\(\int \frac {(f+g x) (A+B x+C x^2)}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) [8]

Optimal result

Integrand size = 39, antiderivative size = 646 \[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \left (C d^2-e (B d-A e)\right ) (e f-d g) \sqrt {a+b x+c x^2}}{e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2}}{3 c e^2}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 b C e^2 (b d-a e) g-c e (3 b d (C e f-C d g+B e g)-a e (3 C e f-5 C d g+3 B e g))+c^2 \left (C \left (6 d^2 e f-8 d^3 g\right )-3 e (B d (e f-2 d g)-A e (e f-d g))\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (C e (b d-a e) g-2 c C d (3 e f-4 d g)+3 c e (B e f-2 B d g+A e g)) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c^2 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2*(C*d^2-e*(-A*e+B*d))*(-d*g+e*f)*(c*x^2+b*x+a)^(1/2)/e^2/(a*e^2-b*d*e+c* 
d^2)/(e*x+d)^(1/2)+2/3*C*g*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^2+1/3*2^( 
1/2)*(-4*a*c+b^2)^(1/2)*(2*b*C*e^2*(-a*e+b*d)*g-c*e*(3*b*d*(B*e*g-C*d*g+C* 
e*f)-a*e*(3*B*e*g-5*C*d*g+3*C*e*f))+c^2*(C*(-8*d^3*g+6*d^2*e*f)-3*e*(B*d*( 
-2*d*g+e*f)-A*e*(-d*g+e*f))))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2) 
)^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*( 
-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^2/e^3/(a*e^ 
2-b*d*e+c*d^2)/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(c*x^2+b 
*x+a)^(1/2)+2/3*2^(1/2)*(-4*a*c+b^2)^(1/2)*(C*e*(-a*e+b*d)*g-2*c*C*d*(-4*d 
*g+3*e*f)+3*c*e*(A*e*g-2*B*d*g+B*e*f))*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^( 
1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2* 
c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-( 
b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^2/e^3/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.51 (sec) , antiderivative size = 11109, normalized size of antiderivative = 17.20 \[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 3.29 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {2181, 27, 2184, 27, 1269, 1172, 321, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2184

\(\displaystyle -\frac {\frac {2 \int -\frac {b^2 C e g d^2+b c (C d (3 e f-4 d g)-3 e (B e f-B d g+A e g)) d+e \left (3 A c e (c d f+a e g)-a \left (a C g e^2-3 B c (e f-d g) e+c C d (3 e f-2 d g)\right )\right )+\left (\left (2 C d^2 (3 e f-4 d g)-3 e (B d (e f-2 d g)-A e (e f-d g))\right ) c^2-e (3 b d (C e f-C d g+B e g)-a e (3 C e f-5 C d g+3 B e g)) c+2 b C e^2 (b d-a e) g\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}-\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{3 c e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g) \left (C d^2-e (B d-A e)\right )}{e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {b^2 C e g d^2+b c (C d (3 e f-4 d g)-3 e (B e f-B d g+A e g)) d+e \left (3 A c e (c d f+a e g)-a \left (a C g e^2-3 B c (e f-d g) e+c C d (3 e f-2 d g)\right )\right )+\left (\left (C \left (6 d^2 e f-8 d^3 g\right )-3 e (B d (e f-2 d g)-A e (e f-d g))\right ) c^2-e (3 b d (C e f-C d g+B e g)-a e (3 C e f-5 C d g+3 B e g)) c+2 b C e^2 (b d-a e) g\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}-\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{3 c e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g) \left (C d^2-e (B d-A e)\right )}{e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {-\frac {\frac {\left (-c e (3 b d (B e g-C d g+C e f)-a e (3 B e g-5 C d g+3 C e f))+2 b C e^2 g (b d-a e)+c^2 \left (2 C d^2 (3 e f-4 d g)-3 e (B d (e f-2 d g)-A e (e f-d g))\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (a e^2-b d e+c d^2\right ) (C e g (b d-a e)+3 c e (A e g-2 B d g+B e f)-2 c C d (3 e f-4 d g)) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{3 c e^2}-\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{3 c e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g) \left (C d^2-e (B d-A e)\right )}{e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2 \left (C d^2-e (B d-A e)\right ) \sqrt {c x^2+b x+a} (e f-d g)}{e^2 \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {-\frac {2 C \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a} g}{3 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) (C e (b d-a e) g-2 c C d (3 e f-4 d g)+3 c e (B e f-2 B d g+A e g)) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C d^2 (3 e f-4 d g)-3 e (B d (e f-2 d g)-A e (e f-d g))\right ) c^2-e (3 b d (C e f-C d g+B e g)-a e (3 C e f-5 C d g+3 B e g)) c+2 b C e^2 (b d-a e) g\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}}{3 c e^2}}{c d^2-b e d+a e^2}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 b d (B e g-C d g+C e f)-a e (3 B e g-5 C d g+3 C e f))+2 b C e^2 g (b d-a e)+c^2 \left (2 C d^2 (3 e f-4 d g)-3 e (B d (e f-2 d g)-A e (e f-d g))\right )\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (C e g (b d-a e)+3 c e (A e g-2 B d g+B e f)-2 c C d (3 e f-4 d g))}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c e^2}-\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{3 c e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g) \left (C d^2-e (B d-A e)\right )}{e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) \left (-c e (3 b d (B e g-C d g+C e f)-a e (3 B e g-5 C d g+3 C e f))+2 b C e^2 g (b d-a e)+c^2 \left (2 C d^2 (3 e f-4 d g)-3 e (B d (e f-2 d g)-A e (e f-d g))\right )\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right ) (C e g (b d-a e)+3 c e (A e g-2 B d g+B e f)-2 c C d (3 e f-4 d g))}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{3 c e^2}-\frac {2 C g \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}{3 c e^2}}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (e f-d g) \left (C d^2-e (B d-A e)\right )}{e^2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1191\) vs. \(2(588)=1176\).

Time = 10.05 (sec) , antiderivative size = 1192, normalized size of antiderivative = 1.85

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1192\)
risch	\(\text {Expression too large to display}\)	\(1746\)
default	\(\text {Expression too large to display}\)	\(15929\)

Input:

int((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETU 
RNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2*(c*e*x^ 
2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)/e^3*(A*d*e^2*g-A*e^3*f-B*d^2*e*g+B*d*e^2* 
f+C*d^3*g-C*d^2*e*f)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2/3*C*g/e^2/c*(c* 
e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*((A*e^2*g-B*d*e*g+B*e^2*f+C 
*d^2*g-C*d*e*f)/e^3+1/e^3*(b*e-c*d)*(A*d*e^2*g-A*e^3*f-B*d^2*e*g+B*d*e^2*f 
+C*d^3*g-C*d^2*e*f)/(a*e^2-b*d*e+c*d^2)-b/e^2/(a*e^2-b*d*e+c*d^2)*(A*d*e^2 
*g-A*e^3*f-B*d^2*e*g+B*d*e^2*f+C*d^3*g-C*d^2*e*f)-2/3*C*g/e^2/c*(1/2*a*e+1 
/2*b*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b 
^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(- 
4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+( 
-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/ 
2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2 
*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2* 
(1/e^2*(B*e*g-C*d*g+C*e*f)-1/e^2*c*(A*d*e^2*g-A*e^3*f-B*d^2*e*g+B*d*e^2*f+ 
C*d^3*g-C*d^2*e*f)/(a*e^2-b*d*e+c*d^2)-2/3*C*g/e^2/c*(b*e+c*d))*(d/e-1/2*( 
b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2 
)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))) 
^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/ 
c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b 
+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1317 vs. \(2 (598) = 1196\).

Time = 0.16 (sec) , antiderivative size = 1317, normalized size of antiderivative = 2.04 \[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="fricas")
 

Output:

-2/9*(sqrt(c*e)*(3*(2*C*c^3*d^4*e - (2*C*b*c^2 + B*c^3)*d^3*e^2 - (C*b^2*c 
 + 2*A*c^3 - 2*(2*C*a + B*b)*c^2)*d^2*e^3 + (C*a*b*c - (3*B*a - A*b)*c^2)* 
d*e^4)*f - (8*C*c^3*d^5 - (7*C*b*c^2 + 6*B*c^3)*d^4*e - (2*C*b^2*c - 3*A*c 
^3 - (11*C*a + 6*B*b)*c^2)*d^3*e^2 - (2*C*b^3 + 6*(2*B*a + A*b)*c^2 - (7*C 
*a*b + 3*B*b^2)*c)*d^2*e^3 + (2*C*a*b^2 + 9*A*a*c^2 - 3*(C*a^2 + B*a*b)*c) 
*d*e^4)*g + (3*(2*C*c^3*d^3*e^2 - (2*C*b*c^2 + B*c^3)*d^2*e^3 - (C*b^2*c + 
 2*A*c^3 - 2*(2*C*a + B*b)*c^2)*d*e^4 + (C*a*b*c - (3*B*a - A*b)*c^2)*e^5) 
*f - (8*C*c^3*d^4*e - (7*C*b*c^2 + 6*B*c^3)*d^3*e^2 - (2*C*b^2*c - 3*A*c^3 
 - (11*C*a + 6*B*b)*c^2)*d^2*e^3 - (2*C*b^3 + 6*(2*B*a + A*b)*c^2 - (7*C*a 
*b + 3*B*b^2)*c)*d*e^4 + (2*C*a*b^2 + 9*A*a*c^2 - 3*(C*a^2 + B*a*b)*c)*e^5 
)*g)*x)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c 
^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2 
*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*sqrt( 
c*e)*(3*(2*C*c^3*d^3*e^2 - (C*b*c^2 + B*c^3)*d^2*e^3 + (C*a*c^2 + A*c^3)*d 
*e^4)*f - (8*C*c^3*d^4*e - 3*(C*b*c^2 + 2*B*c^3)*d^3*e^2 - (2*C*b^2*c - 3* 
A*c^3 - (5*C*a + 3*B*b)*c^2)*d^2*e^3 + (2*C*a*b*c - 3*B*a*c^2)*d*e^4)*g + 
(3*(2*C*c^3*d^2*e^3 - (C*b*c^2 + B*c^3)*d*e^4 + (C*a*c^2 + A*c^3)*e^5)*f - 
 (8*C*c^3*d^3*e^2 - 3*(C*b*c^2 + 2*B*c^3)*d^2*e^3 - (2*C*b^2*c - 3*A*c^3 - 
 (5*C*a + 3*B*b)*c^2)*d*e^4 + (2*C*a*b*c - 3*B*a*c^2)*e^5)*g)*x)*weierstra 
ssZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*...
 

Sympy [F]

\[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right ) \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

Integral((f + g*x)*(A + B*x + C*x**2)/((d + e*x)**(3/2)*sqrt(a + b*x + c*x 
**2)), x)
 

Maxima [F]

\[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/ 
2)), x)
 

Giac [F]

\[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(3/ 
2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (C\,x^2+B\,x+A\right )}{{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int(((f + g*x)*(A + B*x + C*x^2))/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2) 
),x)
                                                                                    
                                                                                    
 

Output:

int(((f + g*x)*(A + B*x + C*x^2))/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(1/2) 
), x)
 

Reduce [F]

\[ \int \frac {(f+g x) \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right ) \left (C \,x^{2}+B x +A \right )}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((g*x+f)*(C*x^2+B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)