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

Optimal. Leaf size=451 \[ -\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )+8 \left (2 a^2 f^2 (B f+2 C e)-b^2 e \left (C e^2-2 f (5 A f+B e)\right )\right )\right )}{120 b^4 f}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^4}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+a^2 \left (a^2 C f^2+2 b^2 e (2 B f+C e)\right )\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x}}{6 b^2 f} \]

[Out]

((2*A*(4*b^4*e^2 + a^2*b^2*f^2) + a^2*(a^2*C*f^2 + 2*b^2*e*(C*e + 2*B*f)))*x*Sqr
t[a + b*x]*Sqrt[a*c - b*c*x])/(16*b^4) + ((C*e - 2*B*f)*Sqrt[a + b*x]*Sqrt[a*c -
b*c*x]*(e + f*x)^2*(a^2 - b^2*x^2))/(10*b^2*f) - (C*Sqrt[a + b*x]*Sqrt[a*c - b*
c*x]*(e + f*x)^3*(a^2 - b^2*x^2))/(6*b^2*f) - (Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(
8*(2*a^2*f^2*(2*C*e + B*f) - b^2*e*(C*e^2 - 2*f*(B*e + 5*A*f))) + 3*f*(5*a^2*C*f
^2 - b^2*(2*C*e^2 - 2*f*(2*B*e + 5*A*f)))*x)*(a^2 - b^2*x^2))/(120*b^4*f) + (a^2
*Sqrt[c]*(2*A*(4*b^4*e^2 + a^2*b^2*f^2) + a^2*(a^2*C*f^2 + 2*b^2*e*(C*e + 2*B*f)
))*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]]
)/(16*b^5*Sqrt[a^2*c - b^2*c*x^2])

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Rubi [A]  time = 2.0392, antiderivative size = 450, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175 \[ -\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (8 \left (2 a^2 f^2 (B f+2 C e)-\frac{1}{8} b^2 \left (8 C e^3-16 e f (5 A f+B e)\right )\right )+3 f x \left (5 a^2 C f^2-b^2 \left (2 C e^2-2 f (5 A f+2 B e)\right )\right )\right )}{120 b^4 f}+\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x} (C e-2 B f)}{10 b^2 f}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^3 \sqrt{a c-b c x}}{6 b^2 f}+\frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (a^4 C f^2+2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)\right )}{16 b^4}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^4 C f^2+2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)\right )}{16 b^5 \sqrt{a^2 c-b^2 c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*f^2))*x*Sqrt[
a + b*x]*Sqrt[a*c - b*c*x])/(16*b^4) + ((C*e - 2*B*f)*Sqrt[a + b*x]*Sqrt[a*c - b
*c*x]*(e + f*x)^2*(a^2 - b^2*x^2))/(10*b^2*f) - (C*Sqrt[a + b*x]*Sqrt[a*c - b*c*
x]*(e + f*x)^3*(a^2 - b^2*x^2))/(6*b^2*f) - (Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(8*
(2*a^2*f^2*(2*C*e + B*f) - (b^2*(8*C*e^3 - 16*e*f*(B*e + 5*A*f)))/8) + 3*f*(5*a^
2*C*f^2 - b^2*(2*C*e^2 - 2*f*(2*B*e + 5*A*f)))*x)*(a^2 - b^2*x^2))/(120*b^4*f) +
(a^2*Sqrt[c]*(a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*
f^2))*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^
2]])/(16*b^5*Sqrt[a^2*c - b^2*c*x^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((f*x+e)**2*(C*x**2+B*x+A)*(b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.687751, size = 281, normalized size = 0.62 \[ \frac{1}{16} \sqrt{c (a-b x)} \left (\frac{a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right ) \left (a^4 C f^2+2 A \left (a^2 b^2 f^2+4 b^4 e^2\right )+2 a^2 b^2 e (2 B f+C e)\right )}{b^5 \sqrt{a-b x}}-\frac{\sqrt{a+b x} \left (a^4 f (32 B f+64 C e+15 C f x)+2 a^2 b^2 \left (5 A f (16 e+3 f x)+B \left (40 e^2+30 e f x+8 f^2 x^2\right )+C x \left (15 e^2+16 e f x+5 f^2 x^2\right )\right )-4 b^4 x \left (5 A \left (6 e^2+8 e f x+3 f^2 x^2\right )+x \left (2 B \left (10 e^2+15 e f x+6 f^2 x^2\right )+C x \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )\right )}{15 b^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c*(a - b*x)]*(-(Sqrt[a + b*x]*(a^4*f*(64*C*e + 32*B*f + 15*C*f*x) + 2*a^2*
b^2*(5*A*f*(16*e + 3*f*x) + C*x*(15*e^2 + 16*e*f*x + 5*f^2*x^2) + B*(40*e^2 + 30
*e*f*x + 8*f^2*x^2)) - 4*b^4*x*(5*A*(6*e^2 + 8*e*f*x + 3*f^2*x^2) + x*(2*B*(10*e
^2 + 15*e*f*x + 6*f^2*x^2) + C*x*(15*e^2 + 24*e*f*x + 10*f^2*x^2)))))/(15*b^4) +
(a^2*(a^4*C*f^2 + 2*a^2*b^2*e*(C*e + 2*B*f) + 2*A*(4*b^4*e^2 + a^2*b^2*f^2))*Ar
cTan[(b*x)/(Sqrt[a - b*x]*Sqrt[a + b*x])])/(b^5*Sqrt[a - b*x])))/16

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Maple [B]  time = 0.024, size = 987, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/240*(b*x+a)^(1/2)*(-c*(b*x-a))^(1/2)*(-32*C*x^2*a^2*b^2*e*f*(b^2*c)^(1/2)*(-c*
(b^2*x^2-a^2))^(1/2)+15*C*f^2*a^6*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1
/2))-60*a^2*x*(-c*(b^2*x^2-a^2))^(1/2)*B*e*f*(b^2*c)^(1/2)*b^2-32*B*a^4*f^2*(b^2
*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-30*a^2*x*(-c*(b^2*x^2-a^2))^(1/2)*A*f^2*(b^2*
c)^(1/2)*b^2-30*a^2*x*(-c*(b^2*x^2-a^2))^(1/2)*C*e^2*(b^2*c)^(1/2)*b^2+160*A*x^2
*b^4*e*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-16*B*x^2*a^2*b^2*f^2*(b^2*c)^(1/
2)*(-c*(b^2*x^2-a^2))^(1/2)-160*A*a^2*b^2*e*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(
1/2)+60*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*B*e*f*b^2+96*C*x^
4*b^4*e*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+120*B*x^3*b^4*e*f*(b^2*c)^(1/2)
*(-c*(b^2*x^2-a^2))^(1/2)-10*C*x^3*a^2*b^2*f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^
(1/2)+120*A*e^2*x*(-c*(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)*b^4-15*C*f^2*a^4*x*(-c*
(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)+40*C*x^5*b^4*f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a
^2))^(1/2)+48*B*x^4*b^4*f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+60*A*x^3*b^4*
f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+60*C*x^3*b^4*e^2*(b^2*c)^(1/2)*(-c*(b
^2*x^2-a^2))^(1/2)+80*B*x^2*b^4*e^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-80*B*
a^2*b^2*e^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-64*C*a^4*e*f*(b^2*c)^(1/2)*(-
c*(b^2*x^2-a^2))^(1/2)+30*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))
*A*f^2*b^2+120*A*e^2*c*a^2*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b^4+
30*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*C*e^2*b^2)/(-c*(b^2*x^
2-a^2))^(1/2)/(b^2*c)^(1/2)/b^4

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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(sqrt(-b*c*x + a*c)*(C*x^2 + B*x + A)*sqrt(b*x + a)*(f*x + e)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283302, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (4 \, B a^{4} b^{2} e f + 2 \,{\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} +{\left (C a^{6} + 2 \, A a^{4} b^{2}\right )} f^{2}\right )} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \,{\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \,{\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f -{\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \,{\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \,{\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \,{\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \,{\left (4 \, B a^{2} b^{3} e f + 2 \,{\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} +{\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{480 \, b^{5}}, \frac{15 \,{\left (4 \, B a^{4} b^{2} e f + 2 \,{\left (C a^{4} b^{2} + 4 \, A a^{2} b^{4}\right )} e^{2} +{\left (C a^{6} + 2 \, A a^{4} b^{2}\right )} f^{2}\right )} \sqrt{c} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) +{\left (40 \, C b^{5} f^{2} x^{5} - 80 \, B a^{2} b^{3} e^{2} - 32 \, B a^{4} b f^{2} + 48 \,{\left (2 \, C b^{5} e f + B b^{5} f^{2}\right )} x^{4} + 10 \,{\left (6 \, C b^{5} e^{2} + 12 \, B b^{5} e f -{\left (C a^{2} b^{3} - 6 \, A b^{5}\right )} f^{2}\right )} x^{3} - 32 \,{\left (2 \, C a^{4} b + 5 \, A a^{2} b^{3}\right )} e f + 16 \,{\left (5 \, B b^{5} e^{2} - B a^{2} b^{3} f^{2} - 2 \,{\left (C a^{2} b^{3} - 5 \, A b^{5}\right )} e f\right )} x^{2} - 15 \,{\left (4 \, B a^{2} b^{3} e f + 2 \,{\left (C a^{2} b^{3} - 4 \, A b^{5}\right )} e^{2} +{\left (C a^{4} b + 2 \, A a^{2} b^{3}\right )} f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{240 \, b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/480*(15*(4*B*a^4*b^2*e*f + 2*(C*a^4*b^2 + 4*A*a^2*b^4)*e^2 + (C*a^6 + 2*A*a^4
*b^2)*f^2)*sqrt(-c)*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*sqrt(
-c)*x - a^2*c) + 2*(40*C*b^5*f^2*x^5 - 80*B*a^2*b^3*e^2 - 32*B*a^4*b*f^2 + 48*(2
*C*b^5*e*f + B*b^5*f^2)*x^4 + 10*(6*C*b^5*e^2 + 12*B*b^5*e*f - (C*a^2*b^3 - 6*A*
b^5)*f^2)*x^3 - 32*(2*C*a^4*b + 5*A*a^2*b^3)*e*f + 16*(5*B*b^5*e^2 - B*a^2*b^3*f
^2 - 2*(C*a^2*b^3 - 5*A*b^5)*e*f)*x^2 - 15*(4*B*a^2*b^3*e*f + 2*(C*a^2*b^3 - 4*A
*b^5)*e^2 + (C*a^4*b + 2*A*a^2*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^
5, 1/240*(15*(4*B*a^4*b^2*e*f + 2*(C*a^4*b^2 + 4*A*a^2*b^4)*e^2 + (C*a^6 + 2*A*a
^4*b^2)*f^2)*sqrt(c)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(b*x + a))) + (4
0*C*b^5*f^2*x^5 - 80*B*a^2*b^3*e^2 - 32*B*a^4*b*f^2 + 48*(2*C*b^5*e*f + B*b^5*f^
2)*x^4 + 10*(6*C*b^5*e^2 + 12*B*b^5*e*f - (C*a^2*b^3 - 6*A*b^5)*f^2)*x^3 - 32*(2
*C*a^4*b + 5*A*a^2*b^3)*e*f + 16*(5*B*b^5*e^2 - B*a^2*b^3*f^2 - 2*(C*a^2*b^3 - 5
*A*b^5)*e*f)*x^2 - 15*(4*B*a^2*b^3*e*f + 2*(C*a^2*b^3 - 4*A*b^5)*e^2 + (C*a^4*b
+ 2*A*a^2*b^3)*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^5]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- c \left (- a + b x\right )} \sqrt{a + b x} \left (e + f x\right )^{2} \left (A + B x + C x^{2}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(e + f*x)**2*(A + B*x + C*x**2), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out