\(\int \frac {\sqrt {a+b x+c x^2} (A+B x+C x^2)}{\sqrt {d+e x}} \, dx\) [96]

Optimal result

Integrand size = 34, antiderivative size = 654 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{\sqrt {d+e x}} \, dx=-\frac {2 \sqrt {d+e x} \left (5 b c C d-\frac {24 c^2 C d^2}{e}-35 A c^2 e+4 b^2 C e+5 a c C e+7 B c (4 c d-b e)+3 c (6 c C d-7 B c e+4 b C e) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 e^2}+\frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (5 c (2 c d-b e) (3 b C d-7 A c e+a C e)-\frac {(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )}{e}\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (4 b^2 C e^2+c e (8 b C d-7 b B e-10 a C e)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^3 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/105*(e*x+d)^(1/2)*(5*b*c*C*d-24*c^2*C*d^2/e-35*A*c^2*e+4*b^2*C*e+5*a*c* 
C*e+7*B*c*(-b*e+4*c*d)+3*c*(-7*B*c*e+4*C*b*e+6*C*c*d)*x)*(c*x^2+b*x+a)^(1/ 
2)/c^2/e^2+2/7*C*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2)/c/e+1/105*2^(1/2)*(-4*a 
*c+b^2)^(1/2)*(5*c*(-b*e+2*c*d)*(-7*A*c*e+C*a*e+3*C*b*d)-(-7*B*c*e+4*C*b*e 
+6*C*c*d)*(8*c^2*d^2-2*b^2*e^2-3*c*e*(-2*a*e+b*d))/e)*(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^3/e^3/(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)+2/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(4*b^ 
2*C*e^2+c*e*(-7*B*b*e-10*C*a*e+8*C*b*d)+c^2*(48*C*d^2-14*e*(-5*A*e+4*B*d)) 
)*(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^3/ 
e^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 36.16 (sec) , antiderivative size = 9965, normalized size of antiderivative = 15.24 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{\sqrt {d+e x}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (warning: unable to verify)

Time = 2.09 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2184, 27, 1231, 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.

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

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 \int -\frac {e (3 b C d-7 A c e+a C e+(6 c C d-7 B c e+4 b C e) x) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{7 c e^2}+\frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\int \frac {(3 b C d-7 A c e+a C e+(6 c C d-7 B c e+4 b C e) x) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{7 c e}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {2 \int \frac {5 c e (b d-2 a e) (3 b C d-7 A c e+a C e)-2 (6 c C d-7 B c e+4 b C e) \left (\frac {1}{2} b d (4 c d-b e)-\frac {1}{2} a e (2 c d+b e)\right )+\left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{7 c e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {\int \frac {5 c e (b d-2 a e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (-d e b^2+4 c d^2 b-a e^2 b-2 a c d e\right )+\left (5 c e (2 c d-b e) (3 b C d-7 A c e+a C e)-(6 c C d-7 B c e+4 b C e) \left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}}{7 c e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {\frac {\left (a e^2-b d e+c d^2\right ) \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}+\frac {\left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}}{15 c e^2}}{7 c e}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e^2}}{7 c e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 c e^2}}{7 c e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 C \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}}{7 c e}-\frac {\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} (5 c e (a C e-7 A c e+3 b C d)+3 c e x (4 b C e-7 B c e+6 c C d)-(4 c d-b e) (4 b C e-7 B c e+6 c C d))}{15 c e^2}-\frac {\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \left (c e (-10 a C e-7 b B e+8 b C d)+c^2 \left (48 C d^2-14 e (4 B d-5 A e)\right )+4 b^2 C e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (5 c e (2 c d-b e) (a C e-7 A c e+3 b C d)-\left (-3 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right ) (4 b C e-7 B c e+6 c C d)\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}}{15 c e^2}}{7 c e}\)

Input:

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

Output:

(2*C*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/(7*c*e) - ((2*Sqrt[d + e*x]*(5 
*c*e*(3*b*C*d - 7*A*c*e + a*C*e) - (4*c*d - b*e)*(6*c*C*d - 7*B*c*e + 4*b* 
C*e) + 3*c*e*(6*c*C*d - 7*B*c*e + 4*b*C*e)*x)*Sqrt[a + b*x + c*x^2])/(15*c 
*e^2) - ((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(5*c*e*(2*c*d - b*e)*(3*b*C*d - 7*A*c* 
e + a*C*e) - (6*c*C*d - 7*B*c*e + 4*b*C*e)*(8*c^2*d^2 - 2*b^2*e^2 - 3*c*e* 
(b*d - 2*a*e)))*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)*(4*b^2*C*e 
^2 + c*e*(8*b*C*d - 7*b*B*e - 10*a*C*e) + c^2*(48*C*d^2 - 14*e*(4*B*d - 5* 
A*e)))*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])) 
/(15*c*e^2))/(7*c*e)
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 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. \(1201\) vs. \(2(594)=1188\).

Time = 3.56 (sec) , antiderivative size = 1202, normalized size of antiderivative = 1.84

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1202\)
risch	\(\text {Expression too large to display}\)	\(4505\)
default	\(\text {Expression too large to display}\)	\(12761\)

Input:

int((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(1/2),x,method=_RETURNVERBOS 
E)
 

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)*(2/7*C/e*x 
^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/5*(B*c+b*C-2/7*C/e*(3 
*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(A* 
c+B*b+C*a-2/7*C/e*(5/2*a*e+5/2*b*d)-2/5*(B*c+b*C-2/7*C/e*(3*b*e+3*c*d))/c/ 
e*(2*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(A* 
a-2/5*(B*c+b*C-2/7*C/e*(3*b*e+3*c*d))/c/e*a*d-2/3*(A*c+B*b+C*a-2/7*C/e*(5/ 
2*a*e+5/2*b*d)-2/5*(B*c+b*C-2/7*C/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e* 
(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*(A*b+B*a-4/7*C/e*a*d-2/5*(B*c+b*C-2/7*C/e*(3*b*e+3*c*d))/c/e*(3/ 
2*a*e+3/2*b*d)-2/3*(A*c+B*b+C*a-2/7*C/e*(5/2*a*e+5/2*b*d)-2/5*(B*c+b*C-2/7 
*C/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(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*...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a+b x+c x^2} \left (A+B x+C x^2\right )}{\sqrt {d+e x}} \, dx =\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(1/2),x, algorithm="fr 
icas")
 

Output:

2/315*((48*C*c^4*d^4 - 8*(5*C*b*c^3 + 7*B*c^4)*d^3*e - (10*C*b^2*c^2 - 70* 
A*c^4 - (62*C*a + 49*B*b)*c^3)*d^2*e^2 - (5*C*b^3*c + 14*(6*B*a + 5*A*b)*c 
^3 - 2*(11*C*a*b + 7*B*b^2)*c^2)*d*e^3 - (8*C*b^4 - 210*A*a*c^3 + (30*C*a^ 
2 + 63*B*a*b + 35*A*b^2)*c^2 - (41*C*a*b^2 + 14*B*b^3)*c)*e^4)*sqrt(c*e)*w 
eierstrassPInverse(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*(48*C*c^4*d^3* 
e - 8*(2*C*b*c^3 + 7*B*c^4)*d^2*e^2 - (9*C*b^2*c^2 - 70*A*c^4 - (26*C*a + 
21*B*b)*c^3)*d*e^3 - (8*C*b^3*c + 7*(6*B*a + 5*A*b)*c^3 - (29*C*a*b + 14*B 
*b^2)*c^2)*e^4)*sqrt(c*e)*weierstrassZeta(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), 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*(15*C*c^4*e^4*x^2 + 24*C*c^4*d^2*e^2 - ( 
5*C*b*c^3 + 28*B*c^4)*d*e^3 - (4*C*b^2*c^2 - 35*A*c^4 - (10*C*a + 7*B*b)*c 
^3)*e^4 - 3*(6*C*c^4*d*e^3 - (C*b*c^3 + 7*B*c^4)*e^4)*x)*sqrt(c*x^2 + b*x 
+ a)*sqrt(e*x + d))/(c^4*e^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((c*x^2+b*x+a)^(1/2)*(C*x^2+B*x+A)/(e*x+d)^(1/2),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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