\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^7} \, dx\) [242]

Optimal result

Integrand size = 37, antiderivative size = 260 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {7 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^4}-\frac {14 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^3 (d+e x)^2}-\frac {14 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{15 e^2 (d+e x)^4}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{5 e (d+e x)^6}-\frac {7 c^{5/2} d^{5/2} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{9/2}} \] Output:

7*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^4-14/3*c^2*d^2*(a*d*e+ 
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^3/(e*x+d)^2-14/15*c*d*(a*d*e+(a*e^2+c*d 
^2)*x+c*d*e*x^2)^(5/2)/e^2/(e*x+d)^4-2/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2) 
^(7/2)/e/(e*x+d)^6-7*c^(5/2)*d^(5/2)*(-a*e^2+c*d^2)*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))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^7} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {e} \left (-6 a^3 e^6-2 a^2 c d e^4 (7 d+16 e x)-2 a c^2 d^2 e^2 \left (35 d^2+84 d e x+58 e^2 x^2\right )+c^3 d^3 \left (105 d^3+245 d^2 e x+161 d e^2 x^2+15 e^3 x^3\right )\right )}{(d+e x)^3}-\frac {105 c^{5/2} d^{5/2} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{15 e^{9/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*(-6*a^3*e^6 - 2*a^2*c*d*e^4*(7*d 
+ 16*e*x) - 2*a*c^2*d^2*e^2*(35*d^2 + 84*d*e*x + 58*e^2*x^2) + c^3*d^3*(10 
5*d^3 + 245*d^2*e*x + 161*d*e^2*x^2 + 15*e^3*x^3)))/(d + e*x)^3 - (105*c^( 
5/2)*d^(5/2)*(c*d^2 - a*e^2)*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])))/(15*e^(9/2))
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1130, 1130, 1125, 27, 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.

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(5*e*(d + e*x)^6) + (7* 
c*d*((-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(3*e*(d + e*x)^4) 
+ (5*c*d*((-2*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
])/(d + e*x) - (c*d*(-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] + (3*(c* 
d^2 - a*e^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])])/(2*Sqrt[c]*Sqrt[d]*Sqrt[e 
])))/e^2))/(3*e)))/(5*e)
 

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 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 1125

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 
2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[(1/Sqrt[a + b*x + c*x^2])*Expan 
dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x 
), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && ILtQ[m, 0] && EqQ[m + p, -3/2]
 

rule 1130

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 + p + 1))), x] 
- Simp[c*(p/(e^2*(m + p + 1)))   Int[(d + e*x)^(m + 2)*(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] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + 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. \(909\) vs. \(2(232)=464\).

Time = 5.47 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.50

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 )^{7}}+\frac {4 d e c \left (-\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}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {2 d e c \left (-\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}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {8 d e c \left (\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}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {10 d e c \left (\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}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 d e c \left (\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 )}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{7}}\)	\(910\)

Input:

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

Output:

1/e^7*(-2/5/(a*e^2-c*d^2)/(x+d/e)^7*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e) 
)^(9/2)+4/5*d*e*c/(a*e^2-c*d^2)*(-2/3/(a*e^2-c*d^2)/(x+d/e)^6*(d*e*c*(x+d/ 
e)^2+(a*e^2-c*d^2)*(x+d/e))^(9/2)+2*d*e*c/(a*e^2-c*d^2)*(-2/(a*e^2-c*d^2)/ 
(x+d/e)^5*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(9/2)+8*d*e*c/(a*e^2-c*d 
^2)*(2/(a*e^2-c*d^2)/(x+d/e)^4*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(9/ 
2)-10*d*e*c/(a*e^2-c*d^2)*(2/3/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e)^2+(a 
*e^2-c*d^2)*(x+d/e))^(9/2)-4*d*e*c/(a*e^2-c*d^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)*(1/1 
2*(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*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 = 1.16 (sec) , antiderivative size = 798, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/60*(105*(c^3*d^7 - a*c^2*d^5*e^2 + (c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^3 + 
 3*(c^3*d^5*e^2 - a*c^2*d^3*e^4)*x^2 + 3*(c^3*d^6*e - a*c^2*d^4*e^3)*x)*sq 
rt(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*(2 
*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)* 
sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(15*c^3*d^3*e^3*x^3 + 105*c 
^3*d^6 - 70*a*c^2*d^4*e^2 - 14*a^2*c*d^2*e^4 - 6*a^3*e^6 + (161*c^3*d^4*e^ 
2 - 116*a*c^2*d^2*e^4)*x^2 + (245*c^3*d^5*e - 168*a*c^2*d^3*e^3 - 32*a^2*c 
*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^7*x^3 + 3*d*e^6 
*x^2 + 3*d^2*e^5*x + d^3*e^4), 1/30*(105*(c^3*d^7 - a*c^2*d^5*e^2 + (c^3*d 
^4*e^3 - a*c^2*d^2*e^5)*x^3 + 3*(c^3*d^5*e^2 - a*c^2*d^3*e^4)*x^2 + 3*(c^3 
*d^6*e - a*c^2*d^4*e^3)*x)*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*x 
^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) + 2*(15*c^3*d^3*e^3*x^3 + 105*c 
^3*d^6 - 70*a*c^2*d^4*e^2 - 14*a^2*c*d^2*e^4 - 6*a^3*e^6 + (161*c^3*d^4*e^ 
2 - 116*a*c^2*d^2*e^4)*x^2 + (245*c^3*d^5*e - 168*a*c^2*d^3*e^3 - 32*a^2*c 
*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^7*x^3 + 3*d*e^6 
*x^2 + 3*d^2*e^5*x + d^3*e^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^7,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*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 931, normalized size of antiderivative = 3.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*e**7 - 56*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a**2*c*d**2*e**5 - 128*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c* 
d*e**6*x - 280*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**3 - 672*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x - 464*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*a*c**2*d**2*e**5*x**2 + 420*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c** 
3*d**6*e + 980*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e**2*x + 644*sqrt 
(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x**2 + 60*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*c**3*d**3*e**4*x**3 + 420*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*c**2*d**5*e**2 + 1260*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*c**2*d**4*e 
**3*x + 1260*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*c**2*d**3*e**4*x**2 + 
420*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*c**2*d**2*e**5*x**3 - 420*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**3*d**7 - 1260*sqrt(e)*sqrt(d)*sqrt(c)*l 
og((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 
 - c*d**2))*c**3*d**6*e*x - 1260*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...