\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^2} \, dx\) [225]

Optimal result

Integrand size = 37, antiderivative size = 308 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {5 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d e^3}-\frac {5 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{96 c d e^2 (d+e x)}+\frac {\left (\frac {d}{e}-\frac {a e}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 (d+e x)^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4 c d (d+e x)^3}-\frac {5 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{64 c^{3/2} d^{3/2} e^{7/2}} \] Output:

5/64*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e^3-5/96 
*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e^2/(e*x+d)+ 
1/24*(d/e-a*e/c/d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^2+1/4*( 
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+d)^3-5/64*(-a*e^2+c*d^2)^4 
*arctanh(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/( 
e*x+d))/c^(3/2)/d^(3/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (15 a^3 e^6+a^2 c d e^4 (73 d+118 e x)+a c^2 d^2 e^2 \left (-55 d^2+36 d e x+136 e^2 x^2\right )+c^3 d^3 \left (15 d^3-10 d^2 e x+8 d e^2 x^2+48 e^3 x^3\right )\right )-\frac {15 \left (c d^2-a e^2\right )^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{192 c^{3/2} d^{3/2} e^{7/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(15*a^3*e^6 + a^2* 
c*d*e^4*(73*d + 118*e*x) + a*c^2*d^2*e^2*(-55*d^2 + 36*d*e*x + 136*e^2*x^2 
) + c^3*d^3*(15*d^3 - 10*d^2*e*x + 8*d*e^2*x^2 + 48*e^3*x^3)) - (15*(c*d^2 
 - a*e^2)^4*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c* 
d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(192*c^(3/2)*d^(3/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1127, 1134, 1160, 1087, 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.

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

\(\Big \downarrow \) 1127

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{2 e}\right )}{8 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{2 e}\right )}{8 e}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1127

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

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 [A] (verified)

Time = 1.86 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.32

method	result	size
default	\(\frac {\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {10 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{2}}\)	\(408\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^2,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^2*(2/3/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e)) 
^(7/2)-10/3*d*e*c/(a*e^2-c*d^2)*(1/5*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e 
))^(5/2)+1/2*(a*e^2-c*d^2)*(1/8*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c 
*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/d/e/c*(1/4*(2 
*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^ 
(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d* 
e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))) 
)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 672, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\left [\frac {15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e - 55 \, a c^{3} d^{5} e^{3} + 73 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (c^{4} d^{5} e^{3} + 17 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{6} e^{2} - 18 \, a c^{3} d^{4} e^{4} - 59 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{768 \, c^{2} d^{2} e^{4}}, \frac {15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (48 \, c^{4} d^{4} e^{4} x^{3} + 15 \, c^{4} d^{7} e - 55 \, a c^{3} d^{5} e^{3} + 73 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7} + 8 \, {\left (c^{4} d^{5} e^{3} + 17 \, a c^{3} d^{3} e^{5}\right )} x^{2} - 2 \, {\left (5 \, c^{4} d^{6} e^{2} - 18 \, a c^{3} d^{4} e^{4} - 59 \, a^{2} c^{2} d^{2} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{384 \, c^{2} d^{2} e^{4}}\right ] \] Input:

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

Output:

[1/768*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^ 
6 + a^4*e^8)*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*(48*c^4*d^4*e^4* 
x^3 + 15*c^4*d^7*e - 55*a*c^3*d^5*e^3 + 73*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^ 
7 + 8*(c^4*d^5*e^3 + 17*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 - 18*a*c^3*d 
^4*e^4 - 59*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x 
))/(c^2*d^2*e^4), 1/384*(15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 
 - 4*a^3*c*d^2*e^6 + a^4*e^8)*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*(48*c^4*d^4*e^4*x^ 
3 + 15*c^4*d^7*e - 55*a*c^3*d^5*e^3 + 73*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7 
+ 8*(c^4*d^5*e^3 + 17*a*c^3*d^3*e^5)*x^2 - 2*(5*c^4*d^6*e^2 - 18*a*c^3*d^4 
*e^4 - 59*a^2*c^2*d^2*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) 
/(c^2*d^2*e^4)]
 

Sympy [A] (verification not implemented)

Time = 137.23 (sec) , antiderivative size = 1416, normalized size of antiderivative = 4.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

a**2*e**2*Piecewise(((x/2 + (a*e**2/4 + c*d**2/4)/(c*d*e))*sqrt(a*d*e + c* 
d*e*x**2 + x*(a*e**2 + c*d**2)) + (a*d*e/2 - (a*e**2/4 + c*d**2/4)*(a*e**2 
 + c*d**2)/(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*(a*d*e + x*(a*e**2 + c*d**2)) 
**(3/2)/(3*(a*e**2 + c*d**2)), Ne(a*e**2 + c*d**2, 0)), (x*sqrt(a*d*e), Tr 
ue)) + 2*a*c*d*e*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e**2 + c* 
d**2)*(a*d*e/3 - (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e) 
)/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sq 
rt(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)) + (x**2/3 + x*(a*e**2/6 + c*d**2/6)/(2*c*d*e) + (a*d*e/3 
- (a*e**2/6 + c*d**2/6)*(3*a*e**2/2 + 3*c*d**2/2)/(2*c*d*e))/(c*d*e))*sqrt 
(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0)), (2*(-a*d*e*(a*d 
*e + x*(a*e**2 + c*d**2))**(3/2)/3 + (a*d*e + x*(a*e**2 + c*d**2))**(5/2)/ 
5)/(a*e**2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), (x**2*sqrt(a*d*e)/2, Tru 
e)) + c**2*d**2*Piecewise(((-a*(a*d*e/4 - (a*e**2/8 + c*d**2/8)*(5*a*e*...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1319 vs. \(2 (276) = 552\).

Time = 0.89 (sec) , antiderivative size = 1319, normalized size of antiderivative = 4.28 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/192*(15*(c^4*d^8*sgn(1/(e*x + d))*sgn(e) - 4*a*c^3*d^6*e^2*sgn(1/(e*x + 
d))*sgn(e) + 6*a^2*c^2*d^4*e^4*sgn(1/(e*x + d))*sgn(e) - 4*a^3*c*d^2*e^6*s 
gn(1/(e*x + d))*sgn(e) + a^4*e^8*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(c*d* 
e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))/sqrt(-c*d*e))/(sqrt(-c*d*e)*c*d*e 
^3*abs(e)) + (15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^7*d^1 
1*e^3*sgn(1/(e*x + d))*sgn(e) - 60*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/ 
(e*x + d))*a*c^6*d^9*e^5*sgn(1/(e*x + d))*sgn(e) + 90*sqrt(c*d*e - c*d^2*e 
/(e*x + d) + a*e^3/(e*x + d))*a^2*c^5*d^7*e^7*sgn(1/(e*x + d))*sgn(e) - 60 
*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^3*c^4*d^5*e^9*sgn(1/( 
e*x + d))*sgn(e) + 15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^ 
4*c^3*d^3*e^11*sgn(1/(e*x + d))*sgn(e) - 55*(c*d*e - c*d^2*e/(e*x + d) + a 
*e^3/(e*x + d))^(3/2)*c^6*d^10*e^2*sgn(1/(e*x + d))*sgn(e) + 220*(c*d*e - 
c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c^5*d^8*e^4*sgn(1/(e*x + d))* 
sgn(e) - 330*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^2*c^4*d 
^6*e^6*sgn(1/(e*x + d))*sgn(e) + 220*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e 
*x + d))^(3/2)*a^3*c^3*d^4*e^8*sgn(1/(e*x + d))*sgn(e) - 55*(c*d*e - c*d^2 
*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^4*c^2*d^2*e^10*sgn(1/(e*x + d))*sg 
n(e) + 73*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*c^5*d^9*e*sg 
n(1/(e*x + d))*sgn(e) - 292*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^ 
(5/2)*a*c^4*d^7*e^3*sgn(1/(e*x + d))*sgn(e) + 438*(c*d*e - c*d^2*e/(e*x...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.88 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{7}+73 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{5}+118 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{6} x -55 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{3}+36 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{4} x +136 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{5} x^{2}+15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{7} e -10 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{6} e^{2} x +8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{5} e^{3} x^{2}+48 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{4} d^{4} e^{4} x^{3}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{4} e^{8}+60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} c \,d^{2} e^{6}-90 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c^{2} d^{4} e^{4}+60 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{3} d^{6} e^{2}-15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{4} d^{8}}{192 c^{2} d^{2} e^{4}} \] Input:

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

Output:

(15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**7 + 73*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a**2*c**2*d**3*e**5 + 118*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a** 
2*c**2*d**2*e**6*x - 55*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**5*e**3 + 
 36*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**4*e**4*x + 136*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a*c**3*d**3*e**5*x**2 + 15*sqrt(d + e*x)*sqrt(a*e + c*d 
*x)*c**4*d**7*e - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**6*e**2*x + 8* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**4*d**5*e**3*x**2 + 48*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*c**4*d**4*e**4*x**3 - 15*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 
))*a**4*e**8 + 60*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))*a**3*c*d**2*e**6 - 
90*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))*a**2*c**2*d**4*e**4 + 60*sqrt(e)*s 
qrt(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))*a*c**3*d**6*e**2 - 15*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**4*d**8)/(192*c**2*d**2*e**4)