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

Optimal. Leaf size=501 \[ -\frac{\left (a^2-b^2 x^2\right ) (e+f x)^2 \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{60 b^4 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+a^2 \left (3 a^2 f^2 (B f+3 C e)+4 b^2 e^2 (3 B f+C e)\right )\right )}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (a^2-b^2 x^2\right ) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^4}{5 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left (a^2-b^2 x^2\right ) \left (b^2 f x \left (a^2 f^2 (45 B f+71 C e)-2 b^2 e \left (3 C e^2-5 f (10 A f+3 B e)\right )\right )+4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )+b^4 \left (-e^2\right ) \left (3 C e^2-5 f (16 A f+3 B e)\right )\right )\right )}{120 b^6 f \sqrt{a+b x} \sqrt{a c-b c x}} \]

[Out]

-((16*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(3*B*e + 4*A*f)))*(e + f*x)^2*(a^2 - b^2*x^
2))/(60*b^4*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((C*e - 5*B*f)*(e + f*x)^3*(a^2
- b^2*x^2))/(20*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*(e + f*x)^4*(a^2 -
b^2*x^2))/(5*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(16*a^4*C*f^4 + 4*a^2*
b^2*f^2*(13*C*e^2 + 5*f*(3*B*e + A*f)) - b^4*e^2*(3*C*e^2 - 5*f*(3*B*e + 16*A*f)
)) + b^2*f*(a^2*f^2*(71*C*e + 45*B*f) - 2*b^2*e*(3*C*e^2 - 5*f*(3*B*e + 10*A*f))
)*x)*(a^2 - b^2*x^2))/(120*b^6*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((4*A*(2*b^4
*e^3 + 3*a^2*b^2*e*f^2) + a^2*(3*a^2*f^2*(3*C*e + B*f) + 4*b^2*e^2*(C*e + 3*B*f)
))*Sqrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^5
*Sqrt[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

_______________________________________________________________________________________

Rubi [A]  time = 2.67029, antiderivative size = 496, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (a^2-b^2 x^2\right ) (e+f x)^2 \left (-\frac{16 a^2 C f^2}{b^2}-5 f (4 A f+3 B e)+3 C e^2\right )}{60 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\left (a^2-b^2 x^2\right ) (e+f x)^3 (C e-5 B f)}{20 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{C \left (a^2-b^2 x^2\right ) (e+f x)^4}{5 b^2 f \sqrt{a+b x} \sqrt{a c-b c x}}-\frac{\left (a^2-b^2 x^2\right ) \left (b^2 f x \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )+4 \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )+b^4 \left (-e^2\right ) \left (3 C e^2-5 f (16 A f+3 B e)\right )\right )\right )}{120 b^6 f \sqrt{a+b x} \sqrt{a c-b c x}}+\frac{\sqrt{a^2 c-b^2 c x^2} \tan ^{-1}\left (\frac{b \sqrt{c} x}{\sqrt{a^2 c-b^2 c x^2}}\right ) \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right )}{8 b^5 \sqrt{c} \sqrt{a+b x} \sqrt{a c-b c x}} \]

Antiderivative was successfully verified.

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

[Out]

((3*C*e^2 - (16*a^2*C*f^2)/b^2 - 5*f*(3*B*e + 4*A*f))*(e + f*x)^2*(a^2 - b^2*x^2
))/(60*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((C*e - 5*B*f)*(e + f*x)^3*(a^2
- b^2*x^2))/(20*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - (C*(e + f*x)^4*(a^2 - b
^2*x^2))/(5*b^2*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) - ((4*(16*a^4*C*f^4 + 4*a^2*b
^2*f^2*(13*C*e^2 + 5*f*(3*B*e + A*f)) - b^4*e^2*(3*C*e^2 - 5*f*(3*B*e + 16*A*f))
) + b^2*f*(a^2*f^2*(71*C*e + 45*B*f) - b^2*(6*C*e^3 - 10*e*f*(3*B*e + 10*A*f)))*
x)*(a^2 - b^2*x^2))/(120*b^6*f*Sqrt[a + b*x]*Sqrt[a*c - b*c*x]) + ((3*a^4*f^2*(3
*C*e + B*f) + 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^2*e*f^2))*S
qrt[a^2*c - b^2*c*x^2]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(8*b^5*Sqr
t[c]*Sqrt[a + b*x]*Sqrt[a*c - b*c*x])

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.901538, size = 289, normalized size = 0.58 \[ \frac{\frac{(b x-a) \sqrt{a+b x} \left (64 a^4 C f^3+a^2 b^2 f \left (5 f (16 A f+48 B e+9 B f x)+C \left (240 e^2+135 e f x+32 f^2 x^2\right )\right )+2 b^4 \left (10 A f \left (18 e^2+9 e f x+2 f^2 x^2\right )+15 B \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )+3 C x \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )\right )\right )}{15 b^6}+\frac{\sqrt{a-b x} \tan ^{-1}\left (\frac{b x}{\sqrt{a-b x} \sqrt{a+b x}}\right ) \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right )}{b^5}}{8 \sqrt{c (a-b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(((-a + b*x)*Sqrt[a + b*x]*(64*a^4*C*f^3 + a^2*b^2*f*(5*f*(48*B*e + 16*A*f + 9*B
*f*x) + C*(240*e^2 + 135*e*f*x + 32*f^2*x^2)) + 2*b^4*(10*A*f*(18*e^2 + 9*e*f*x
+ 2*f^2*x^2) + 15*B*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + 3*C*x*(10*e^3
+ 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3))))/(15*b^6) + ((3*a^4*f^2*(3*C*e + B*f)
+ 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^2*e*f^2))*Sqrt[a - b*x
]*ArcTan[(b*x)/(Sqrt[a - b*x]*Sqrt[a + b*x])])/b^5)/(8*Sqrt[c*(a - b*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.04, size = 965, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^3*(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)/c*(-24*C*x^4*b^4*f^3*(b^2*c)^(1/2)*(-c*(b
^2*x^2-a^2))^(1/2)-30*B*x^3*b^4*f^3*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-90*C*
x^3*b^4*e*f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+180*f^2*A*e*c*a^2*arctan((b
^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b^4+120*c*arctan((b^2*c)^(1/2)*x/(-c*(b^
2*x^2-a^2))^(1/2))*A*e^3*b^6-40*A*x^2*b^4*f^3*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(
1/2)+45*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*B*f^3*b^2+180*B*f
*e^2*c*a^2*arctan((b^2*c)^(1/2)*x/(-c*(b^2*x^2-a^2))^(1/2))*b^4-120*B*x^2*b^4*e*
f^2*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)+135*a^4*c*arctan((b^2*c)^(1/2)*x/(-c*
(b^2*x^2-a^2))^(1/2))*C*e*f^2*b^2+60*e^3*C*c*a^2*arctan((b^2*c)^(1/2)*x/(-c*(b^2
*x^2-a^2))^(1/2))*b^4-32*C*x^2*a^2*b^2*f^3*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2
)-120*C*x^2*b^4*e^2*f*(b^2*c)^(1/2)*(-c*(b^2*x^2-a^2))^(1/2)-180*f^2*A*e*x*(-c*(
b^2*x^2-a^2))^(1/2)*b^4*(b^2*c)^(1/2)-45*a^2*x*(-c*(b^2*x^2-a^2))^(1/2)*B*f^3*b^
2*(b^2*c)^(1/2)-180*B*f*e^2*x*(-c*(b^2*x^2-a^2))^(1/2)*b^4*(b^2*c)^(1/2)-135*a^2
*x*(-c*(b^2*x^2-a^2))^(1/2)*C*e*f^2*b^2*(b^2*c)^(1/2)-60*e^3*C*x*(-c*(b^2*x^2-a^
2))^(1/2)*b^4*(b^2*c)^(1/2)-80*a^2*(-c*(b^2*x^2-a^2))^(1/2)*A*f^3*b^2*(b^2*c)^(1
/2)-360*(-c*(b^2*x^2-a^2))^(1/2)*A*e^2*f*b^4*(b^2*c)^(1/2)-240*a^2*(-c*(b^2*x^2-
a^2))^(1/2)*B*e*f^2*b^2*(b^2*c)^(1/2)-120*(-c*(b^2*x^2-a^2))^(1/2)*B*e^3*b^4*(b^
2*c)^(1/2)-64*a^4*(-c*(b^2*x^2-a^2))^(1/2)*C*f^3*(b^2*c)^(1/2)-240*a^2*(-c*(b^2*
x^2-a^2))^(1/2)*C*e^2*f*b^2*(b^2*c)^(1/2))/(-c*(b^2*x^2-a^2))^(1/2)/b^6/(b^2*c)^
(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.288362, size = 1, normalized size = 0. \[ \left [-\frac{2 \,{\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \,{\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \,{\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \,{\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \,{\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} +{\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \,{\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \,{\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a} \sqrt{-c} - 15 \,{\left (12 \, B a^{2} b^{3} c e^{2} f + 3 \, B a^{4} b c f^{3} + 4 \,{\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} c e^{3} + 3 \,{\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} c e f^{2}\right )} \log \left (2 \, \sqrt{-b c x + a c} \sqrt{b x + a} b x +{\left (2 \, b^{2} x^{2} - a^{2}\right )} \sqrt{-c}\right )}{240 \, b^{6} \sqrt{-c} c}, -\frac{{\left (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \,{\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \,{\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \,{\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \,{\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} +{\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \,{\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \,{\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt{-b c x + a c} \sqrt{b x + a} \sqrt{c} - 15 \,{\left (12 \, B a^{2} b^{3} c e^{2} f + 3 \, B a^{4} b c f^{3} + 4 \,{\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} c e^{3} + 3 \,{\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} c e f^{2}\right )} \arctan \left (\frac{b \sqrt{c} x}{\sqrt{-b c x + a c} \sqrt{b x + a}}\right )}{120 \, b^{6} c^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*(f*x + e)^3/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)),x, algorithm="fricas")

[Out]

[-1/240*(2*(24*C*b^4*f^3*x^4 + 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^
2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*a^2*b^2)*f^3 + 30*(3*C*b^4*e*f^2 + B*
b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^4)*f^3)
*x^2 + 15*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A
*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(-c) - 15*(12*B*a^2*b^3*c*e
^2*f + 3*B*a^4*b*c*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*c*e^3 + 3*(3*C*a^4*b + 4*A*a^2*
b^3)*c*e*f^2)*log(2*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*b*x + (2*b^2*x^2 - a^2)*sqr
t(-c)))/(b^6*sqrt(-c)*c), -1/120*((24*C*b^4*f^3*x^4 + 120*B*b^4*e^3 + 240*B*a^2*
b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*a^2*b^2)*f^3 +
30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*
a^2*b^2 + 5*A*b^4)*f^3)*x^2 + 15*(4*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3
+ 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a)*sqrt(c)
- 15*(12*B*a^2*b^3*c*e^2*f + 3*B*a^4*b*c*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*c*e^3 + 3
*(3*C*a^4*b + 4*A*a^2*b^3)*c*e*f^2)*arctan(b*sqrt(c)*x/(sqrt(-b*c*x + a*c)*sqrt(
b*x + a))))/(b^6*c^(3/2))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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((C*x^2 + B*x + A)*(f*x + e)^3/(sqrt(-b*c*x + a*c)*sqrt(b*x + a)),x, algorithm="giac")

[Out]

Timed out