\(\int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx\) [653]

Optimal result

Integrand size = 30, antiderivative size = 540 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {a+b x+c x^2}}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 (8 c d-b e+6 c e x) \sqrt {a+b x+c x^2}}{3 e^2 (d+e x)^{3/2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^3 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {16 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*b*d))*(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*(6*c*e*x-b*e+8*c*d)*(c*x^2+b*x+a)^(1/2) 
/e^2/(e*x+d)^(3/2)+1/3*2^(1/2)*(-4*a*c+b^2)^(1/2)*(16*c^2*d^2+b^2*e^2-4*c* 
e*(-3*a*e+4*b*d))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Elli 
pticE(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))/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)-16/3 
*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*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))/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 = 34.10 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.50 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\frac {4 e^2 (a+x (b+c x)) \left (-b e^3 (a+b x)-2 c^2 d^2 (4 d+5 e x)+c e (-2 a e (2 d+3 e x)+b d (7 d+10 e x))\right )+\frac {(d+e x) \left (4 e^2 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) (a+x (b+c x))-i (d+e x)^{3/2} \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (-b^3 e^3+b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (a c e^3-4 c d e \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 c \left (4 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-2 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{6 e^4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^{3/2} \sqrt {a+x (b+c x)}} \] Input:

Integrate[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^(5/2),x]
 

Output:

(4*e^2*(a + x*(b + c*x))*(-(b*e^3*(a + b*x)) - 2*c^2*d^2*(4*d + 5*e*x) + c 
*e*(-2*a*e*(2*d + 3*e*x) + b*d*(7*d + 10*e*x))) + ((d + e*x)*(4*e^2*Sqrt[( 
c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^ 
2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*(a + x*(b + c*x)) - I*(d + e*x)^ 
(3/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 
4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4 
*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*EllipticE[I*Ar 
cSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4* 
a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2 
*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (-(b^3*e^3) + b^2*e^2*(2*c*d + S 
qrt[(b^2 - 4*a*c)*e^2]) + 4*b*(a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) 
+ 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + a*e^2*(-2*c*d + 3*Sqrt[(b^2 - 4*a 
*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2* 
c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + S 
qrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])))/Sqrt[ 
(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/(6*e 
^4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (warning: unable to verify)

Time = 1.43 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1229, 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.

\(\displaystyle \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {2 \int \frac {c \left (7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c \int \frac {7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {c \left (\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {c \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{3 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {a+b x+c x^2} \left (e x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-c d e (7 b d-4 a e)+a b e^3+8 c^2 d^3\right )}{3 e^2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(d + e*x)^(5/2),x]
 

Output:

(-2*(8*c^2*d^3 + a*b*e^3 - c*d*e*(7*b*d - 4*a*e) + e*(10*c^2*d^2 + b^2*e^2 
 - 2*c*e*(5*b*d - 3*a*e))*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*(c*d^2 - b*d*e 
+ a*e^2)*(d + e*x)^(3/2)) - (c*(-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*c^2*d^2 + 
 b^2*e^2 - 4*c*e*(4*b*d - 3*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)/S 
qrt[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])) + (16*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)* 
(c*d^2 - b*d*e + a*e^2)*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*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 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1091\) vs. \(2(480)=960\).

Time = 4.45 (sec) , antiderivative size = 1092, normalized size of antiderivative = 2.02

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1092\)
default	\(\text {Expression too large to display}\)	\(8612\)

Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
 

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)*(-2/3*(b*e 
-2*c*d)/e^4*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(x+d/e)^2-2/3* 
(c*e*x^2+b*e*x+a*e)/e^3/(a*e^2-b*d*e+c*d^2)*(6*a*c*e^2+b^2*e^2-10*b*c*d*e+ 
10*c^2*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(c*(3*b*e-4*c*d)/e^3-1/3 
*(b*e-2*c*d)/e^3*c-1/3/e^3*(b*e-c*d)*(6*a*c*e^2+b^2*e^2-10*b*c*d*e+10*c^2* 
d^2)/(a*e^2-b*d*e+c*d^2)+1/3*b/e^2/(a*e^2-b*d*e+c*d^2)*(6*a*c*e^2+b^2*e^2- 
10*b*c*d*e+10*c^2*d^2))*(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*(2*c^2/e^2+1/3*c/e^2*(6*a*c*e^2+b^2*e^2-10*b*c*d*e+10*c^2*d 
^2)/(a*e^2-b*d*e+c*d^2))*(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))/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))+1/2/c*(-b+(-4*a*c+b...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.55 \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(5/2),x, algorithm="fricas 
")
 

Output:

-2/9*((16*c^3*d^5 - 24*b*c^2*d^4*e + 6*(b^2*c + 4*a*c^2)*d^3*e^2 + (b^3 - 
12*a*b*c)*d^2*e^3 + (16*c^3*d^3*e^2 - 24*b*c^2*d^2*e^3 + 6*(b^2*c + 4*a*c^ 
2)*d*e^4 + (b^3 - 12*a*b*c)*e^5)*x^2 + 2*(16*c^3*d^4*e - 24*b*c^2*d^3*e^2 
+ 6*(b^2*c + 4*a*c^2)*d^2*e^3 + (b^3 - 12*a*b*c)*d*e^4)*x)*sqrt(c*e)*weier 
strassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/2 
7*(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*(16*c^3*d^4*e - 16 
*b*c^2*d^3*e^2 + (b^2*c + 12*a*c^2)*d^2*e^3 + (16*c^3*d^2*e^3 - 16*b*c^2*d 
*e^4 + (b^2*c + 12*a*c^2)*e^5)*x^2 + 2*(16*c^3*d^3*e^2 - 16*b*c^2*d^2*e^3 
+ (b^2*c + 12*a*c^2)*d*e^4)*x)*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), weierstrassPI 
nverse(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*(8*c^3*d^3*e^2 - 7*b*c^2* 
d^2*e^3 + 4*a*c^2*d*e^4 + a*b*c*e^5 + (10*c^3*d^2*e^3 - 10*b*c^2*d*e^4 + ( 
b^2*c + 6*a*c^2)*e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^2*d^4*e^4 
 - b*c*d^3*e^5 + a*c*d^2*e^6 + (c^2*d^2*e^6 - b*c*d*e^7 + a*c*e^8)*x^2 + 2 
*(c^2*d^3*e^5 - b*c*d^2*e^6 + a*c*d*e^7)*x)
 

Sympy [F]

\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (b + 2 c x\right ) \sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((2*c*x+b)*(c*x**2+b*x+a)**(1/2)/(e*x+d)**(5/2),x)
 

Output:

Integral((b + 2*c*x)*sqrt(a + b*x + c*x**2)/(d + e*x)**(5/2), x)
 

Maxima [F]

\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(5/2),x, algorithm="maxima 
")
 

Output:

integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d)^(5/2), x)
 

Giac [F]

\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^2 + b*x + a)*(2*c*x + b)/(e*x + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (b+2\,c\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(5/2),x)
 

Output:

int(((b + 2*c*x)*(a + b*x + c*x^2)^(1/2))/(d + e*x)^(5/2), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(b+2 c x) \sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\text {too large to display} \] Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(1/2)/(e*x+d)^(5/2),x)
 

Output:

( - 8*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*a*c*e - 2*sqrt(d + e*x)*sqrt(a 
+ b*x + c*x**2)*b**2*e + 12*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b*c*d + 8 
*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b*c*e*x - 8*sqrt(d + e*x)*sqrt(a + b 
*x + c*x**2)*c**2*d*x - 4*int((sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*x**2)/ 
(a*b*d**3*e + 3*a*b*d**2*e**2*x + 3*a*b*d*e**3*x**2 + a*b*e**4*x**3 - a*c* 
d**4 - 3*a*c*d**3*e*x - 3*a*c*d**2*e**2*x**2 - a*c*d*e**3*x**3 + b**2*d**3 
*e*x + 3*b**2*d**2*e**2*x**2 + 3*b**2*d*e**3*x**3 + b**2*e**4*x**4 - b*c*d 
**4*x - 2*b*c*d**3*e*x**2 + 2*b*c*d*e**3*x**4 + b*c*e**4*x**5 - c**2*d**4* 
x**2 - 3*c**2*d**3*e*x**3 - 3*c**2*d**2*e**2*x**4 - c**2*d*e**3*x**5),x)*a 
*b*c**2*d**2*e**3 - 8*int((sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*x**2)/(a*b 
*d**3*e + 3*a*b*d**2*e**2*x + 3*a*b*d*e**3*x**2 + a*b*e**4*x**3 - a*c*d**4 
 - 3*a*c*d**3*e*x - 3*a*c*d**2*e**2*x**2 - a*c*d*e**3*x**3 + b**2*d**3*e*x 
 + 3*b**2*d**2*e**2*x**2 + 3*b**2*d*e**3*x**3 + b**2*e**4*x**4 - b*c*d**4* 
x - 2*b*c*d**3*e*x**2 + 2*b*c*d*e**3*x**4 + b*c*e**4*x**5 - c**2*d**4*x**2 
 - 3*c**2*d**3*e*x**3 - 3*c**2*d**2*e**2*x**4 - c**2*d*e**3*x**5),x)*a*b*c 
**2*d*e**4*x - 4*int((sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*x**2)/(a*b*d**3 
*e + 3*a*b*d**2*e**2*x + 3*a*b*d*e**3*x**2 + a*b*e**4*x**3 - a*c*d**4 - 3* 
a*c*d**3*e*x - 3*a*c*d**2*e**2*x**2 - a*c*d*e**3*x**3 + b**2*d**3*e*x + 3* 
b**2*d**2*e**2*x**2 + 3*b**2*d*e**3*x**3 + b**2*e**4*x**4 - b*c*d**4*x - 2 
*b*c*d**3*e*x**2 + 2*b*c*d*e**3*x**4 + b*c*e**4*x**5 - c**2*d**4*x**2 -...