\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x)^{5/2}} \, dx\) [645]

Optimal result

Integrand size = 24, antiderivative size = 479 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 (8 c d-3 b e+2 c e x) \sqrt {a+b x+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+b x+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 c e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/3*(2*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+a)^(1/2)/e^3/(e*x+d)^(1/2)-2/3*(c*x^2 
+b*x+a)^(3/2)/e/(e*x+d)^(3/2)-8/3*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^4/(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)*(-4*a*c+b^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/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 = 28.41 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 (a+x (b+c x)) \left (-e (3 b d+a e+4 b e x)+c \left (8 d^2+10 d e x+e^2 x^2\right )\right )}{e^3 (d+e x)^2}-\frac {(d+e x) \left (-\frac {16 e^2 (-2 c d+b e) \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}}} (a+x (b+c x))}{(d+e x)^2}-\frac {4 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 ) \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 {d+e x}}+\frac {i \sqrt {2} \left (b^2 e^2-4 a c e^2+8 c d \sqrt {\left (b^2-4 a c\right ) e^2}-4 b e \sqrt {\left (b^2-4 a c\right ) e^2}\right ) \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 {d+e x}}\right )}{e^5 \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}}}}\right )}{3 \sqrt {a+x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1161, 1230, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*(a + b*x + c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + ((2*(8*c*d - 3*b*e + 
2*c*e*x)*Sqrt[a + b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((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)/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)])/(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[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))/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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (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. \(1042\) vs. \(2(419)=838\).

Time = 10.45 (sec) , antiderivative size = 1043, normalized size of antiderivative = 2.18

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1043\)
risch	\(\text {Expression too large to display}\)	\(3544\)
default	\(\text {Expression too large to display}\)	\(5874\)

Input:

int((c*x^2+b*x+a)^(3/2)/(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*(a*e 
^2-b*d*e+c*d^2)/e^5*(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 
)^2-8/3*(c*e*x^2+b*e*x+a*e)*(b*e-2*c*d)/e^4/((x+d/e)*(c*e*x^2+b*e*x+a*e))^ 
(1/2)+2/3*c/e^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*((2*a*c* 
e^2+b^2*e^2-4*b*c*d*e+3*c^2*d^2)/e^4-1/3*(a*e^2-b*d*e+c*d^2)/e^4*c-4/3*(b* 
e-2*c*d)/e^4*(b*e-c*d)+4/3*b/e^3*(b*e-2*c*d)-2/3*c/e^3*(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)*Ellipt 
icF(((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/e^3* 
(b*e-c*d)+4/3*(b*e-2*c*d)/e^3*c-2/3*c/e^3*(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*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 [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left ({\left (16 \, c^{2} d^{4} - 16 \, b c d^{3} e + {\left (b^{2} + 12 \, a c\right )} d^{2} e^{2} + {\left (16 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + {\left (b^{2} + 12 \, a c\right )} e^{4}\right )} x^{2} + 2 \, {\left (16 \, c^{2} d^{3} e - 16 \, b c d^{2} e^{2} + {\left (b^{2} + 12 \, 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 ) + 24 \, {\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} e^{4} x^{2} + 8 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} - a c e^{4} + 2 \, {\left (5 \, c^{2} d e^{3} - 2 \, b c e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{9 \, {\left (c e^{7} x^{2} + 2 \, c d e^{6} x + c d^{2} e^{5}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 10*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*a*b*e**2 + 10*sqrt(d + e*x)*sq 
rt(a + b*x + c*x**2)*a*c*d*e + 12*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b** 
2*d*e + 8*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b**2*e**2*x - 18*sqrt(d + e 
*x)*sqrt(a + b*x + c*x**2)*b*c*d**2 - 20*sqrt(d + e*x)*sqrt(a + b*x + c*x* 
*2)*b*c*d*e*x + 2*sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*b*c*e**2*x**2 + 12* 
sqrt(d + e*x)*sqrt(a + b*x + c*x**2)*c**2*d**2*x - 2*sqrt(d + e*x)*sqrt(a 
+ b*x + c*x**2)*c**2*d*e*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*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**4*d**2*e**4 + 6*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 
**4*d*e**5*x + 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*c*d**4 -...