\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [224]

Optimal result

Integrand size = 37, antiderivative size = 373 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^2 d^2 e^2 (d+e x)}+\frac {\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{80 c^2 d^2 e (d+e x)^2}+\frac {3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{40 c^2 d^2 (d+e x)^3}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{5 c d (d+e x)^2}-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{128 c^{5/2} d^{5/2} e^{7/2}} \] Output:

3/128*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e^3 
-1/64*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e^2 
/(e*x+d)+1/80*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2 
/d^2/e/(e*x+d)^2+3/40*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/ 
2)/c^2/d^2/(e*x+d)^3+1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+ 
d)^2-3/128*(-a*e^2+c*d^2)^5*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^(5/2)/d^(5/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-15 a^4 e^8+10 a^3 c d e^6 (7 d+e x)+2 a^2 c^2 d^2 e^4 \left (64 d^2+233 d e x+124 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (-35 d^3+23 d^2 e x+256 d e^2 x^2+168 e^3 x^3\right )+c^4 d^4 \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )}{(a e+c d x) (d+e x)}-\frac {15 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 c^{5/2} d^{5/2} e^{7/2}} \] Input:

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

Output:

(((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-15*a^4*e^8 + 
10*a^3*c*d*e^6*(7*d + e*x) + 2*a^2*c^2*d^2*e^4*(64*d^2 + 233*d*e*x + 124*e 
^2*x^2) + 2*a*c^3*d^3*e^2*(-35*d^3 + 23*d^2*e*x + 256*d*e^2*x^2 + 168*e^3* 
x^3) + c^4*d^4*(15*d^4 - 10*d^3*e*x + 8*d^2*e^2*x^2 + 176*d*e^3*x^3 + 128* 
e^4*x^4)))/((a*e + c*d*x)*(d + e*x)) - (15*(c*d^2 - a*e^2)^5*ArcTanh[(Sqrt 
[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(3 
/2)*(d + e*x)^(3/2))))/(640*c^(5/2)*d^(5/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1131, 1087, 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.

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{2 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{2 e}\)

Input:

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

Output:

(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*e) - ((c*d^2 - a*e^2)*((( 
c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/ 
(8*c*d*e) - (3*(c*d^2 - a*e^2)^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))))/(16*c*d*e)))/(2*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 1131

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

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{2}}{e}\)	\(327\)

Input:

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

Output:

1/e*(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 = 0.13 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2 
*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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*(128*c^5*d^5*e^5*x^4 + 15*c^5*d^9*e - 70*a*c^4*d^7*e^3 + 128*a^2 
*c^3*d^5*e^5 + 70*a^3*c^2*d^3*e^7 - 15*a^4*c*d*e^9 + 16*(11*c^5*d^6*e^4 + 
21*a*c^4*d^4*e^6)*x^3 + 8*(c^5*d^7*e^3 + 64*a*c^4*d^5*e^5 + 31*a^2*c^3*d^3 
*e^7)*x^2 - 2*(5*c^5*d^8*e^2 - 23*a*c^4*d^6*e^4 - 233*a^2*c^3*d^4*e^6 - 5* 
a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3* 
e^4), 1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3 
*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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*(128* 
c^5*d^5*e^5*x^4 + 15*c^5*d^9*e - 70*a*c^4*d^7*e^3 + 128*a^2*c^3*d^5*e^5 + 
70*a^3*c^2*d^3*e^7 - 15*a^4*c*d*e^9 + 16*(11*c^5*d^6*e^4 + 21*a*c^4*d^4*e^ 
6)*x^3 + 8*(c^5*d^7*e^3 + 64*a*c^4*d^5*e^5 + 31*a^2*c^3*d^3*e^7)*x^2 - 2*( 
5*c^5*d^8*e^2 - 23*a*c^4*d^6*e^4 - 233*a^2*c^3*d^4*e^6 - 5*a^3*c^2*d^2*e^8 
)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^4)]
 

Sympy [A] (verification not implemented)

Time = 35.97 (sec) , antiderivative size = 3516, normalized size of antiderivative = 9.43 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \] Input:

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

Output:

a**2*d*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*sqr 
t(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), 
True)) + a**2*e**3*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)* 
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)) + (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))*sq 
rt(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, T 
rue)) + 2*a*c*d**2*e*Piecewise(((-a*(a*e**2/6 + c*d**2/6)/(2*c) - (a*e*...
 

Maxima [F(-2)]

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

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

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 [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {1}{640} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c^{2} d^{2} e x + \frac {11 \, c^{6} d^{7} e^{4} + 21 \, a c^{5} d^{5} e^{6}}{c^{4} d^{4} e^{4}}\right )} x + \frac {c^{6} d^{8} e^{3} + 64 \, a c^{5} d^{6} e^{5} + 31 \, a^{2} c^{4} d^{4} e^{7}}{c^{4} d^{4} e^{4}}\right )} x - \frac {5 \, c^{6} d^{9} e^{2} - 23 \, a c^{5} d^{7} e^{4} - 233 \, a^{2} c^{4} d^{5} e^{6} - 5 \, a^{3} c^{3} d^{3} e^{8}}{c^{4} d^{4} e^{4}}\right )} x + \frac {15 \, c^{6} d^{10} e - 70 \, a c^{5} d^{8} e^{3} + 128 \, a^{2} c^{4} d^{6} e^{5} + 70 \, a^{3} c^{3} d^{4} e^{7} - 15 \, a^{4} c^{2} d^{2} e^{9}}{c^{4} d^{4} e^{4}}\right )} + \frac {3 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{256 \, \sqrt {c d e} c^{2} d^{2} e^{3}} \] Input:

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

Output:

1/640*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*c^2*d^2*e*x 
+ (11*c^6*d^7*e^4 + 21*a*c^5*d^5*e^6)/(c^4*d^4*e^4))*x + (c^6*d^8*e^3 + 64 
*a*c^5*d^6*e^5 + 31*a^2*c^4*d^4*e^7)/(c^4*d^4*e^4))*x - (5*c^6*d^9*e^2 - 2 
3*a*c^5*d^7*e^4 - 233*a^2*c^4*d^5*e^6 - 5*a^3*c^3*d^3*e^8)/(c^4*d^4*e^4))* 
x + (15*c^6*d^10*e - 70*a*c^5*d^8*e^3 + 128*a^2*c^4*d^6*e^5 + 70*a^3*c^3*d 
^4*e^7 - 15*a^4*c^2*d^2*e^9)/(c^4*d^4*e^4)) + 3/256*(c^5*d^10 - 5*a*c^4*d^ 
8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^ 
10)*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*e^3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d*e**9 + 70*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**3*c**2*d**3*e**7 + 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**3*c**2*d**2*e**8*x + 128*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**5* 
e**5 + 466*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**4*e**6*x + 248*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**7*x**2 - 70*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a*c**4*d**7*e**3 + 46*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c 
**4*d**6*e**4*x + 512*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**5*e**5*x** 
2 + 336*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e**6*x**3 + 15*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*c**5*d**9*e - 10*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
c**5*d**8*e**2*x + 8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**7*e**3*x**2 + 
 176*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**6*e**4*x**3 + 128*sqrt(d + e* 
x)*sqrt(a*e + c*d*x)*c**5*d**5*e**5*x**4 + 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**5*e**10 - 75*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*c*d**2* 
e**8 + 150*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**2*d**4*e**6 - 150* 
sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*s 
qrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c**3*d**6*e**4 + 75*sqrt(e)*sqrt 
(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e...