\(\int \frac {(b+2 c x) (d+e x)^{3/2}}{(a+b x+c x^2)^{5/2}} \, dx\) [673]

Optimal result

Integrand size = 30, antiderivative size = 436 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 e (b+2 c x) \sqrt {d+e x}}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} e \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} e (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2)-2*e*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c 
+b^2)/(c*x^2+b*x+a)^(1/2)+2*2^(1/2)*e*(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)^(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)*e*(-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))/c/(-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 = 17.93 (sec) , antiderivative size = 928, normalized size of antiderivative = 2.13 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left ((a+x (b+c x)) \left (-\frac {2 (d+e x)}{3 (a+x (b+c x))^2}-\frac {2 e (b+2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}\right )+\frac {(d+e x) \left (\frac {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}}} (a+x (b+c x))}{(d+e x)^2}-\frac {i \sqrt {2} \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} \sqrt {\left (b^2-4 a c\right ) e^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 {d+e x}}\right )}{\left (b^2-4 a c\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}}}}\right )}{\sqrt {a+x (b+c x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1222, 1163, 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.

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

\(\Big \downarrow \) 1222

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

\(\Big \downarrow \) 1163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(-2*(d + e*x)^(3/2))/(3*(a + b*x + c*x^2)^(3/2)) + e*((-2*(b + 2*c*x)*Sqrt 
[d + e*x])/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) + (e*((2*Sqrt[2]*Sqrt[b^2 
 - 4*a*c]*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellip 
ticE[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]*(2*c*d - b*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))]*E 
llipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sq 
rt[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])))/(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 1163

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^m*(b + 2*c*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 - 1 
)*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 
1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ 
m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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 1222

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)/(2*c*(p + 1))), x] - Simp[e*g*(m/(2*c*(p + 1)))   Int[(d + e*x)^(m - 1)* 
(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ 
[2*c*f - b*g, 0] && LtQ[p, -1] && GtQ[m, 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. \(911\) vs. \(2(382)=764\).

Time = 6.53 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.09

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\left (-\frac {2 e x}{3 c^{2}}-\frac {2 d}{3 c^{2}}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {2 e x}{4 a c -b^{2}}-\frac {b e}{\left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (c e x +c d \right )}}-\frac {2 e^{2} b \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 )}{\left (4 a c -b^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}-\frac {4 c \,e^{2} \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 )}{\left (4 a c -b^{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}}\)	\(912\)
default	\(\text {Expression too large to display}\)	\(4740\)

Input:

int((2*c*x+b)*(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*e*x 
/c^2-2/3*d/c^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)*(-2*e/(4*a*c-b^2)*x-b*e/(4*a*c-b^2)/c)/((a/c+b/c*x+x 
^2)*(c*e*x+c*d))^(1/2)-2*e^2/(4*a*c-b^2)*b*(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))-4*c*e^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)*((-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 = 0.15 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.54 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-2/3*(((2*c^3*d - b*c^2*e)*x^4 + 2*a^2*c*d - a^2*b*e + 2*(2*b*c^2*d - b^2* 
c*e)*x^3 + (2*(b^2*c + 2*a*c^2)*d - (b^3 + 2*a*b*c)*e)*x^2 + 2*(2*a*b*c*d 
- a*b^2*e)*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)) + 6*(c^3*e*x^4 + 2*b*c^2*e*x^3 + 2*a*b*c*e*x + a^2*c*e + (b^2*c + 2 
*a*c^2)*e*x^2)*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))) + (6*c^3*e*x^3 + 9*b*c^2*e*x^2 + 3*a*b*c*e + 
2*(2*b^2*c + a*c^2)*e*x + (b^2*c - 4*a*c^2)*d)*sqrt(c*x^2 + b*x + a)*sqrt( 
e*x + d))/(a^2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 
4*a*b*c^3)*x^3 + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^ 
2*b*c^2)*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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