\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^8} \, dx\) [243]

Optimal result

Integrand size = 37, antiderivative size = 255 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=-\frac {2 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^4 (d+e x)}-\frac {2 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e^3 (d+e x)^3}-\frac {2 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e^2 (d+e x)^5}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 e (d+e x)^7}+\frac {2 c^{7/2} d^{7/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{9/2}} \] Output:

-2*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^4/(e*x+d)-2/3*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)^3-2/5*c*d*(a*d*e+(a*e 
^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e^2/(e*x+d)^5-2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^(7/2)/e/(e*x+d)^7+2*c^(7/2)*d^(7/2)*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 = 0.39 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {e} \left (15 a^3 e^6+3 a^2 c d e^4 (7 d+22 e x)+a c^2 d^2 e^2 \left (35 d^2+112 d e x+122 e^2 x^2\right )+c^3 d^3 \left (105 d^3+350 d^2 e x+406 d e^2 x^2+176 e^3 x^3\right )\right )}{(d+e x)^4}+\frac {105 c^{7/2} d^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{105 e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^8,x]
 

Output:

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(15*a^3*e^6 + 3*a^2*c*d*e^4*( 
7*d + 22*e*x) + a*c^2*d^2*e^2*(35*d^2 + 112*d*e*x + 122*e^2*x^2) + c^3*d^3 
*(105*d^3 + 350*d^2*e*x + 406*d*e^2*x^2 + 176*e^3*x^3)))/(d + e*x)^4) + (1 
05*c^(7/2)*d^(7/2)*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])))/(105*e^(9/2))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1130, 1130, 1130, 1125, 25, 27, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx\)

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^8,x]
 

Output:

(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(7*e*(d + e*x)^7) + (c* 
d*((-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*e*(d + e*x)^5) + 
(c*d*((-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*e*(d + e*x)^3) 
 + (c*d*((-2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e*(d + e*x)) + 
(Sqrt[c]*Sqrt[d]*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sq 
rt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/e^(3/2)))/e))/e))/e
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(990\) vs. \(2(227)=454\).

Time = 7.24 (sec) , antiderivative size = 991, normalized size of antiderivative = 3.89

method	result	size
default	\(\text {Expression too large to display}\)	\(991\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^8,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^8*(-2/7/(a*e^2-c*d^2)/(x+d/e)^8*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e) 
)^(9/2)+2/7*d*e*c/(a*e^2-c*d^2)*(-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)*(1/12*(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*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 = 3.53 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.92 \[ \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)^8} \, dx=\left [\frac {105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {\frac {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 \, {\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {c d}{e}} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (176 \, c^{3} d^{3} e^{3} x^{3} + 105 \, c^{3} d^{6} + 35 \, a c^{2} d^{4} e^{2} + 21 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 2 \, {\left (203 \, c^{3} d^{4} e^{2} + 61 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (175 \, c^{3} d^{5} e + 56 \, a c^{2} d^{3} e^{3} + 33 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{210 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}}, -\frac {105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {-\frac {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 {-\frac {c d}{e}}}{2 \, {\left (c^{2} d^{2} e x^{2} + a c d^{2} e + {\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )}}\right ) + 2 \, {\left (176 \, c^{3} d^{3} e^{3} x^{3} + 105 \, c^{3} d^{6} + 35 \, a c^{2} d^{4} e^{2} + 21 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 2 \, {\left (203 \, c^{3} d^{4} e^{2} + 61 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (175 \, c^{3} d^{5} e + 56 \, a c^{2} d^{3} e^{3} + 33 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{105 \, {\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^8,x, algorithm=" 
fricas")
 

Output:

[1/210*(105*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c 
^3*d^6*e*x + c^3*d^7)*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*(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*(17 
6*c^3*d^3*e^3*x^3 + 105*c^3*d^6 + 35*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 15 
*a^3*e^6 + 2*(203*c^3*d^4*e^2 + 61*a*c^2*d^2*e^4)*x^2 + 2*(175*c^3*d^5*e + 
 56*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a 
*e^2)*x))/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4), 
 -1/105*(105*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4* 
c^3*d^6*e*x + c^3*d^7)*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*(176*c^3*d^3*e^3*x^3 + 105*c^3* 
d^6 + 35*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 15*a^3*e^6 + 2*(203*c^3*d^4*e^ 
2 + 61*a*c^2*d^2*e^4)*x^2 + 2*(175*c^3*d^5*e + 56*a*c^2*d^3*e^3 + 33*a^2*c 
*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^8*x^4 + 4*d*e^7 
*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5*x + d^4*e^4)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**8,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^8,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. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^8,x, algorithm=" 
giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{1,[0,0,20]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[8,8 
]%%%}+%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{{\left (d+e\,x\right )}^8} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^8,x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^8, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^8,x)
 

Output:

(2*( - 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*e**7 - 21*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**2*c*d**2*e**5 - 66*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2* 
c*d*e**6*x - 35*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**3 - 112*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x - 122*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a*c**2*d**2*e**5*x**2 - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c* 
*3*d**6*e - 350*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e**2*x - 406*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x**2 - 176*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*c**3*d**3*e**4*x**3 + 105*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 + 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**6*e*x + 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**2))*c**3*d**5*e**2*x**2 + 420*sqrt(e)*sq 
rt(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**4*e**3*x**3 + 105*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**3*e**4*x**4 + 56*sqrt(e)*sqrt(d)*sqrt(c)*c**3*d**7 
 + 224*sqrt(e)*sqrt(d)*sqrt(c)*c**3*d**6*e*x + 336*sqrt(e)*sqrt(d)*sqrt(c) 
*c**3*d**5*e**2*x**2 + 224*sqrt(e)*sqrt(d)*sqrt(c)*c**3*d**4*e**3*x**3 + 5 
6*sqrt(e)*sqrt(d)*sqrt(c)*c**3*d**3*e**4*x**4))/(105*e**5*(d**4 + 4*d**...