\(\int \frac {(d+e x)^2}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [249]

Optimal result

Integrand size = 37, antiderivative size = 180 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2}+\frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d}+\frac {3 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.86 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c} \sqrt {d} (a e+c d x) (d+e x) \left (-3 a e^2+c d (5 d+2 e x)\right )+\frac {3 \left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {e}}}{4 c^{5/2} d^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {1134, 1160, 1092, 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.

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1134

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

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(156)=312\).

Time = 1.13 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.46

method	result	size
default	\(\frac {d^{2} \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{\sqrt {d e c}}+e^{2} \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )+2 d e \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )\)	\(442\)

Input:

int((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.32 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (2 \, c^{2} d^{2} e^{2} x + 5 \, c^{2} d^{3} e - 3 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, c^{3} d^{3} e}, -\frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} e^{2} x + 5 \, c^{2} d^{3} e - 3 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, c^{3} d^{3} e}\right ] \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (173) = 346\).

Time = 0.77 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.79 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\begin {cases} \left (\frac {e x}{2 c d} + \frac {2 d e - \frac {e \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d}}{c d e}\right ) \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} + \left (- \frac {a e^{2}}{2 c} + d^{2} - \frac {\left (a e^{2} + c d^{2}\right ) \left (2 d e - \frac {e \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c d e \neq 0 \\\frac {2 \left (\frac {c^{2} d^{6} \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )}}{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}} + \frac {2 c d^{3} e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{3 \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )} + \frac {e^{2} \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}{5 \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}\right )}{a e^{2} + c d^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}}{\sqrt {a d e}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise(((e*x/(2*c*d) + (2*d*e - e*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d))/(c 
*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)) + (-a*e**2/(2*c) + d 
**2 - (a*e**2 + c*d**2)*(2*d*e - e*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d))/(2*c 
*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d 
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + 
c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e))*log(x - (-a 
*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))** 
2), True)), Ne(c*d*e, 0)), (2*(c**2*d**6*sqrt(a*d*e + x*(a*e**2 + c*d**2)) 
/(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4) + 2*c*d**3*e*(a*d*e + x*(a*e**2 
 + c*d**2))**(3/2)/(3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)) + e**2*(a 
*d*e + x*(a*e**2 + c*d**2))**(5/2)/(5*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2* 
d**4)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), (Piecewise((d**2*x, Eq 
(e, 0)), ((d + e*x)**3/(3*e), True))/sqrt(a*d*e), True))
 

Maxima [F(-2)]

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

integrate((e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(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-c*d^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{4} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (\frac {2 \, e x}{c d} + \frac {5 \, c d^{2} e - 3 \, a e^{3}}{c^{2} d^{2} e}\right )} - \frac {3 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{8 \, \sqrt {c d e} c^{2} d^{2}} \] Input:

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

Output:

1/4*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*e*x/(c*d) + (5*c*d^2*e 
- 3*a*e^3)/(c^2*d^2*e)) - 3/8*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*log(abs( 
-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + 
 a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^2*d^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{3}+5 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} x +3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}-6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4}}{4 c^{3} d^{3} e} \] Input:

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

Output:

( - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d*e**3 + 5*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*c**2*d**3*e + 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**2*e**2* 
x + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqr 
t(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*e**4 - 6*sqrt(e)*sqrt(d)*s 
qrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqr 
t(a*e**2 - c*d**2))*a*c*d**2*e**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e) 
*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2)) 
*c**2*d**4)/(4*c**3*d**3*e)