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

Optimal. Leaf size=540 \[ -\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac{(d e-c f) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac{C (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{4 b d f} \]

[Out]

-((8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f
^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2
*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*Sqrt[c + d*x]*Sqrt
[e + f*x])/(64*d^3*f^4) + (C*(a + b*x)^2*(c + d*x)^(3/2)*Sqrt[e + f*x])/(4*b*d*f
) - ((c + d*x)^(3/2)*Sqrt[e + f*x]*(24*a^2*C*d^2*f^2 + 8*a*b*d*f*(5*C*d*e + 3*c*
C*f - 6*B*d*f) + b^2*(8*d*f*(5*B*d*e + 3*B*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c
*d*e*f + 15*c^2*f^2)) + 4*b*d*f*(4*a*C*d*f + b*(7*C*d*e + 5*c*C*f - 8*B*d*f))*x)
)/(96*b*d^3*f^3) + ((d*e - c*f)*(8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*
(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d
*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f +
c^2*f^2))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(7/2
)*f^(9/2))

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Rubi [A]  time = 1.79833, antiderivative size = 540, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147 \[ -\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (24 a^2 C d^2 f^2+4 b d f x (4 a C d f+b (-8 B d f+5 c C f+7 C d e))+8 a b d f (-6 B d f+3 c C f+5 C d e)+b^2 \left (8 d f (-6 A d f+3 B c f+5 B d e)-C \left (15 c^2 f^2+22 c d e f+35 d^2 e^2\right )\right )\right )}{96 b d^3 f^3}-\frac{\sqrt{c+d x} \sqrt{e+f x} \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^3 f^4}+\frac{(d e-c f) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (8 a d f \left (2 d f (-4 A d f+B c f+3 B d e)-C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )\right )}{64 d^{7/2} f^{9/2}}+\frac{C (a+b x)^2 (c+d x)^{3/2} \sqrt{e+f x}}{4 b d f} \]

Antiderivative was successfully verified.

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

[Out]

-((8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f
^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2
*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*Sqrt[c + d*x]*Sqrt
[e + f*x])/(64*d^3*f^4) + (C*(a + b*x)^2*(c + d*x)^(3/2)*Sqrt[e + f*x])/(4*b*d*f
) - ((c + d*x)^(3/2)*Sqrt[e + f*x]*(24*a^2*C*d^2*f^2 + 8*a*b*d*f*(5*C*d*e + 3*c*
C*f - 6*B*d*f) + b^2*(8*d*f*(5*B*d*e + 3*B*c*f - 6*A*d*f) - C*(35*d^2*e^2 + 22*c
*d*e*f + 15*c^2*f^2)) + 4*b*d*f*(4*a*C*d*f + b*(7*C*d*e + 5*c*C*f - 8*B*d*f))*x)
)/(96*b*d^3*f^3) + ((d*e - c*f)*(8*a*d*f*(2*d*f*(3*B*d*e + B*c*f - 4*A*d*f) - C*
(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d
*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f +
c^2*f^2))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(64*d^(7/2
)*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)*(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 = 1.5515, size = 470, normalized size = 0.87 \[ \frac{3 (d e-c f) \log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (b \left (8 d f \left (2 A d f (c f+3 d e)-B \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )+C \left (5 c^3 f^3+9 c^2 d e f^2+15 c d^2 e^2 f+35 d^3 e^3\right )\right )-8 a d f \left (2 d f (4 A d f-B c f-3 B d e)+C \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )\right )-2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x} \left (b \left (C \left (-15 c^3 f^3+c^2 d f^2 (10 f x-17 e)+c d^2 f \left (-25 e^2+12 e f x-8 f^2 x^2\right )+d^3 \left (105 e^3-70 e^2 f x+56 e f^2 x^2-48 f^3 x^3\right )\right )-8 d f \left (6 A d f (c f-3 d e+2 d f x)+B \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )-8 a d f \left (6 d f (4 A d f+B (c f-3 d e+2 d f x))+C \left (-3 c^2 f^2+2 c d f (f x-2 e)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )}{384 d^{7/2} f^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]*(-8*a*d*f*(6*d*f*(4*A*d*f + B*(-
3*d*e + c*f + 2*d*f*x)) + C*(-3*c^2*f^2 + 2*c*d*f*(-2*e + f*x) + d^2*(15*e^2 - 1
0*e*f*x + 8*f^2*x^2))) + b*(C*(-15*c^3*f^3 + c^2*d*f^2*(-17*e + 10*f*x) + c*d^2*
f*(-25*e^2 + 12*e*f*x - 8*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 + c*f + 2*d*f*x) + B*(-3*c^2*f^2 + 2*c*d*f
*(-2*e + f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^2))))) + 3*(d*e - c*f)*(-8*a*d*
f*(2*d*f*(-3*B*d*e - B*c*f + 4*A*d*f) + C*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2)) + b
*(C*(35*d^3*e^3 + 15*c*d^2*e^2*f + 9*c^2*d*e*f^2 + 5*c^3*f^3) + 8*d*f*(2*A*d*f*(
3*d*e + c*f) - B*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))))*Log[d*e + c*f + 2*d*f*x +
2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]])/(384*d^(7/2)*f^(9/2))

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Maple [B]  time = 0.043, size = 2002, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(d*x+c)^(1/2)*(f*x+e)^(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*A*a*f^4*d^3+96*C*x^3*b*d^3*f^3*((d*x+c)*(f*
x+e))^(1/2)*(f*d)^(1/2)+128*B*x^2*b*d^3*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+
128*C*x^2*a*d^3*f^3*((d*x+c)*(f*x+e))^(1/2)*(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*C*a*e^2*f^2*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*C*b*e^
3*f*d^3-288*((d*x+c)*(f*x+e))^(1/2)*A*b*e*f^2*d^3*(f*d)^(1/2)-288*((d*x+c)*(f*x+
e))^(1/2)*B*a*e*f^2*d^3*(f*d)^(1/2)+240*((d*x+c)*(f*x+e))^(1/2)*B*b*e^2*f*d^3*(f
*d)^(1/2)+240*((d*x+c)*(f*x+e))^(1/2)*C*a*e^2*f*d^3*(f*d)^(1/2)-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*A*b*e*f^3*d^3-9
6*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*
B*a*e*f^3*d^3+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*B*b*e^2*f^2*d^3+96*A*b*((d*x+c)*(f*x+e))^(1/2)*c*f^3*d^2*(f*d)^(1
/2)+96*B*a*((d*x+c)*(f*x+e))^(1/2)*c*f^3*d^2*(f*d)^(1/2)-48*((d*x+c)*(f*x+e))^(1
/2)*c^2*B*b*f^3*d*(f*d)^(1/2)-48*((d*x+c)*(f*x+e))^(1/2)*c^2*C*a*f^3*d*(f*d)^(1/
2)+192*A*b*((d*x+c)*(f*x+e))^(1/2)*x*f^3*d^3*(f*d)^(1/2)+192*B*a*((d*x+c)*(f*x+e
))^(1/2)*x*f^3*d^3*(f*d)^(1/2)+24*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*B*b*f^3*d^2+24*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*C*a*f^3*d^2-12*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*e*f^3*d-
18*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*C*b*f^2*d^2-15*C*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))*c^4*f^4+16*C*x^2*b*c*d^2*f^3*((d*x+c)*(f*x+e))^(1/2)*(f*d)
^(1/2)+32*((d*x+c)*(f*x+e))^(1/2)*x*c*C*a*f^3*d^2*(f*d)^(1/2)-160*((d*x+c)*(f*x+
e))^(1/2)*x*e*C*a*f^2*d^3*(f*d)^(1/2)-20*C*b*((d*x+c)*(f*x+e))^(1/2)*x*c^2*f^3*d
*(f*d)^(1/2)-112*C*x^2*b*d^3*e*f^2*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+140*((d*x
+c)*(f*x+e))^(1/2)*x*e^2*C*b*f*d^3*(f*d)^(1/2)-64*((d*x+c)*(f*x+e))^(1/2)*c*e*B*
b*f^2*d^2*(f*d)^(1/2)-64*((d*x+c)*(f*x+e))^(1/2)*c*e*C*a*f^2*d^2*(f*d)^(1/2)+34*
((d*x+c)*(f*x+e))^(1/2)*c^2*C*b*e*f^2*d*(f*d)^(1/2)+50*((d*x+c)*(f*x+e))^(1/2)*c
*e^2*C*b*f*d^2*(f*d)^(1/2)+32*((d*x+c)*(f*x+e))^(1/2)*x*c*B*b*f^3*d^2*(f*d)^(1/2
)-160*((d*x+c)*(f*x+e))^(1/2)*x*e*B*b*f^2*d^3*(f*d)^(1/2)+105*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))*d^4*e^4*C*b-24*((d*x+c
)*(f*x+e))^(1/2)*x*c*C*b*e*f^2*d^2*(f*d)^(1/2)-120*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))*d^4*e^3*B*b*f-120*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))*d^4*e^3*C*a*f+384*(
(d*x+c)*(f*x+e))^(1/2)*A*a*f^3*d^3*(f*d)^(1/2)-210*((d*x+c)*(f*x+e))^(1/2)*C*b*e
^3*d^3*(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))*d^4*e*A*a*f^3+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))*d^4*e^2*A*b*f^2+30*C*b*((d*x+c)*(f*x+e))^(1/2)*
c^3*f^3*(f*d)^(1/2)-48*A*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))*c^2*f^4*d^2-48*B*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))*c^2*f^4*d^2+24*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*B*b*f^4*d+24*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*a*f^4*d+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))*d^4*
e^2*B*a*f^2)/((d*x+c)*(f*x+e))^(1/2)/f^4/d^3/(f*d)^(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)*sqrt(d*x + c)/sqrt(f*x + e),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 22.9898, 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)*sqrt(d*x + c)/sqrt(f*x + e),x, algorithm="fricas")

[Out]

[1/768*(4*(48*C*b*d^3*f^3*x^3 - 105*C*b*d^3*e^3 + 5*(5*C*b*c*d^2 + 24*(C*a + B*b
)*d^3)*e^2*f + (17*C*b*c^2*d - 32*(C*a + B*b)*c*d^2 - 144*(B*a + A*b)*d^3)*e*f^2
+ 3*(5*C*b*c^3 + 64*A*a*d^3 - 8*(C*a + B*b)*c^2*d + 16*(B*a + A*b)*c*d^2)*f^3 -
8*(7*C*b*d^3*e*f^2 - (C*b*c*d^2 + 8*(C*a + B*b)*d^3)*f^3)*x^2 + 2*(35*C*b*d^3*e
^2*f - 2*(3*C*b*c*d^2 + 20*(C*a + B*b)*d^3)*e*f^2 - (5*C*b*c^2*d - 8*(C*a + B*b)
*c*d^2 - 48*(B*a + A*b)*d^3)*f^3)*x)*sqrt(d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 3*(
35*C*b*d^4*e^4 - 20*(C*b*c*d^3 + 2*(C*a + B*b)*d^4)*e^3*f - 6*(C*b*c^2*d^2 - 4*(
C*a + B*b)*c*d^3 - 8*(B*a + A*b)*d^4)*e^2*f^2 - 4*(C*b*c^3*d + 16*A*a*d^4 - 2*(C
*a + B*b)*c^2*d^2 + 8*(B*a + A*b)*c*d^3)*e*f^3 - (5*C*b*c^4 - 64*A*a*c*d^3 - 8*(
C*a + B*b)*c^3*d + 16*(B*a + A*b)*c^2*d^2)*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^3*f^4), 1/384*(2*(48*C*
b*d^3*f^3*x^3 - 105*C*b*d^3*e^3 + 5*(5*C*b*c*d^2 + 24*(C*a + B*b)*d^3)*e^2*f + (
17*C*b*c^2*d - 32*(C*a + B*b)*c*d^2 - 144*(B*a + A*b)*d^3)*e*f^2 + 3*(5*C*b*c^3
+ 64*A*a*d^3 - 8*(C*a + B*b)*c^2*d + 16*(B*a + A*b)*c*d^2)*f^3 - 8*(7*C*b*d^3*e*
f^2 - (C*b*c*d^2 + 8*(C*a + B*b)*d^3)*f^3)*x^2 + 2*(35*C*b*d^3*e^2*f - 2*(3*C*b*
c*d^2 + 20*(C*a + B*b)*d^3)*e*f^2 - (5*C*b*c^2*d - 8*(C*a + B*b)*c*d^2 - 48*(B*a
+ A*b)*d^3)*f^3)*x)*sqrt(-d*f)*sqrt(d*x + c)*sqrt(f*x + e) + 3*(35*C*b*d^4*e^4
- 20*(C*b*c*d^3 + 2*(C*a + B*b)*d^4)*e^3*f - 6*(C*b*c^2*d^2 - 4*(C*a + B*b)*c*d^
3 - 8*(B*a + A*b)*d^4)*e^2*f^2 - 4*(C*b*c^3*d + 16*A*a*d^4 - 2*(C*a + B*b)*c^2*d
^2 + 8*(B*a + A*b)*c*d^3)*e*f^3 - (5*C*b*c^4 - 64*A*a*c*d^3 - 8*(C*a + B*b)*c^3*
d + 16*(B*a + A*b)*c^2*d^2)*f^4)*arctan(1/2*(2*d*f*x + d*e + c*f)*sqrt(-d*f)/(sq
rt(d*x + c)*sqrt(f*x + e)*d*f)))/(sqrt(-d*f)*d^3*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)*(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 = 0.28132, size = 994, normalized size = 1.84 \[ \frac{{\left (\sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e}{\left (2 \,{\left (d x + c\right )}{\left (4 \,{\left (d x + c\right )}{\left (\frac{6 \,{\left (d x + c\right )} C b}{d^{4} f} - \frac{17 \, C b c d^{12} f^{6} - 8 \, C a d^{13} f^{6} - 8 \, B b d^{13} f^{6} + 7 \, C b d^{13} f^{5} e}{d^{16} f^{7}}\right )} + \frac{59 \, C b c^{2} d^{12} f^{6} - 56 \, C a c d^{13} f^{6} - 56 \, B b c d^{13} f^{6} + 48 \, B a d^{14} f^{6} + 48 \, A b d^{14} f^{6} + 50 \, C b c d^{13} f^{5} e - 40 \, C a d^{14} f^{5} e - 40 \, B b d^{14} f^{5} e + 35 \, C b d^{14} f^{4} e^{2}}{d^{16} f^{7}}\right )} - \frac{3 \,{\left (5 \, C b c^{3} d^{12} f^{6} - 8 \, C a c^{2} d^{13} f^{6} - 8 \, B b c^{2} d^{13} f^{6} + 16 \, B a c d^{14} f^{6} + 16 \, A b c d^{14} f^{6} - 64 \, A a d^{15} f^{6} + 9 \, C b c^{2} d^{13} f^{5} e - 16 \, C a c d^{14} f^{5} e - 16 \, B b c d^{14} f^{5} e + 48 \, B a d^{15} f^{5} e + 48 \, A b d^{15} f^{5} e + 15 \, C b c d^{14} f^{4} e^{2} - 40 \, C a d^{15} f^{4} e^{2} - 40 \, B b d^{15} f^{4} e^{2} + 35 \, C b d^{15} f^{3} e^{3}\right )}}{d^{16} f^{7}}\right )} \sqrt{d x + c} + \frac{3 \,{\left (5 \, C b c^{4} f^{4} - 8 \, C a c^{3} d f^{4} - 8 \, B b c^{3} d f^{4} + 16 \, B a c^{2} d^{2} f^{4} + 16 \, A b c^{2} d^{2} f^{4} - 64 \, A a c d^{3} f^{4} + 4 \, C b c^{3} d f^{3} e - 8 \, C a c^{2} d^{2} f^{3} e - 8 \, B b c^{2} d^{2} f^{3} e + 32 \, B a c d^{3} f^{3} e + 32 \, A b c d^{3} f^{3} e + 64 \, A a d^{4} f^{3} e + 6 \, C b c^{2} d^{2} f^{2} e^{2} - 24 \, C a c d^{3} f^{2} e^{2} - 24 \, B b c d^{3} f^{2} e^{2} - 48 \, B a d^{4} f^{2} e^{2} - 48 \, A b d^{4} f^{2} e^{2} + 20 \, C b c d^{3} f e^{3} + 40 \, C a d^{4} f e^{3} + 40 \, B b d^{4} f e^{3} - 35 \, C b d^{4} e^{4}\right )}{\rm ln}\left ({\left | -\sqrt{d f} \sqrt{d x + c} + \sqrt{{\left (d x + c\right )} d f - c d f + d^{2} e} \right |}\right )}{\sqrt{d f} d^{3} f^{4}}\right )} d}{192 \,{\left | d \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(sqrt((d*x + c)*d*f - c*d*f + d^2*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c
)*C*b/(d^4*f) - (17*C*b*c*d^12*f^6 - 8*C*a*d^13*f^6 - 8*B*b*d^13*f^6 + 7*C*b*d^1
3*f^5*e)/(d^16*f^7)) + (59*C*b*c^2*d^12*f^6 - 56*C*a*c*d^13*f^6 - 56*B*b*c*d^13*
f^6 + 48*B*a*d^14*f^6 + 48*A*b*d^14*f^6 + 50*C*b*c*d^13*f^5*e - 40*C*a*d^14*f^5*
e - 40*B*b*d^14*f^5*e + 35*C*b*d^14*f^4*e^2)/(d^16*f^7)) - 3*(5*C*b*c^3*d^12*f^6
- 8*C*a*c^2*d^13*f^6 - 8*B*b*c^2*d^13*f^6 + 16*B*a*c*d^14*f^6 + 16*A*b*c*d^14*f
^6 - 64*A*a*d^15*f^6 + 9*C*b*c^2*d^13*f^5*e - 16*C*a*c*d^14*f^5*e - 16*B*b*c*d^1
4*f^5*e + 48*B*a*d^15*f^5*e + 48*A*b*d^15*f^5*e + 15*C*b*c*d^14*f^4*e^2 - 40*C*a
*d^15*f^4*e^2 - 40*B*b*d^15*f^4*e^2 + 35*C*b*d^15*f^3*e^3)/(d^16*f^7))*sqrt(d*x
+ c) + 3*(5*C*b*c^4*f^4 - 8*C*a*c^3*d*f^4 - 8*B*b*c^3*d*f^4 + 16*B*a*c^2*d^2*f^4
+ 16*A*b*c^2*d^2*f^4 - 64*A*a*c*d^3*f^4 + 4*C*b*c^3*d*f^3*e - 8*C*a*c^2*d^2*f^3
*e - 8*B*b*c^2*d^2*f^3*e + 32*B*a*c*d^3*f^3*e + 32*A*b*c*d^3*f^3*e + 64*A*a*d^4*
f^3*e + 6*C*b*c^2*d^2*f^2*e^2 - 24*C*a*c*d^3*f^2*e^2 - 24*B*b*c*d^3*f^2*e^2 - 48
*B*a*d^4*f^2*e^2 - 48*A*b*d^4*f^2*e^2 + 20*C*b*c*d^3*f*e^3 + 40*C*a*d^4*f*e^3 +
40*B*b*d^4*f*e^3 - 35*C*b*d^4*e^4)*ln(abs(-sqrt(d*f)*sqrt(d*x + c) + sqrt((d*x +
c)*d*f - c*d*f + d^2*e)))/(sqrt(d*f)*d^3*f^4))*d/abs(d)