\(\int \frac {(b+2 c x) (a+b x+c x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [656]

Optimal result

Integrand size = 30, antiderivative size = 572 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \left (128 c^2 d^2+51 b^2 e^2-4 c e (44 b d-5 a e)-48 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{35 e^4}+\frac {2 (16 c d-7 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{7 e^2 \sqrt {d+e x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2+27 b^2 e^2-4 c e (32 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2/35*(e*x+d)^(1/2)*(128*c^2*d^2+51*b^2*e^2-4*c*e*(-5*a*e+44*b*d)-48*c*e*(- 
b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^4+2/7*(2*c*e*x-7*b*e+16*c*d)*(c*x^2+b* 
x+a)^(3/2)/e^2/(e*x+d)^(1/2)-1/35*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*d)* 
(128*c^2*d^2+3*b^2*e^2-4*c*e*(-29*a*e+32*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/e^5/(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)+4/35*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(128*c^2*d^2+27 
*b^2*e^2-4*c*e*(-5*a*e+32*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^5/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.33 (sec) , antiderivative size = 5373, normalized size of antiderivative = 9.39 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1230, 27, 1231, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {6 \int \frac {\left (-7 e b^2+16 c d b-4 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{2 \sqrt {d+e x}}dx}{7 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \int \frac {\left (-7 e b^2+16 c d b-4 a c e+16 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{\sqrt {d+e x}}dx}{7 e^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (-\frac {2 \int -\frac {c \left (51 d e^2 b^3-2 \left (27 a e^3+88 c d^2 e\right ) b^2+4 c d \left (32 c d^2+45 a e^2\right ) b-8 a c e \left (8 c d^2+5 a e^2\right )+(2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 c e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (\frac {\int \frac {51 d e^2 b^3-2 \left (27 a e^3+88 c d^2 e\right ) b^2+4 c d \left (32 c d^2+45 a e^2\right ) b-8 a c e \left (8 c d^2+5 a e^2\right )+(2 c d-b e) \left (128 c^2 d^2+3 b^2 e^2-4 c e (32 b d-29 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt {d+e x}}-\frac {3 \left (\frac {\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2+27 b^2 e^2-128 b c d e+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}}{15 e^2}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{15 e^2}\right )}{7 e^2}\)

Input:

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

Output:

(2*(16*c*d - 7*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x 
]) - (3*((-2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a 
*e) - 48*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(15*e^2) + ((Sqrt[2]* 
Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 
 29*a*e))*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)])/(c*e*Sq 
rt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2 
]) - (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^2*d^2 - 1 
28*b*c*d*e + 27*b^2*e^2 + 20*a*c*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqr 
t[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]))/(15*e^2)))/(7*e^2)
 

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 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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[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. \(1719\) vs. \(2(512)=1024\).

Time = 7.69 (sec) , antiderivative size = 1720, normalized size of antiderivative = 3.01

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1720\)
risch	\(\text {Expression too large to display}\)	\(1897\)
default	\(\text {Expression too large to display}\)	\(6527\)

Input:

int((2*c*x+b)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^(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 
^2+b*e*x+a*e)*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^5/(( 
x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+4/7*c^2/e^2*x^2*(c*e*x^3+b*e*x^2+c*d*x^2 
+a*e*x+b*d*x+a*d)^(1/2)+2/5*(c^2/e^2*(5*b*e-2*c*d)-4/7*c^2/e^2*(3*b*e+3*c* 
d))/c/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*(c/e^3*(4*a* 
c*e^2+4*b^2*e^2-5*b*c*d*e+2*c^2*d^2)-4/7*c^2/e^2*(5/2*a*e+5/2*b*d)-2/5*(c^ 
2/e^2*(5*b*e-2*c*d)-4/7*c^2/e^2*(3*b*e+3*c*d))/c/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*((2*a^2*c*e^4+2*a*b^2*e^4-6* 
a*b*c*d*e^3+4*a*c^2*d^2*e^2-b^3*d*e^3+4*b^2*c*d^2*e^2-5*b*c^2*d^3*e+2*c^3* 
d^4)/e^5-(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)/e^5*(b*e-c* 
d)+b/e^4*(a*b*e^3-2*a*c*d*e^2-b^2*d*e^2+3*b*c*d^2*e-2*c^2*d^3)-2/5*(c^2/e^ 
2*(5*b*e-2*c*d)-4/7*c^2/e^2*(3*b*e+3*c*d))/c/e*a*d-2/3*(c/e^3*(4*a*c*e^2+4 
*b^2*e^2-5*b*c*d*e+2*c^2*d^2)-4/7*c^2/e^2*(5/2*a*e+5/2*b*d)-2/5*(c^2/e^2*( 
5*b*e-2*c*d)-4/7*c^2/e^2*(3*b*e+3*c*d))/c/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 
)*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 = 0.11 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.45 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/105*((256*c^4*d^5 - 512*b*c^3*d^4*e + 2*(139*b^2*c^2 + 212*a*c^3)*d^3*e^ 
2 - 2*(11*b^3*c + 212*a*b*c^2)*d^2*e^3 - (3*b^4 - 46*a*b^2*c - 120*a^2*c^2 
)*d*e^4 + (256*c^4*d^4*e - 512*b*c^3*d^3*e^2 + 2*(139*b^2*c^2 + 212*a*c^3) 
*d^2*e^3 - 2*(11*b^3*c + 212*a*b*c^2)*d*e^4 - (3*b^4 - 46*a*b^2*c - 120*a^ 
2*c^2)*e^5)*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)) + 3*(256*c^4*d^4*e - 384*b*c^3*d^3*e^2 + 2*(67*b^2*c^2 + 116*a*c^3 
)*d^2*e^3 - (3*b^3*c + 116*a*b*c^2)*d*e^4 + (256*c^4*d^3*e^2 - 384*b*c^3*d 
^2*e^3 + 2*(67*b^2*c^2 + 116*a*c^3)*d*e^4 - (3*b^3*c + 116*a*b*c^2)*e^5)*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*(10*c^4*e^5*x^3 + 128*c^4*d^3*e^2 - 176*b*c^3*d^2*e^3 
- 35*a*b*c^2*e^5 + (51*b^2*c^2 + 100*a*c^3)*d*e^4 - (16*c^4*d*e^4 - 23*b*c 
^3*e^5)*x^2 + 2*(16*c^4*d^2*e^3 - 24*b*c^3*d*e^4 + (8*b^2*c^2 + 15*a*c^3)* 
e^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^2*e^7*x + c^2*d*e^6)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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