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

Optimal. Leaf size=300 \[ \frac{x \sqrt{a+b x} \sqrt{a c-b c x} \left (a^2 (B f+C e)+4 A b^2 e\right )}{8 b^2}-\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (4 \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )-3 b^2 f x (3 C e-5 B f)\right )}{60 b^4 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+4 A b^2 e\right )}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x}}{5 b^2 f} \]

[Out]

((4*A*b^2*e + a^2*(C*e + B*f))*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/(8*b^2) - (C*S
qrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2*(a^2 - b^2*x^2))/(5*b^2*f) - (Sqrt[a
+ b*x]*Sqrt[a*c - b*c*x]*(4*(2*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(B*e + A*f))) - 3*
b^2*f*(3*C*e - 5*B*f)*x)*(a^2 - b^2*x^2))/(60*b^4*f) + (a^2*Sqrt[c]*(4*A*b^2*e +
a^2*(C*e + 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]])/(8*b^3*Sqrt[a^2*c - b^2*c*x^2])

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Rubi [A]  time = 0.865239, antiderivative size = 297, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{1}{8} x \sqrt{a+b x} \sqrt{a c-b c x} \left (\frac{a^2 (B f+C e)}{b^2}+4 A e\right )-\frac{\sqrt{a+b x} \left (a^2-b^2 x^2\right ) \sqrt{a c-b c x} \left (4 \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )-3 b^2 f x (3 C e-5 B f)\right )}{60 b^4 f}+\frac{a^2 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (a^2 (B f+C e)+4 A b^2 e\right )}{8 b^3 \sqrt{a^2 c-b^2 c x^2}}-\frac{C \sqrt{a+b x} \left (a^2-b^2 x^2\right ) (e+f x)^2 \sqrt{a c-b c x}}{5 b^2 f} \]

Antiderivative was successfully verified.

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

[Out]

((4*A*e + (a^2*(C*e + B*f))/b^2)*x*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])/8 - (C*Sqrt[
a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2*(a^2 - b^2*x^2))/(5*b^2*f) - (Sqrt[a + b*
x]*Sqrt[a*c - b*c*x]*(4*(2*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(B*e + A*f))) - 3*b^2*
f*(3*C*e - 5*B*f)*x)*(a^2 - b^2*x^2))/(60*b^4*f) + (a^2*Sqrt[c]*(4*A*b^2*e + a^2
*(C*e + 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]])/(8*b^3*Sqrt[a^2*c - b^2*c*x^2])

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Rubi in Sympy [A]  time = 93.5357, size = 286, normalized size = 0.95 \[ - \frac{C \sqrt{a + b x} \left (a^{2} - b^{2} x^{2}\right ) \left (e + f x\right )^{2} \sqrt{a c - b c x}}{5 b^{2} f} + \frac{a^{2} \sqrt{c} \sqrt{a + b x} \sqrt{a c - b c x} \left (4 A b^{2} e + B a^{2} f + C a^{2} e\right ) \operatorname{atan}{\left (\frac{b \sqrt{c} x}{\sqrt{a^{2} c - b^{2} c x^{2}}} \right )}}{8 b^{3} \sqrt{a^{2} c - b^{2} c x^{2}}} + \frac{x \sqrt{a + b x} \sqrt{a c - b c x} \left (4 A b^{2} e + B a^{2} f + C a^{2} e\right )}{8 b^{2}} - \frac{\sqrt{a + b x} \left (a^{2} - b^{2} x^{2}\right ) \sqrt{a c - b c x} \left (4 b^{2} e \left (5 B f - 3 C e\right ) + 3 b^{2} f x \left (5 B f - 3 C e\right ) + 4 f^{2} \left (5 A b^{2} + 2 C a^{2}\right )\right )}{60 b^{4} f} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-C*sqrt(a + b*x)*(a**2 - b**2*x**2)*(e + f*x)**2*sqrt(a*c - b*c*x)/(5*b**2*f) +
a**2*sqrt(c)*sqrt(a + b*x)*sqrt(a*c - b*c*x)*(4*A*b**2*e + B*a**2*f + C*a**2*e)*
atan(b*sqrt(c)*x/sqrt(a**2*c - b**2*c*x**2))/(8*b**3*sqrt(a**2*c - b**2*c*x**2))
+ x*sqrt(a + b*x)*sqrt(a*c - b*c*x)*(4*A*b**2*e + B*a**2*f + C*a**2*e)/(8*b**2)
- sqrt(a + b*x)*(a**2 - b**2*x**2)*sqrt(a*c - b*c*x)*(4*b**2*e*(5*B*f - 3*C*e)
+ 3*b**2*f*x*(5*B*f - 3*C*e) + 4*f**2*(5*A*b**2 + 2*C*a**2))/(60*b**4*f)

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Mathematica [A]  time = 0.428941, size = 178, normalized size = 0.59 \[ \frac{1}{8} \sqrt{c (a-b x)} \left (\frac{a^2 \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right ) \left (a^2 (B f+C e)+4 A b^2 e\right )}{b^3 \sqrt{a-b x}}-\frac{\sqrt{a+b x} \left (16 a^4 C f+a^2 b^2 (40 A f+5 B (8 e+3 f x)+C x (15 e+8 f x))-2 b^4 x (10 A (3 e+2 f x)+x (5 B (4 e+3 f x)+3 C x (5 e+4 f x)))\right )}{15 b^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c*(a - b*x)]*(-(Sqrt[a + b*x]*(16*a^4*C*f + a^2*b^2*(40*A*f + 5*B*(8*e + 3
*f*x) + C*x*(15*e + 8*f*x)) - 2*b^4*x*(10*A*(3*e + 2*f*x) + x*(5*B*(4*e + 3*f*x)
+ 3*C*x*(5*e + 4*f*x)))))/(15*b^4) + (a^2*(4*A*b^2*e + a^2*(C*e + B*f))*ArcTan[
(b*x)/(Sqrt[a - b*x]*Sqrt[a + b*x])])/(b^3*Sqrt[a - b*x])))/8

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Maple [B]  time = 0.018, size = 588, normalized size = 2. \[{\frac{1}{120\,{b}^{4}}\sqrt{bx+a}\sqrt{-c \left ( bx-a \right ) } \left ( 24\,C{x}^{4}{b}^{4}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+30\,B{x}^{3}{b}^{4}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+30\,C{x}^{3}{b}^{4}e\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+60\,eAc{a}^{2}\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ){b}^{4}+40\,A{x}^{2}{b}^{4}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+15\,{a}^{4}c\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ) Bf{b}^{2}+40\,B{x}^{2}{b}^{4}e\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+15\,{a}^{4}c\arctan \left ({\frac{\sqrt{{b}^{2}c}x}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}} \right ) Ce{b}^{2}-8\,C{x}^{2}{a}^{2}{b}^{2}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }+60\,eAx\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{{b}^{2}c}{b}^{4}-15\,{a}^{2}x\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }Bf\sqrt{{b}^{2}c}{b}^{2}-15\,{a}^{2}x\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }Ce\sqrt{{b}^{2}c}{b}^{2}-40\,A{a}^{2}{b}^{2}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }-40\,B{a}^{2}{b}^{2}e\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }-16\,C{a}^{4}f\sqrt{{b}^{2}c}\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) } \right ){\frac{1}{\sqrt{-c \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }}}{\frac{1}{\sqrt{{b}^{2}c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(b*x+a)^(1/2)*(-c*(b*x-a))^(1/2)*(24*C*x^4*b^4*f*(b^2*c)^(1/2)*(-c*(b^2*x^
2-a^2))^(1/2)+30*B*x^3*b^4*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+30*C*x^3*b^4
*e*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+60*e*A*c*a^2*arctan((b^2*c)^(1/2)*x/(-
c*(b^2*x^2-a^2))^(1/2))*b^4+40*A*x^2*b^4*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2
)+15*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*B*f*b^2+40*B*x^2*b^4
*e*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+15*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b
^2*x^2-a^2))^(1/2))*C*e*b^2-8*C*x^2*a^2*b^2*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(
1/2)+60*e*A*x*(-c*(b^2*x^2-a^2))^(1/2)*(b^2*c)^(1/2)*b^4-15*a^2*x*(-c*(b^2*x^2-a
^2))^(1/2)*B*f*(b^2*c)^(1/2)*b^2-15*a^2*x*(-c*(b^2*x^2-a^2))^(1/2)*C*e*(b^2*c)^(
1/2)*b^2-40*A*a^2*b^2*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-40*B*a^2*b^2*e*(b
^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-16*C*a^4*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))
^(1/2))/(-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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255654, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (B a^{4} b f +{\left (C a^{4} b + 4 \, A a^{2} b^{3}\right )} e\right )} \sqrt{-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b \sqrt{-c} x - a^{2} c\right ) + 2 \,{\left (24 \, C b^{4} f x^{4} - 40 \, B a^{2} b^{2} e + 30 \,{\left (C b^{4} e + B b^{4} f\right )} x^{3} + 8 \,{\left (5 \, B b^{4} e -{\left (C a^{2} b^{2} - 5 \, A b^{4}\right )} f\right )} x^{2} - 8 \,{\left (2 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f - 15 \,{\left (B a^{2} b^{2} f +{\left (C a^{2} b^{2} - 4 \, A b^{4}\right )} e\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{240 \, b^{4}}, \frac{15 \,{\left (B a^{4} b f +{\left (C a^{4} b + 4 \, A a^{2} b^{3}\right )} e\right )} \sqrt{c} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right ) +{\left (24 \, C b^{4} f x^{4} - 40 \, B a^{2} b^{2} e + 30 \,{\left (C b^{4} e + B b^{4} f\right )} x^{3} + 8 \,{\left (5 \, B b^{4} e -{\left (C a^{2} b^{2} - 5 \, A b^{4}\right )} f\right )} x^{2} - 8 \,{\left (2 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f - 15 \,{\left (B a^{2} b^{2} f +{\left (C a^{2} b^{2} - 4 \, A b^{4}\right )} e\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a}}{120 \, b^{4}}\right ] \]

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),x, algorithm="fricas")

[Out]

[1/240*(15*(B*a^4*b*f + (C*a^4*b + 4*A*a^2*b^3)*e)*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*(24*C*b^4*f*x^4 - 40*
B*a^2*b^2*e + 30*(C*b^4*e + B*b^4*f)*x^3 + 8*(5*B*b^4*e - (C*a^2*b^2 - 5*A*b^4)*
f)*x^2 - 8*(2*C*a^4 + 5*A*a^2*b^2)*f - 15*(B*a^2*b^2*f + (C*a^2*b^2 - 4*A*b^4)*e
)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^4, 1/120*(15*(B*a^4*b*f + (C*a^4*b + 4*
A*a^2*b^3)*e)*sqrt(c)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(b*x + a))) + (
24*C*b^4*f*x^4 - 40*B*a^2*b^2*e + 30*(C*b^4*e + B*b^4*f)*x^3 + 8*(5*B*b^4*e - (C
*a^2*b^2 - 5*A*b^4)*f)*x^2 - 8*(2*C*a^4 + 5*A*a^2*b^2)*f - 15*(B*a^2*b^2*f + (C*
a^2*b^2 - 4*A*b^4)*e)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^4]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x+e)*(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)*(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),x, algorithm="giac")

[Out]

Timed out