\(\int \frac {(d+e x)^{3/2}}{(a+b x+c x^2)^{5/2}} \, dx\) [672]

Optimal result

Integrand size = 24, antiderivative size = 522 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (8 b c d-5 b^2 e+4 a c e+8 c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} (2 c d-b e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(e*x+d)^(1/2)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^( 
3/2)+2/3*(e*x+d)^(1/2)*(8*b*c*d-5*b^2*e+4*a*c*e+8*c*(-b*e+2*c*d)*x)/(-4*a* 
c+b^2)^2/(c*x^2+b*x+a)^(1/2)-8/3*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))/(-4*a*c+b^2)^(3/2)/(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)*(16*c^2*d^2+3*b^2*e^2-4*c*e*(-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/(-4*a*c+b^2)^(3/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 19.88 (sec) , antiderivative size = 1141, normalized size of antiderivative = 2.19 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)) 
/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (2*(-8*b*c*d + 5*b^2*e - 4*a*c*e 
- 16*c^2*d*x + 8*b*c*e*x))/(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2))))/(a + x* 
(b + c*x))^(5/2) - ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-16*(-2*c*d + 
 b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e 
^2])]*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)) 
)/(d + e*x)) + ((4*I)*Sqrt[2]*(-2*c*d + b*e)*(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*(-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*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 
[d + e*x] - (I*Sqrt[2]*(-(b^2*e^2) + 4*a*c*e^2 - 8*c*d*Sqrt[(b^2 - 4*a*c)* 
e^2] + 4*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*(-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])]*EllipticF[I*ArcSinh[(Sqrt[2]*...
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1164, 27, 1235, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b 
*x + c*x^2)^(3/2)) - ((-2*Sqrt[d + e*x]*(8*b*c*d - 5*b^2*e + 4*a*c*e + 8*c 
*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (e*((8*Sqrt[2]* 
Sqrt[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)/Sqr 
t[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)])/(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]*(16*c^2*d^2 + 3*b^2* 
e^2 - 4*c*e*(4*b*d - 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[Sqr 
t[(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]*Sqr 
t[a + b*x + c*x^2])))/(b^2 - 4*a*c))/(3*(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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -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. \(1063\) vs. \(2(462)=924\).

Time = 1.90 (sec) , antiderivative size = 1064, normalized size of antiderivative = 2.04

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1064\)
default	\(\text {Expression too large to display}\)	\(8889\)

Input:

int((e*x+d)^(3/2)/(c*x^2+b*x+a)^(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/c^2 
*(b*e-2*c*d)/(4*a*c-b^2)*x-2/3*(2*a*e-b*d)/c^2/(4*a*c-b^2))*(c*e*x^3+b*e*x 
^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(8/3*(b* 
e-2*c*d)/(4*a*c-b^2)^2*x-1/3*(4*a*c*e-5*b^2*e+8*b*c*d)/c/(4*a*c-b^2)^2)/(( 
a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(2/3*(4*a*c*e^2-b^2*e^2-8*b*c*d*e+16*c 
^2*d^2)/(4*a*c-b^2)^2-1/3*e*(4*a*c*e-5*b^2*e+8*b*c*d)/(4*a*c-b^2)^2+16/3*c 
*d*(b*e-2*c*d)/(4*a*c-b^2)^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))+16/3*(b*e-2*c*d)*c*e/(4*a*c-b^2)^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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1005 vs. \(2 (470) = 940\).

Time = 0.18 (sec) , antiderivative size = 1005, normalized size of antiderivative = 1.93 \[ \int \frac {(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/9*((16*a^2*c^2*d^2 - 16*a^2*b*c*d*e + (16*c^4*d^2 - 16*b*c^3*d*e + (b^2* 
c^2 + 12*a*c^3)*e^2)*x^4 + 2*(16*b*c^3*d^2 - 16*b^2*c^2*d*e + (b^3*c + 12* 
a*b*c^2)*e^2)*x^3 + (a^2*b^2 + 12*a^3*c)*e^2 + (16*(b^2*c^2 + 2*a*c^3)*d^2 
 - 16*(b^3*c + 2*a*b*c^2)*d*e + (b^4 + 14*a*b^2*c + 24*a^2*c^2)*e^2)*x^2 + 
 2*(16*a*b*c^2*d^2 - 16*a*b^2*c*d*e + (a*b^3 + 12*a^2*b*c)*e^2)*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)) + 24*(2*a^2*c^ 
2*d*e - a^2*b*c*e^2 + (2*c^4*d*e - b*c^3*e^2)*x^4 + 2*(2*b*c^3*d*e - b^2*c 
^2*e^2)*x^3 + (2*(b^2*c^2 + 2*a*c^3)*d*e - (b^3*c + 2*a*b*c^2)*e^2)*x^2 + 
2*(2*a*b*c^2*d*e - a*b^2*c*e^2)*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), weierstra 
ssPInverse(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*(2*c^4*d*e - b*c^3 
*e^2)*x^3 - (b^3*c - 12*a*b*c^2)*d*e - (3*a*b^2*c + 4*a^2*c^2)*e^2 + (24*b 
*c^3*d*e - (13*b^2*c^2 - 4*a*c^3)*e^2)*x^2 + 2*(3*(b^2*c^2 + 4*a*c^3)*d*e 
- 2*(b^3*c + 2*a*b*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/((b^4 
*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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