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

Optimal. Leaf size=364 \[ \frac{\sqrt{c+d x} \sqrt{e+f x} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right )}{b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right )}{b^3 \sqrt{b c-a d} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (4 a C d f+b (-2 B d f-c C f+C d e))}{b^3 \sqrt{d} f^{3/2}} \]

[Out]

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

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Rubi [A]  time = 2.62415, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{c+d x} \sqrt{e+f x} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right )}{b^2 f (b c-a d) (b e-a f)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c+d x} \sqrt{b e-a f}}{\sqrt{e+f x} \sqrt{b c-a d}}\right ) \left (4 a^3 C d f-a^2 b (2 B d f+3 c C f+5 C d e)+a b^2 (B c f+3 B d e+4 c C e)-b^3 (-A c f+A d e+2 B c e)\right )}{b^3 \sqrt{b c-a d} (b e-a f)^{3/2}}-\frac{(c+d x)^{3/2} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b (a+b x) (b c-a d) (b e-a f)}-\frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{d} \sqrt{e+f x}}\right ) (4 a C d f+b (-2 B d f-c C f+C d e))}{b^3 \sqrt{d} f^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 96.4609, size = 332, normalized size = 0.91 \[ \frac{C \sqrt{c + d x} \sqrt{e + f x}}{b^{2} f} + \frac{C \left (c f - d e\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{b^{2} \sqrt{d} f^{\frac{3}{2}}} + \frac{\left (c f - d e\right ) \left (A b^{2} - B a b + C a^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x} \sqrt{a f - b e}}{\sqrt{e + f x} \sqrt{a d - b c}} \right )}}{b^{2} \sqrt{a d - b c} \left (a f - b e\right )^{\frac{3}{2}}} + \frac{\sqrt{c + d x} \sqrt{e + f x} \left (A b^{2} - B a b + C a^{2}\right )}{b^{2} \left (a + b x\right ) \left (a f - b e\right )} + \frac{2 \sqrt{d} \left (B b - 2 C a\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{e + f x}}{\sqrt{f} \sqrt{c + d x}} \right )}}{b^{3} \sqrt{f}} - \frac{2 \left (B b - 2 C a\right ) \sqrt{a d - b c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x} \sqrt{a f - b e}}{\sqrt{e + f x} \sqrt{a d - b c}} \right )}}{b^{3} \sqrt{a f - b e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*sqrt(c + d*x)*sqrt(e + f*x)/(b**2*f) + C*(c*f - d*e)*atanh(sqrt(d)*sqrt(e + f*
x)/(sqrt(f)*sqrt(c + d*x)))/(b**2*sqrt(d)*f**(3/2)) + (c*f - d*e)*(A*b**2 - B*a*
b + C*a**2)*atanh(sqrt(c + d*x)*sqrt(a*f - b*e)/(sqrt(e + f*x)*sqrt(a*d - b*c)))
/(b**2*sqrt(a*d - b*c)*(a*f - b*e)**(3/2)) + sqrt(c + d*x)*sqrt(e + f*x)*(A*b**2
 - B*a*b + C*a**2)/(b**2*(a + b*x)*(a*f - b*e)) + 2*sqrt(d)*(B*b - 2*C*a)*atanh(
sqrt(d)*sqrt(e + f*x)/(sqrt(f)*sqrt(c + d*x)))/(b**3*sqrt(f)) - 2*(B*b - 2*C*a)*
sqrt(a*d - b*c)*atanh(sqrt(c + d*x)*sqrt(a*f - b*e)/(sqrt(e + f*x)*sqrt(a*d - b*
c)))/(b**3*sqrt(a*f - b*e))

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Mathematica [A]  time = 1.78906, size = 421, normalized size = 1.16 \[ \frac{\frac{\log (a+b x) \left (-4 a^3 C d f+a^2 b (2 B d f+3 c C f+5 C d e)-a b^2 (B c f+3 B d e+4 c C e)+b^3 (-A c f+A d e+2 B c e)\right )}{\sqrt{b c-a d} (b e-a f)^{3/2}}-\frac{\log \left (2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d} \sqrt{b e-a f}-a (c f+d e+2 d f x)+b (2 c e+c f x+d e x)\right ) \left (-4 a^3 C d f+a^2 b (2 B d f+3 c C f+5 C d e)-a b^2 (B c f+3 B d e+4 c C e)+b^3 (-A c f+A d e+2 B c e)\right )}{\sqrt{b c-a d} (b e-a f)^{3/2}}+2 b \sqrt{c+d x} \sqrt{e+f x} \left (\frac{a (b B-a C)-A b^2}{(a+b x) (b e-a f)}+\frac{C}{f}\right )+\frac{\log \left (2 \sqrt{d} \sqrt{f} \sqrt{c+d x} \sqrt{e+f x}+c f+d e+2 d f x\right ) (b (2 B d f+c C f-C d e)-4 a C d f)}{\sqrt{d} f^{3/2}}}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.063, size = 3670, normalized size = 10.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-2*A*b^4*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/
b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+2*C*x*b^4*e*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)+2*B*a*b^3*f*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-4*C*a
^2*b^2*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(
f*d)^(1/2)+2*C*a*b^3*e*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f
*x+e))^(1/2)*(f*d)^(1/2)+A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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)
)*x*b^4*c*f^2*(f*d)^(1/2)-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))*x*b^4*d*e^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2
)+A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b^3*c*f^2*(f*d)^(1/
2)-2*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*b^2*d*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-2*B*ln((-2*a*d*f*
x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b*d*f^2*(f*d)^(1/2)+B*ln((-2*a*d*f
*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b^2*c*f^2*(f*d)^(1/2)+4*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^3*b*d*f^2
*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-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*b^2*c*f^2*((a^2*d*f-a*b*c*f-a*b
*d*e+b^2*c*e)/b^2)^(1/2)-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^3*d*e^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
-3*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b*c*f^2*(f*d)^(1
/2)-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)
)*x*a*b^3*c*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+3*B*ln((-2*a*d*f*x
+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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))*x*a*b^3*d*e*f*(f*d)^(1/2)-3*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))*x*a*b^3*d*e*f
*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-5*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*
x+2*((d*x+c)*(f*x+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))*x*a^2*b^2*d*e*f*(f*d)^(1/2)+4*C*ln((-2*a*d*f*x+b*c*f*x
+b*d*e*x+2*((d*x+c)*(f*x+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))*x*a*b^3*c*e*f*(f*d)^(1/2)+4*C*ln((-2*a*d*f*x+b*
c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b^2*c*e*f*(f*d)^(1/2)-2*C*x*a*b^3*f*((
a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*(f*d)^(1/2)-
A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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))*x*b^4*d*e*f*(f*d)^(1/2)
-2*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))
*x*a*b^3*d*f^2*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+2*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))*x*b^4*d*e*f*((a^2*d
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-2*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*
x+c)*(f*x+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))*x*a^2*b^2*d*f^2*(f*d)^(1/2)+B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2
*((d*x+c)*(f*x+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))*x*a*b^3*c*f^2*(f*d)^(1/2)-2*B*ln((-2*a*d*f*x+b*c*f*x+b*d*
e*x+2*((d*x+c)*(f*x+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))*x*b^4*c*e*f*(f*d)^(1/2)+4*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))*x*a^2*b^2*d*f^2*((a^2*d*f-a*
b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+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))*x*b^4*c*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b
^2)^(1/2)+4*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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))*x*a^3*b*d*f
^2*(f*d)^(1/2)-3*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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))*x*a^2*
b^2*c*f^2*(f*d)^(1/2)-A*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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
*b^3*d*e*f*(f*d)^(1/2)+2*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^3*d*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)
+3*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b^2*d*e*f*(f*d)^
(1/2)-2*B*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((d*x+c)*(f*x+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*b^3*c*e*f*(f*
d)^(1/2)-3*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*b^2*d*e*f*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)+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^3*c*e*f*(
(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)-5*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+
2*((d*x+c)*(f*x+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*b*d*e*f*(f*d)^(1/2)+4*C*ln((-2*a*d*f*x+b*c*f*x+b*d*e
*x+2*((d*x+c)*(f*x+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^4*d*f^2*(f*d)^(1/2))/((d*x+c)*(f*x+e))^(1/2)/b^4/(a
*f-b*e)/(f*d)^(1/2)/((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)/f/(b*x+a)

_______________________________________________________________________________________

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)/((b*x + a)^2*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)/((b*x + a)^2*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.667036, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x

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