\(\int \frac {(d+e x)^{5/2}}{(a+b x+c x^2)^{3/2}} \, dx\) [663]

Optimal result

Integrand size = 24, antiderivative size = 513 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^{3/2} (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {2 \sqrt {2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c^2 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.77 (sec) , antiderivative size = 1207, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]
 

Output:

(2*Sqrt[d + e*x]*(b*c*d^2 - 4*a*c*d*e + a*b*e^2 + 2*c^2*d^2*x - 2*b*c*d*e* 
x + b^2*e^2*x - 2*a*c*e^2*x)*(a + b*x + c*x^2))/(c*(-b^2 + 4*a*c)*(a + x*( 
b + c*x))^(3/2)) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*(-4*Sqrt[(c*d^ 
2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c^2*d^2 + 
 b^2*e^2 - c*e*(b*d + 3*a*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + 
 e*x) + (a*e)/(d + e*x)))/(d + e*x)) + (I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 
 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[(Sqrt[(b^2 - 
4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + ( 
2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 
- 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2 
*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*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]))])/Sqrt[d + e*x] - (I*Sqrt[2]*(-(b^3 
*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b*c*e*(4*a*e^2 - d*Sqr 
t[(b^2 - 4*a*c)*e^2]) + c*(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - a*e^2*(8*c*d + 
3*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d 
+ e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b 
*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/( 
d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*...
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1164, 27, 1236, 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.

\(\displaystyle \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

Int[(d + e*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x]
 

Output:

(-2*(d + e*x)^(3/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a 
 + b*x + c*x^2]) + (3*e*((2*(2*c*d - b*e)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x 
^2])/(3*c) + ((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 
 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipt 
icE[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*Sqr 
t[(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]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*Sqr 
t[(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 + Sq 
rt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]))/(3*c)))/( 
b^2 - 4*a*c)
 

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 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[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 1236

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 
1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1 
)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m 
*(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 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. \(2(461)=922\).

Time = 5.10 (sec) , antiderivative size = 1092, normalized size of antiderivative = 2.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1092\)
default	\(\text {Expression too large to display}\)	\(4352\)

Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),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*(c*e*x 
+c*d)*((2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/c^2/(4*a*c-b^2)*x-(a*b*e^2- 
4*a*c*d*e+b*c*d^2)/c^2/(4*a*c-b^2))/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2* 
(-e^2*(b*e-3*c*d)/c^2+(4*a*b*c*e^3-12*a*c^2*d*e^2-b^3*e^3+4*b^2*c*d*e^2-4* 
b*c^2*d^2*e+4*c^3*d^3)/c^2/(4*a*c-b^2)-1/c*e*(a*b*e^2-4*a*c*d*e+b*c*d^2)/( 
4*a*c-b^2)+2/c*d*(2*a*c*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/(4*a*c-b^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*(e^3/c+(2*a*c 
*e^2-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/c*e/(4*a*c-b^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*...
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.55 \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x+d)**(5/2)/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Integral((d + e*x)**(5/2)/(a + b*x + c*x**2)**(3/2), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((e*x + d)^(5/2)/(c*x^2 + b*x + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^(3/2),x)
 

Output:

int((d + e*x)^(5/2)/(a + b*x + c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^{5/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (e x +d \right )^{\frac {5}{2}}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}d x \] Input:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x)
 

Output:

int((e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x)