\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx\) [640]

Optimal result

Integrand size = 24, antiderivative size = 477 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=-\frac {2 \sqrt {a+b x+c x^2}}{3 e (d+e x)^{3/2}}+\frac {2 (2 c d-b e) \sqrt {a+b x+c x^2}}{3 e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^2 \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 29.91 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\frac {4 e^2 (-2 c d+b e) (d+e x) (a+x (b+c x))+4 e^2 (a+x (b+c x)) \left (-e^2 (a+b x)+c d (d+2 e x)\right )+\frac {i \sqrt {2} (2 c d-b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)^{5/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 )}{\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}}}}+\frac {i \sqrt {2} \left (-b^2 e^2+4 a c e^2-2 c d \sqrt {\left (b^2-4 a c\right ) e^2}+b e \sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)^{5/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \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 )}{\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^3 \left (c d^2+e (-b d+a e)\right ) (d+e x)^{3/2} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(4*e^2*(-2*c*d + b*e)*(d + e*x)*(a + x*(b + c*x)) + 4*e^2*(a + x*(b + c*x) 
)*(-(e^2*(a + b*x)) + c*d*(d + 2*e*x)) + (I*Sqrt[2]*(2*c*d - b*e)*(2*c*d - 
 b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x)^(5/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b 
^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x) 
)/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*S 
qrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d 
 + e*x))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[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 + Sqrt[(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])] + (I*Sqrt[2]*(-(b^2*e^2) + 4*a 
*c*e^2 - 2*c*d*Sqrt[(b^2 - 4*a*c)*e^2] + b*e*Sqrt[(b^2 - 4*a*c)*e^2])*(d + 
 e*x)^(5/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e*x + e*Sqr 
t[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c) 
*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + 
 e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e + Sqrt[(b^2 
- 4*a*c)*e^2])*(d + e*x))]*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 + Sqrt[(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...
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1161, 1237, 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.

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

\(\Big \downarrow \) 1161

\(\displaystyle \frac {\int \frac {b+2 c x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 e}-\frac {2 \sqrt {a+b x+c x^2}}{3 e (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*Sqrt[a + b*x + c*x^2])/(3*e*(d + e*x)^(3/2)) + ((2*(2*c*d - b*e)*Sqrt[ 
a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (c*((Sqrt[2]*S 
qrt[b^2 - 4*a*c]*(2*c*d - b*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]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(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])))/ 
(c*d^2 - b*d*e + a*e^2))/(3*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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

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 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(966\) vs. \(2(417)=834\).

Time = 1.38 (sec) , antiderivative size = 967, normalized size of antiderivative = 2.03

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 e^{3} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (c e \,x^{2}+b e x +a e \right ) \left (b e -2 c d \right )}{3 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x +a e \right )}}+\frac {2 \left (\frac {2 c}{3 e^{2}}-\frac {\left (b e -c d \right ) \left (b e -2 c d \right )}{3 e^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {b \left (b e -2 c d \right )}{3 e \left (a \,e^{2}-b d e +c \,d^{2}\right )}\right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}+\frac {2 c \left (b e -2 c d \right ) \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{3 e \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(967\)
default	\(\text {Expression too large to display}\)	\(3645\)

Input:

int((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/e^3* 
(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^2/(a*e^2-b*d*e+c*d^2)*(b*e-2*c*d)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^ 
(1/2)+2*(2/3*c/e^2-1/3/e^2*(b*e-c*d)*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)+1/3*b 
/e/(a*e^2-b*d*e+c*d^2)*(b*e-2*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)*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))+2/3*c/e*(b*e-2*c*d)/(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^2)^(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))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 662, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left ({\left (2 \, c^{2} d^{4} - 2 \, b c d^{3} e - {\left (b^{2} - 6 \, a c\right )} d^{2} e^{2} + {\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} - {\left (b^{2} - 6 \, a c\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{3} e - 2 \, b c d^{2} e^{2} - {\left (b^{2} - 6 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{2} + 2 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (c^{2} d^{2} e^{2} - a c e^{4} + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{9 \, {\left (c^{2} d^{4} e^{3} - b c d^{3} e^{4} + a c d^{2} e^{5} + {\left (c^{2} d^{2} e^{5} - b c d e^{6} + a c e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{3} e^{4} - b c d^{2} e^{5} + a c d e^{6}\right )} x\right )}} \] Input:

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

Output:

2/9*((2*c^2*d^4 - 2*b*c*d^3*e - (b^2 - 6*a*c)*d^2*e^2 + (2*c^2*d^2*e^2 - 2 
*b*c*d*e^3 - (b^2 - 6*a*c)*e^4)*x^2 + 2*(2*c^2*d^3*e - 2*b*c*d^2*e^2 - (b^ 
2 - 6*a*c)*d*e^3)*x)*sqrt(c*e)*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*(2*c^2*d^3*e - b*c*d^2*e^2 + (2*c^2*d*e^3 - b*c*e^4)*x^2 
 + 2*(2*c^2*d^2*e^2 - b*c*d*e^3)*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), weierstr 
assPInverse(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*(c^2*d^2*e^2 - a*c*e 
^4 + (2*c^2*d*e^3 - b*c*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^2* 
d^4*e^3 - b*c*d^3*e^4 + a*c*d^2*e^5 + (c^2*d^2*e^5 - b*c*d*e^6 + a*c*e^7)* 
x^2 + 2*(c^2*d^3*e^4 - b*c*d^2*e^5 + a*c*d*e^6)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((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)/(e*x + d)^(5/2), x)
 

Giac [F]

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

integrate((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)/(e*x + d)^(5/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b + 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)*b**2*c*d**2*e**2 + 2*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)*b**2*c*d*e**3*x + 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)*b**2*c*e**4*x**2 - 3*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...