\(\int \frac {(b+2 c x) (d+e x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx\) [660]

Optimal result

Integrand size = 30, antiderivative size = 487 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 (2 c d-b e) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{5 c}+\frac {4}{5} (d+e x)^{3/2} \sqrt {a+b x+c x^2}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c^2 e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{5 c^2 e \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 32.47 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.42 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\frac {4 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (a+x (b+c x))}{\sqrt {d+e x}}+2 c \sqrt {d+e x} (4 c d-b e+2 c e x) (a+x (b+c x))-\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )+\left (b^3 e^3-b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (-4 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (-c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{e^2 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{5 c^2 \sqrt {a+x (b+c x)}} \] Input:

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

Output:

((4*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(a + x*(b + c*x)))/Sqrt[d + e* 
x] + 2*c*Sqrt[d + e*x]*(4*c*d - b*e + 2*c*e*x)*(a + x*(b + c*x)) - (I*(d + 
 e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 
4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((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))*EllipticE[I*ArcSinh[(S 
qrt[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]))] + (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*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-(c*d^2*S 
qrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^2])))*Elli 
pticF[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]))]))/(e^2*Sqrt[(c*d^2 + e*( 
-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]))/(5*c^2*Sqrt[a + 
 x*(b + c*x)])
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1236, 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.

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(4*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/5 + (3*((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))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/S 
qrt[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]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(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]))/(3*c)))/5
 

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 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. \(955\) vs. \(2(427)=854\).

Time = 5.26 (sec) , antiderivative size = 956, normalized size of antiderivative = 1.96

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {4 e x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{5}+\frac {2 \left (b \,e^{2}+4 d e c -\frac {4 e \left (2 b e +2 c d \right )}{5}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}{3 c e}+\frac {2 \left (b \,d^{2}-\frac {4 a d e}{5}-\frac {2 \left (b \,e^{2}+4 d e c -\frac {4 e \left (2 b e +2 c d \right )}{5}\right ) \left (\frac {a e}{2}+\frac {b d}{2}\right )}{3 c e}\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 \left (2 b d e +2 c \,d^{2}-\frac {4 e \left (\frac {3 a e}{2}+\frac {3 b d}{2}\right )}{5}-\frac {2 \left (b \,e^{2}+4 d e c -\frac {4 e \left (2 b e +2 c d \right )}{5}\right ) \left (b e +c d \right )}{3 c e}\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 )}{\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}}\)	\(956\)
risch	\(\text {Expression too large to display}\)	\(1684\)
default	\(\text {Expression too large to display}\)	\(4356\)

Input:

int((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),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)*(4/5*e*x*( 
c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(b*e^2+4*d*e*c-4/5*e*(2 
*b*e+2*c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(b*d^2- 
4/5*a*d*e-2/3*(b*e^2+4*d*e*c-4/5*e*(2*b*e+2*c*d))/c/e*(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)*Ellipti 
cF(((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*b*d*e+2 
*c*d^2-4/5*e*(3/2*a*e+3/2*b*d)-2/3*(b*e^2+4*d*e*c-4/5*e*(2*b*e+2*c*d))/c/e 
*(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))*Ellip 
ticF(((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 = 0.09 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.99 \[ \int \frac {(b+2 c x) (d+e x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (c^{3} d^{2} e - b c^{2} d e^{2} + {\left (b^{2} c - 3 \, a c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (2 \, c^{3} e^{3} x + 4 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{15 \, c^{3} e^{2}} \] Input:

integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas 
")
 

Output:

-2/15*((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)*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*d^2*e - b*c^2*d*e^2 + (b^2*c - 3*a*c^2)*e^3)*sqrt(c*e)*weie 
rstrassZeta(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*(2*c^3*e^3*x + 4*c^3*d*e^2 - b*c^2*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(e*x 
 + d))/(c^3*e^2)
 

Sympy [F]

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

integrate((2*c*x+b)*(e*x+d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate((2*c*x+b)*(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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