\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^2} \, dx\) [237]

Optimal result

Integrand size = 37, antiderivative size = 436 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=-\frac {7 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^2 d^2 e^4}+\frac {7 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{768 c^2 d^2 e^3 (d+e x)}-\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{960 c^2 d^2 e^2 (d+e x)^2}+\frac {\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{160 c^2 d^2 e (d+e x)^3}+\frac {\left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{20 c^2 d^2 (d+e x)^4}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{6 c d (d+e x)^3}+\frac {7 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{512 c^{5/2} d^{5/2} e^{9/2}} \] Output:

-7/512*(-a*e^2+c*d^2)^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2/e^ 
4+7/768*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2/e 
^3/(e*x+d)-7/960*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/ 
c^2/d^2/e^2/(e*x+d)^2+1/160*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(7/2)/c^2/d^2/e/(e*x+d)^3+1/20*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*e*x^2)^(9/2)/c^2/d^2/(e*x+d)^4+1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^( 
9/2)/c/d/(e*x+d)^3+7/512*(-a*e^2+c*d^2)^6*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^(9/2)
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+35 a^4 c d e^8 (17 d+2 e x)+2 a^3 c^2 d^2 e^6 \left (843 d^2+2876 d e x+1508 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (-231 d^3+150 d^2 e x+1588 d e^2 x^2+1032 e^3 x^3\right )+a c^4 d^4 e^2 \left (595 d^4-392 d^3 e x+312 d^2 e^2 x^2+6560 d e^3 x^3+4736 e^4 x^4\right )+c^5 d^5 \left (-105 d^5+70 d^4 e x-56 d^3 e^2 x^2+48 d^2 e^3 x^3+1664 d e^4 x^4+1280 e^5 x^5\right )\right )}{(a e+c d x) (d+e x)}+\frac {105 \left (c d^2-a e^2\right )^6 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 c^{5/2} d^{5/2} e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^2,x]
 

Output:

(((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 
+ 35*a^4*c*d*e^8*(17*d + 2*e*x) + 2*a^3*c^2*d^2*e^6*(843*d^2 + 2876*d*e*x 
+ 1508*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(-231*d^3 + 150*d^2*e*x + 1588*d*e^2*x 
^2 + 1032*e^3*x^3) + a*c^4*d^4*e^2*(595*d^4 - 392*d^3*e*x + 312*d^2*e^2*x^ 
2 + 6560*d*e^3*x^3 + 4736*e^4*x^4) + c^5*d^5*(-105*d^5 + 70*d^4*e*x - 56*d 
^3*e^2*x^2 + 48*d^2*e^3*x^3 + 1664*d*e^4*x^4 + 1280*e^5*x^5)))/((a*e + c*d 
*x)*(d + e*x)) + (105*(c*d^2 - a*e^2)^6*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))) 
)/(7680*c^(5/2)*d^(5/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1127, 1134, 1160, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 1127

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

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \int (a e+c d x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{12 e}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{2 e}\right )}{12 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{2 e}\right )}{12 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{2 e}\right )}{12 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{2 e}\right )}{12 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{2 e}\right )}{12 e}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(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)^(5/2))/(6*e) - (7*( 
c*d^2 - a*e^2)*((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)))/(12*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 = 1.85 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.16

method	result	size
default	\(\frac {\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {14 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 d e c}-\frac {5 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )}{2}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{2}}\)	\(505\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^2,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^2*(2/5/(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)) 
^(9/2)-14/5*d*e*c/(a*e^2-c*d^2)*(1/7*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e 
))^(7/2)+1/2*(a*e^2-c*d^2)*(1/12*(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))^(5/2)-5/24*(a*e^2-c*d^2)^2/d/e/c*(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*l 
n((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 = 1042, normalized size of antiderivative = 2.39 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^2,x, algorithm=" 
fricas")
                                                                                    
                                                                                    
 

Output:

[1/30720*(105*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c 
^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 - 105*c^6*d^11*e + 
 595*a*c^5*d^9*e^3 - 1386*a^2*c^4*d^7*e^5 + 1686*a^3*c^3*d^5*e^7 + 595*a^4 
*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 37*a*c^5*d^5*e^7)* 
x^4 + 16*(3*c^6*d^8*e^4 + 410*a*c^5*d^6*e^6 + 387*a^2*c^4*d^4*e^8)*x^3 - 8 
*(7*c^6*d^9*e^3 - 39*a*c^5*d^7*e^5 - 1191*a^2*c^4*d^5*e^7 - 377*a^3*c^3*d^ 
3*e^9)*x^2 + 2*(35*c^6*d^10*e^2 - 196*a*c^5*d^8*e^4 + 450*a^2*c^4*d^6*e^6 
+ 2876*a^3*c^3*d^4*e^8 + 35*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + 
(c*d^2 + a*e^2)*x))/(c^3*d^3*e^5), -1/15360*(105*(c^6*d^12 - 6*a*c^5*d^10* 
e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5 
*c*d^2*e^10 + a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 - 
105*c^6*d^11*e + 595*a*c^5*d^9*e^3 - 1386*a^2*c^4*d^7*e^5 + 1686*a^3*c^3*d 
^5*e^7 + 595*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(13*c^6*d^7*e^5 + 37 
*a*c^5*d^5*e^7)*x^4 + 16*(3*c^6*d^8*e^4 + 410*a*c^5*d^6*e^6 + 387*a^2*c^4* 
d^4*e^8)*x^3 - 8*(7*c^6*d^9*e^3 - 39*a*c^5*d^7*e^5 - 1191*a^2*c^4*d^5*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(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. 2611 vs. \(2 (396) = 792\).

Time = 1.22 (sec) , antiderivative size = 2611, normalized size of antiderivative = 5.99 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^2,x, algorithm=" 
giac")
 

Output:

-1/7680*(105*(c^6*d^12*sgn(1/(e*x + d))*sgn(e) - 6*a*c^5*d^10*e^2*sgn(1/(e 
*x + d))*sgn(e) + 15*a^2*c^4*d^8*e^4*sgn(1/(e*x + d))*sgn(e) - 20*a^3*c^3* 
d^6*e^6*sgn(1/(e*x + d))*sgn(e) + 15*a^4*c^2*d^4*e^8*sgn(1/(e*x + d))*sgn( 
e) - 6*a^5*c*d^2*e^10*sgn(1/(e*x + d))*sgn(e) + a^6*e^12*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^2*d^2*e^4*abs(e)) + (105*sqrt(c*d*e - c*d^2*e/(e*x + 
d) + a*e^3/(e*x + d))*c^11*d^17*e^5*sgn(1/(e*x + d))*sgn(e) - 630*sqrt(c*d 
*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a*c^10*d^15*e^7*sgn(1/(e*x + d)) 
*sgn(e) + 1575*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c^9*d 
^13*e^9*sgn(1/(e*x + d))*sgn(e) - 2100*sqrt(c*d*e - c*d^2*e/(e*x + d) + a* 
e^3/(e*x + d))*a^3*c^8*d^11*e^11*sgn(1/(e*x + d))*sgn(e) + 1575*sqrt(c*d*e 
 - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c^7*d^9*e^13*sgn(1/(e*x + d))* 
sgn(e) - 630*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^5*c^6*d^7 
*e^15*sgn(1/(e*x + d))*sgn(e) + 105*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3 
/(e*x + d))*a^6*c^5*d^5*e^17*sgn(1/(e*x + d))*sgn(e) - 595*(c*d*e - c*d^2* 
e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^10*d^16*e^4*sgn(1/(e*x + d))*sgn(e) 
 + 3570*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a*c^9*d^14*e^6 
*sgn(1/(e*x + d))*sgn(e) - 8925*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + 
d))^(3/2)*a^2*c^8*d^12*e^8*sgn(1/(e*x + d))*sgn(e) + 11900*(c*d*e - c*d^2* 
e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*a^3*c^7*d^10*e^10*sgn(1/(e*x + d))...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{{\left (d+e\,x\right )}^2} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^2,x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^2, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1033, normalized size of antiderivative = 2.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^2,x)
 

Output:

( - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d*e**11 + 595*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**4*c**2*d**3*e**9 + 70*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**4*c**2*d**2*e**10*x + 1686*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3* 
d**5*e**7 + 5752*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**4*e**8*x + 3 
016*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**9*x**2 - 1386*sqrt(d 
 + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**7*e**5 + 900*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a**2*c**4*d**6*e**6*x + 9528*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a** 
2*c**4*d**5*e**7*x**2 + 6192*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d** 
4*e**8*x**3 + 595*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**9*e**3 - 392*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**8*e**4*x + 312*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a*c**5*d**7*e**5*x**2 + 6560*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a*c**5*d**6*e**6*x**3 + 4736*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5* 
e**7*x**4 - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**11*e + 70*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*c**6*d**10*e**2*x - 56*sqrt(d + e*x)*sqrt(a*e + c* 
d*x)*c**6*d**9*e**3*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**8*e* 
*4*x**3 + 1664*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**5*x**4 + 1280* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**6*e**6*x**5 + 105*sqrt(e)*sqrt(d)* 
sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sq 
rt(a*e**2 - c*d**2))*a**6*e**12 - 630*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**...