3.78 \(\int \frac{\sqrt{c+d x} \sqrt{e+f x} \left (A+B x+C x^2\right )}{a+b x} \, dx\)
Optimal. Leaf size=450 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (-\left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} (4 b d f (2 A b d f-a C (c f+d e))+(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e)))}{8 b^3 d^2 f^2}-\frac{2 \sqrt{b c-a d} \sqrt{b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^4}-\frac{\sqrt{c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]
[Out]
((4*b*d*f*(2*A*b*d*f - a*C*(d*e + c*f)) + (b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f +
b*(C*d*e + c*C*f - 2*B*d*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^3*d^2*f^2) - ((
2*a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*d
*f^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a
^2*b*d^2*f^2*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B
*d*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e - c*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*
f)^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e +
f*x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*Sqr
t[b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f
*x])])/b^4
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Rubi [A] time = 3.58878, antiderivative size = 453, normalized size of antiderivative =
1.01, number of steps used = 8, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167
\[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) \left (16 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)-2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (-\left (C (d e-c f)^2 (c f+d e)-2 d f \left (B (d e-c f)^2-4 A d f (c f+d e)\right )\right )\right )\right )}{8 b^4 d^{5/2} f^{5/2}}+\frac{\sqrt{c+d x} \sqrt{e+f x} \left (\frac{(4 a d f-b c f+b d e) (2 a C d f+b (-2 B d f+c C f+C d e))}{b d f}-4 a C (c f+d e)+8 A b d f\right )}{8 b^2 d f}-\frac{2 \sqrt{b c-a d} \sqrt{b e-a f} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right )}{b^4}-\frac{\sqrt{c+d x} (e+f x)^{3/2} (2 a C d f+b (-2 B d f+c C f+C d e))}{4 b^2 d f^2}+\frac{C (c+d x)^{3/2} (e+f x)^{3/2}}{3 b d f} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x),x]
[Out]
((8*A*b*d*f - 4*a*C*(d*e + c*f) + ((b*d*e - b*c*f + 4*a*d*f)*(2*a*C*d*f + b*(C*d
*e + c*C*f - 2*B*d*f)))/(b*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(8*b^2*d*f) - ((2*
a*C*d*f + b*(C*d*e + c*C*f - 2*B*d*f))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(4*b^2*d*f
^2) + (C*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*d*f) - ((16*a^3*C*d^3*f^3 - 8*a^2
*b*d^2*f^2*(C*d*e + c*C*f + 2*B*d*f) - 2*a*b^2*d*f*(C*(d*e - c*f)^2 - 4*d*f*(B*d
*e + B*c*f + 2*A*d*f)) - b^3*(C*(d*e - c*f)^2*(d*e + c*f) - 2*d*f*(B*(d*e - c*f)
^2 - 4*A*d*f*(d*e + c*f))))*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*
x])])/(8*b^4*d^(5/2)*f^(5/2)) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[b*c - a*d]*Sqrt[
b*e - a*f]*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x
])])/b^4
_______________________________________________________________________________________
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)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a),x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [A] time = 1.72083, size = 498, normalized size = 1.11 \[ \frac{\frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left (24 a^2 C d^2 f^2-6 a b d f (4 B d f+c C f+C d (e+2 f x))+b^2 \left (6 d f (4 A d f+B c f+B d (e+2 f x))+C \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )\right )}{d^2 f^2}+\frac{3 \log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) \left (-16 a^3 C d^3 f^3+8 a^2 b d^2 f^2 (2 B d f+c C f+C d e)+2 a b^2 d f \left (C (d e-c f)^2-4 d f (2 A d f+B c f+B d e)\right )+b^3 \left (2 d f \left (4 A d f (c f+d e)-B (d e-c f)^2\right )+C (c f+d e) (d e-c f)^2\right )\right )}{d^{5/2} f^{5/2}}+48 \sqrt{b c-a d} \sqrt{b e-a f} \log (a+b x) \left (a (a C-b B)+A b^2\right )-48 \sqrt{b c-a d} \sqrt{b e-a f} \left (a (a C-b B)+A b^2\right ) \log \left (2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d} \sqrt{b e-a f}-a (c f+d e+2 d f x)+b (2 c e+c f x+d e x)\right )}{48 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c + d*x]*Sqrt[e + f*x]*(A + B*x + C*x^2))/(a + b*x),x]
[Out]
((2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(24*a^2*C*d^2*f^2 - 6*a*b*d*f*(c*C*f + 4*B*d*f
+ C*d*(e + 2*f*x)) + b^2*(6*d*f*(B*c*f + 4*A*d*f + B*d*(e + 2*f*x)) + C*(-3*c^2
*f^2 + 2*c*d*f*(e + f*x) + d^2*(-3*e^2 + 2*e*f*x + 8*f^2*x^2)))))/(d^2*f^2) + 48
*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*Log[a + b*x] + (3*(-
16*a^3*C*d^3*f^3 + 8*a^2*b*d^2*f^2*(C*d*e + c*C*f + 2*B*d*f) + 2*a*b^2*d*f*(C*(d
*e - c*f)^2 - 4*d*f*(B*d*e + B*c*f + 2*A*d*f)) + b^3*(C*(d*e - c*f)^2*(d*e + c*f
) + 2*d*f*(-(B*(d*e - c*f)^2) + 4*A*d*f*(d*e + c*f))))*Log[d*e + c*f + 2*d*f*x +
2*Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]])/(d^(5/2)*f^(5/2)) - 48*(A*b^2 +
a*(-(b*B) + a*C))*Sqrt[b*c - a*d]*Sqrt[b*e - a*f]*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)])/(48*b^4)
_______________________________________________________________________________________
Maple [B] time = 0.062, size = 4227, normalized size = 9.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a),x)
[Out]
-1/48*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+
c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e
*d+2*b*c*e)/(b*x+a))*f^3*d^3*a^4*C*(f*d)^(1/2)-16*C*x^2*b^4*d^2*f^2*(f*d)^(1/2)*
(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+6*
C*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*e^2*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2
)^(1/2)*d^2*(f*d)^(1/2)+48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x
+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)
/(b*x+a))*f^3*d^3*a^2*A*b^2*(f*d)^(1/2)+48*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*
x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*a*d^3*f^3*A*b^3*((a^2*d*f-a*b*c*f
-a*b*d*e+b^2*c*e)/b^2)^(1/2)-24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2
)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*f^3*A*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*
e)/b^2)^(1/2)*d^2-24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/
2)+c*f+d*e)/(f*d)^(1/2))*d^3*e*A*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/
2)*f^2-48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2
*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*f^3*d^3
*a^3*B*b*(f*d)^(1/2)-48*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^
(1/2)+c*f+d*e)/(f*d)^(1/2))*a^2*d^3*f^3*B*b^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)
/b^2)^(1/2)+6*B*f^3*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2
)+c*f+d*e)/(f*d)^(1/2))*c^2*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d+
12*C*a*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f
*d)^(1/2))*c*e*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*f^2+12*C*a*
(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*c*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1
/2)*d*(f*d)^(1/2)*f^2+12*C*a*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*e*b^3*((a^2*d*f-a*b
*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d)^(1/2)*f-4*C*(d*f*x^2+c*f*x+d*e*x+c*e)
^(1/2)*c*e*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d*(f*d)^(1/2)*f-4*C
*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*c*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)
^(1/2)*d*(f*d)^(1/2)*f^2-4*C*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*e*b^4*((a^2*d*f-a
*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d)^(1/2)*f-48*ln((-2*a*d*f*x+b*c*f*x+b
*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)
^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*c*e*B*a*b^3*d^2*(f*d)^(1/2)*f^2+24*C*a*(d
*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2
)*d^2*(f*d)^(1/2)*f^2+48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c
*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(
b*x+a))*c*e*C*a^2*b^2*d^2*(f*d)^(1/2)*f^2-24*B*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*x
*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d)^(1/2)*f^2+48*ln((-2
*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a^2*c*f^3*B*b^2*d^2*(f*
d)^(1/2)+48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a
^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a^2*d
^3*e*B*b^2*(f*d)^(1/2)*f^2+24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*
(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*f^3*B*a*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*
e)/b^2)^(1/2)*d^2+24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/
2)+c*f+d*e)/(f*d)^(1/2))*d^3*e*B*a*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(
1/2)*f^2-12*B*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+
d*e)/(f*d)^(1/2))*c*e*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*f^2-
48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a^3*c*f^3*C*b*
d^2*(f*d)^(1/2)-48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1
/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a)
)*a^3*d^3*e*C*b*(f*d)^(1/2)*f^2-24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(
1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*f^3*C*a^2*b^2*((a^2*d*f-a*b*c*f-a*b*d*e
+b^2*c*e)/b^2)^(1/2)*d^2-24*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f
*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*d^3*e*C*a^2*b^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*
e)/b^2)^(1/2)*f^2-6*C*a*f^3*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f
*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^2*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^
(1/2)*d-6*C*a*d^3*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+
c*f+d*e)/(f*d)^(1/2))*e^2*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*f+3*
C*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(
1/2))*c^2*e*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d*f^2+3*C*ln(1/2*(
2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c*e^
2*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*f+48*(d*f*x^2+c*f*x+d*e*
x+c*e)^(1/2)*B*a*b^3*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d)^(1/
2)*f^2-12*B*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*c*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*d*(f*d)^(1/2)*f^2-12*B*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*e*b^4*((a
^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d)^(1/2)*f-48*(d*f*x^2+c*f*x+d
*e*x+c*e)^(1/2)*C*a^2*b^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*d^2*(f*d
)^(1/2)*f^2-48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*
((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*c*e)/(b*x+a))*a*
c*f^3*A*b^3*d^2*(f*d)^(1/2)-48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*(d*f*x^2+c*f*x+d
*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*e*d+2*b*
c*e)/(b*x+a))*a*d^3*e*A*b^3*(f*d)^(1/2)*f^2+48*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*
(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-
a*c*f-a*e*d+2*b*c*e)/(b*x+a))*c*e*A*b^4*d^2*(f*d)^(1/2)*f^2-3*C*f^3*ln(1/2*(2*d*
f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*c^3*b^4*
((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-3*C*d^3*ln(1/2*(2*d*f*x+2*(d*f*x^2
+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*e^3*b^4*((a^2*d*f-a*b*
c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+6*B*d^3*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c
*e)^(1/2)*(f*d)^(1/2)+c*f+d*e)/(f*d)^(1/2))*e^2*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^
2*c*e)/b^2)^(1/2)*f+48*ln(1/2*(2*d*f*x+2*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*(f*d)^(
1/2)+c*f+d*e)/(f*d)^(1/2))*a^3*d^3*f^3*C*b*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^
2)^(1/2)-48*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*A*b^4*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*
c*e)/b^2)^(1/2)*d^2*(f*d)^(1/2)*f^2+6*C*(d*f*x^2+c*f*x+d*e*x+c*e)^(1/2)*c^2*b^4*
((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*(f*d)^(1/2)*f^2)/(d*f*x^2+c*f*x+d*
e*x+c*e)^(1/2)/b^5/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/d^2/(f*d)^(1/2)
/f^2
_______________________________________________________________________________________
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)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [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)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a),x, algorithm="fricas")
[Out]
Timed out
_______________________________________________________________________________________
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((C*x**2+B*x+A)*(d*x+c)**(1/2)*(f*x+e)**(1/2)/(b*x+a),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.429151, 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)*sqrt(d*x + c)*sqrt(f*x + e)/(b*x + a),x, algorithm="giac")
[Out]
Done