\(\int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx\) [634]

Optimal result

Integrand size = 28, antiderivative size = 225 \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {3 \left (b^2-4 a c\right ) e (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.37 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.98 \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) \sqrt {a+x (b+c x)}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}-\frac {3 \left (b^2-4 a c\right ) e (-2 c d+b e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1237, 27, 1228, 1154, 219}

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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x) 
^2) + (((4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e))*Sqrt[a + b*x + c*x^2 
])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(2*c*d - b*e)* 
ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqr 
t[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(3/2)))/(4*(c*d^2 - b*d*e 
 + a*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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(841\) vs. \(2(207)=414\).

Time = 1.89 (sec) , antiderivative size = 842, normalized size of antiderivative = 3.74

method	result	size
default	\(\frac {\left (b e -2 c d \right ) \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {3 \left (b e -2 c d \right ) e \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{4 \left (a \,e^{2}-b d e +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {2 c \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}\)	\(842\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 729 vs. \(2 (207) = 414\).

Time = 0.61 (sec) , antiderivative size = 1500, normalized size of antiderivative = 6.67 \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2 + (2*(b^2*c 
- 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^2 + 2*(2*(b^2*c - 4*a*c^2)*d^2*e 
^2 - (b^3 - 4*a*b*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e 
- 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e 
^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e 
 + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/( 
e^2*x^2 + 2*d*e*x + d^2)) + 4*(8*c^3*d^5 - 18*b*c^2*d^4*e - 2*a^2*b*e^5 + 
(15*b^2*c + 4*a*c^2)*d^3*e^2 - (5*b^3 + 8*a*b*c)*d^2*e^3 + (7*a*b^2 - 4*a^ 
2*c)*d*e^4 + (4*c^3*d^4*e - 8*b*c^2*d^3*e^2 + (7*b^2*c - 4*a*c^2)*d^2*e^3 
- (3*b^3 - 4*a*b*c)*d*e^4 + (3*a*b^2 - 8*a^2*c)*e^5)*x)*sqrt(c*x^2 + b*x + 
 a))/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + 
 a*c^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c 
^3*d^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2) 
*d^4*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c 
^3*d^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^ 
2)*d^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), -1/8 
*(3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2 + (2*(b^2*c - 4*a 
*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^2 + 2*(2*(b^2*c - 4*a*c^2)*d^2*e^2 - 
(b^3 - 4*a*b*c)*d*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c 
*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1068 vs. \(2 (207) = 414\).

Time = 0.23 (sec) , antiderivative size = 1068, normalized size of antiderivative = 4.75 \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-3/4*(2*b^2*c*d*e - 8*a*c^2*d*e - b^3*e^2 + 4*a*b*c*e^2)*arctan(-((sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/( 
(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e 
^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/4*(6*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^3*b^2*c*d*e^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2*d*e^3 
- 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*e^4 + 12*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^3*a*b*c*e^4 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^ 
(7/2)*d^4 - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(5/2)*d^3*e + 34* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^2*c^(3/2)*d^2*e^2 - 40*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^2*a*c^(5/2)*d^2*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^2*b^3*sqrt(c)*d*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a 
*b*c^(3/2)*d*e^3 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(3/2)*e^ 
4 + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^3*d^4 - 24*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))*b^2*c^2*d^3*e - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
*a*c^3*d^3*e + 22*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*c*d^2*e^2 + 8*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b*c^2*d^2*e^2 - 5*(sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))*b^4*d*e^3 - 22*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c 
*d*e^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c^2*d*e^3 + 5*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))*a*b^3*e^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))*a^2*b*c*e^4 + 4*b^2*c^(5/2)*d^4 - 4*b^3*c^(3/2)*d^3*e - 16*a*b*c^(5...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 2088, normalized size of antiderivative = 9.28 \[ \int \frac {b+2 c x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e 
**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b*c*d**2*e**2 - 2 
4*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - 
 b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b*c*d*e**3*x - 12*sqrt 
(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e 
 + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b*c*e**4*x**2 + 24*sqrt(a*e* 
*2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c* 
d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d**3*e + 48*sqrt(a*e**2 - b* 
d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 
 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d**2*e**2*x + 24*sqrt(a*e**2 - b*d* 
e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2 
*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d*e**3*x**2 + 3*sqrt(a*e**2 - b*d*e + 
 c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a* 
e + b*d - b*e*x + 2*c*d*x)*b**3*d**2*e**2 + 6*sqrt(a*e**2 - b*d*e + c*d**2 
)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d 
 - b*e*x + 2*c*d*x)*b**3*d*e**3*x + 3*sqrt(a*e**2 - b*d*e + c*d**2)*log(2* 
sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x 
 + 2*c*d*x)*b**3*e**4*x**2 - 6*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a 
+ b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c* 
d*x)*b**2*c*d**3*e - 12*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*...