^4/f^4/((d*
x+c)*(f*x+e))^(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((C*x^2 + B*x + A)*(b*x + a)^2/(sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 35.095, 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)^2/(sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

[1/768*(4*(48*C*b^2*d^3*f^3*x^3 - 105*C*b^2*d^3*e^3 - 5*(19*C*b^2*c*d^2 - 24*(2*
C*a*b + B*b^2)*d^3)*e^2*f - (95*C*b^2*c^2*d - 112*(2*C*a*b + B*b^2)*c*d^2 + 144*
(C*a^2 + 2*B*a*b + A*b^2)*d^3)*e*f^2 - 3*(35*C*b^2*c^3 - 40*(2*C*a*b + B*b^2)*c^
2*d + 48*(C*a^2 + 2*B*a*b + A*b^2)*c*d^2 - 64*(B*a^2 + 2*A*a*b)*d^3)*f^3 - 8*(7*
C*b^2*d^3*e*f^2 + (7*C*b^2*c*d^2 - 8*(2*C*a*b + B*b^2)*d^3)*f^3)*x^2 + 2*(35*C*b
^2*d^3*e^2*f + 2*(17*C*b^2*c*d^2 - 20*(2*C*a*b + B*b^2)*d^3)*e*f^2 + (35*C*b^2*c
^2*d - 40*(2*C*a*b + B*b^2)*c*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*d^3)*f^3)*x)*sq
rt(d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*
(2*C*a*b + B*b^2)*d^4)*e^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 +
8*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2
)*c^2*d^2 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3
+ (35*C*b^2*c^4 + 128*A*a^2*d^4 - 40*(2*C*a*b + B*b^2)*c^3*d + 48*(C*a^2 + 2*B*a
*b + A*b^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*log(4*(2*d^2*f^2*x + d^2*
e*f + c*d*f^2)*sqrt(d*x + c)*sqrt(f*x + e) + (8*d^2*f^2*x^2 + d^2*e^2 + 6*c*d*e*
f + c^2*f^2 + 8*(d^2*e*f + c*d*f^2)*x)*sqrt(d*f)))/(sqrt(d*f)*d^4*f^4), 1/384*(2
*(48*C*b^2*d^3*f^3*x^3 - 105*C*b^2*d^3*e^3 - 5*(19*C*b^2*c*d^2 - 24*(2*C*a*b + B
*b^2)*d^3)*e^2*f - (95*C*b^2*c^2*d - 112*(2*C*a*b + B*b^2)*c*d^2 + 144*(C*a^2 +
2*B*a*b + A*b^2)*d^3)*e*f^2 - 3*(35*C*b^2*c^3 - 40*(2*C*a*b + B*b^2)*c^2*d + 48*
(C*a^2 + 2*B*a*b + A*b^2)*c*d^2 - 64*(B*a^2 + 2*A*a*b)*d^3)*f^3 - 8*(7*C*b^2*d^3
*e*f^2 + (7*C*b^2*c*d^2 - 8*(2*C*a*b + B*b^2)*d^3)*f^3)*x^2 + 2*(35*C*b^2*d^3*e^
2*f + 2*(17*C*b^2*c*d^2 - 20*(2*C*a*b + B*b^2)*d^3)*e*f^2 + (35*C*b^2*c^2*d - 40
*(2*C*a*b + B*b^2)*c*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*d^3)*f^3)*x)*sqrt(-d*f)*
sqrt(d*x + c)*sqrt(f*x + e) + 3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b
+ B*b^2)*d^4)*e^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2
+ 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^
2 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35*C*
b^2*c^4 + 128*A*a^2*d^4 - 40*(2*C*a*b + B*b^2)*c^3*d + 48*(C*a^2 + 2*B*a*b + A*b
^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*arctan(1/2*(2*d*f*x + d*e + c*f)*
sqrt(-d*f)/(sqrt(d*x + c)*sqrt(f*x + e)*d*f)))/(sqrt(-d*f)*d^4*f^4)]

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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)**2*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.3357, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done