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

Optimal. Leaf size=363 \[ \frac{\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 e \left (f (B e-3 A f)+C e^2\right )\right )}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac{f \left (a^2-b^2 x^2\right ) \left (A+\frac{e (C e-B f)}{f^2}\right )}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac{\sqrt{a^2 c-b^2 c x^2} \left (A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 \left (2 a^2 C f^2+b^2 e (C e-3 B f)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} \left (a^2 f+b^2 e x\right )}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right )}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 1.38057, antiderivative size = 361, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{f \left (a^2-b^2 x^2\right ) \left (A+\frac{e (C e-B f)}{f^2}\right )}{2 \sqrt{a+b x} (e+f x)^2 \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )}+\frac{\left (a^2-b^2 x^2\right ) \left (2 a^2 f^2 (2 C e-B f)-b^2 \left (e f (B e-3 A f)+C e^3\right )\right )}{2 f \sqrt{a+b x} (e+f x) \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^2}+\frac{\sqrt{a^2 c-b^2 c x^2} \left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right ) \tan ^{-1}\left (\frac{\sqrt{c} \left (a^2 f+b^2 e x\right )}{\sqrt{a^2 c-b^2 c x^2} \sqrt{b^2 e^2-a^2 f^2}}\right )}{2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \left (b^2 e^2-a^2 f^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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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((C*x**2+B*x+A)/(f*x+e)**3/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.31546, size = 380, normalized size = 1.05 \[ \frac{\frac{(b x-a) \sqrt{a+b x} \left (a^2 f (f (A f+B (e+2 f x))-C e (3 e+4 f x))+b^2 e \left (-A f (4 e+3 f x)+B e (2 e+f x)+C e^2 x\right )\right )}{(e+f x)^2 \left (b^2 e^2-a^2 f^2\right )^2}+\frac{\sqrt{a-b x} \log (e+f x) \left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right )}{(b e-a f)^2 (a f+b e)^2 \sqrt{a^2 f^2-b^2 e^2}}-\frac{\sqrt{a-b x} \log \left (\sqrt{a-b x} \sqrt{a+b x} \sqrt{a^2 f^2-b^2 e^2}+a^2 f+b^2 e x\right ) \left (2 a^4 C f^2+A \left (a^2 b^2 f^2+2 b^4 e^2\right )+a^2 b^2 e (C e-3 B f)\right )}{(b e-a f)^2 (a f+b e)^2 \sqrt{a^2 f^2-b^2 e^2}}}{2 \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.075, size = 1848, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-3*B*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)
/f^2)^(1/2)*f)/(f*x+e))*x^2*a^2*b^2*c*e*f^3+2*A*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2
*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*b^4*c*e^4+A*a^2*f^4
*(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+B*x*b^2*e^2*f^2*(-c*(b
^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-4*C*x*a^2*e*f^3*(-c*(b^2*x^2-
a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+C*x*b^2*e^3*f*(-c*(b^2*x^2-a^2))^(1/
2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+A*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^
(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x^2*a^2*b^2*c*f^4+2*A*ln(2*(b^
2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x
+e))*x^2*b^4*c*e^2*f^2+4*A*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(
a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x*b^4*c*e^3*f+4*C*ln(2*(b^2*c*e*x+a^2*c*
f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x*a^4*c*e
*f^3+A*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2
)^(1/2)*f)/(f*x+e))*a^2*b^2*c*e^2*f^2-3*B*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a
^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*a^2*b^2*c*e^3*f-3*A*x*b^2
*e*f^3*(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+C*ln(2*(b^2*c*e*
x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x
^2*a^2*b^2*c*e^2*f^2+2*A*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^
2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x*a^2*b^2*c*e*f^3-6*B*ln(2*(b^2*c*e*x+a^2*
c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x*a^2*b
^2*c*e^2*f^2+2*C*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^
2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x*a^2*b^2*c*e^3*f+2*C*ln(2*(b^2*c*e*x+a^2*c*f+(-c*
(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*x^2*a^4*c*f^4+2
*C*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1
/2)*f)/(f*x+e))*a^4*c*e^2*f^2+C*ln(2*(b^2*c*e*x+a^2*c*f+(-c*(b^2*x^2-a^2))^(1/2)
*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)*f)/(f*x+e))*a^2*b^2*c*e^4+2*B*x*a^2*f^4*(-c*(b^
2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)-4*A*b^2*e^2*f^2*(-c*(b^2*x^2-a
^2))^(1/2)*(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+B*a^2*e*f^3*(-c*(b^2*x^2-a^2))^(1/2)*
(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)+2*B*b^2*e^3*f*(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f
^2-b^2*e^2)/f^2)^(1/2)-3*C*a^2*e^2*f^2*(-c*(b^2*x^2-a^2))^(1/2)*(c*(a^2*f^2-b^2*
e^2)/f^2)^(1/2))/c*(-c*(b*x-a))^(1/2)*(b*x+a)^(1/2)/(-c*(b^2*x^2-a^2))^(1/2)/(a*
f+b*e)/(a*f-b*e)/(c*(a^2*f^2-b^2*e^2)/f^2)^(1/2)/(a^2*f^2-b^2*e^2)/(f*x+e)^2/f

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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)/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*(f*x + e)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.95438, 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)/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)*(f*x + e)^3),x, algorithm="fricas")

[Out]

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

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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)/(f*x+e)**3/(b*x+a)**(1/2)/(-b*c*x+a*c)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 1.30891, 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(-b*c*x + a*c)*sqrt(b*x + a)*(f*x + e)^3),x, algorithm="giac")

[Out]

sage0*x