\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^6} \, dx\) [241]

Optimal result

Integrand size = 37, antiderivative size = 277 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {35 c^2 d^2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^2}+\frac {35 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e^3 (d+e x)}-\frac {14 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{4 e^{9/2}} \] Output:

35/4*c^2*d^2*(a-c*d^2/e^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^2+35/ 
6*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)-14/3*c*d*(a* 
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e^2/(e*x+d)^3-2/3*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(7/2)/e/(e*x+d)^5+35/4*c^(3/2)*d^(3/2)*(-a*e^2+c*d^2)^2*arc 
tanh(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 = 0.48 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {e} \left (8 a^3 e^6+8 a^2 c d e^4 (7 d+10 e x)-a c^2 d^2 e^2 \left (175 d^2+238 d e x+39 e^2 x^2\right )+c^3 d^3 \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )\right )}{(d+e x)^2}+\frac {105 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{12 e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^6,x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[e]*(8*a^3*e^6 + 8*a^2*c*d*e^4*(7*d 
 + 10*e*x) - a*c^2*d^2*e^2*(175*d^2 + 238*d*e*x + 39*e^2*x^2) + c^3*d^3*(1 
05*d^3 + 140*d^2*e*x + 21*d*e^2*x^2 - 6*e^3*x^3)))/(d + e*x)^2) + (105*c^( 
3/2)*d^(3/2)*(c*d^2 - a*e^2)^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sq 
rt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(12*e^(9/2) 
)
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1130, 1125, 25, 2192, 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.

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {7 c d \left (\frac {\frac {\int \frac {c^2 d^2 e^4 \left (2 \left (2 c d^2-3 a e^2\right ) \left (c d^2-2 a e^2\right )-c d e \left (7 c d^2-9 a e^2\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \int \frac {2 \left (2 c d^2-3 a e^2\right ) \left (c d^2-2 a e^2\right )-c d e \left (7 c d^2-9 a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (\frac {15}{2} \left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (15 \left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7 c d \left (\frac {\frac {1}{4} c d e^3 \left (\frac {15 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}-\left (7 c d^2-9 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^4 x \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^6}-\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^3 (d+e x)}\right )}{3 e}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3 e (d+e x)^5}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^6,x]
 

Output:

(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(3*e*(d + e*x)^5) + (7* 
c*d*((-2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e 
^3*(d + e*x)) + ((c^2*d^2*e^4*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
])/2 + (c*d*e^3*(-((7*c*d^2 - 9*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2]) + (15*(c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*S 
qrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*S 
qrt[c]*Sqrt[d]*Sqrt[e])))/4)/e^6))/(3*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]
 

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]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(828\) vs. \(2(245)=490\).

Time = 4.04 (sec) , antiderivative size = 829, normalized size of antiderivative = 2.99

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}}}{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}}}{e^{6}}\)	\(829\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^6,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^6*(-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 = 0.77 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,x, algorithm=" 
fricas")
 

Output:

[1/48*(105*(c^3*d^7 - 2*a*c^2*d^5*e^2 + a^2*c*d^3*e^4 + (c^3*d^5*e^2 - 2*a 
*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 2*(c^3*d^6*e - 2*a*c^2*d^4*e^3 + a^2*c*d 
^2*e^5)*x)*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*(6*c^3*d^3*e^3 
*x^3 - 105*c^3*d^6 + 175*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 8*a^3*e^6 - 3* 
(7*c^3*d^4*e^2 - 13*a*c^2*d^2*e^4)*x^2 - 2*(70*c^3*d^5*e - 119*a*c^2*d^3*e 
^3 + 40*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^6* 
x^2 + 2*d*e^5*x + d^2*e^4), -1/24*(105*(c^3*d^7 - 2*a*c^2*d^5*e^2 + a^2*c* 
d^3*e^4 + (c^3*d^5*e^2 - 2*a*c^2*d^3*e^4 + a^2*c*d*e^6)*x^2 + 2*(c^3*d^6*e 
 - 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*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*(6*c^3*d^3*e^3* 
x^3 - 105*c^3*d^6 + 175*a*c^2*d^4*e^2 - 56*a^2*c*d^2*e^4 - 8*a^3*e^6 - 3*( 
7*c^3*d^4*e^2 - 13*a*c^2*d^2*e^4)*x^2 - 2*(70*c^3*d^5*e - 119*a*c^2*d^3*e^ 
3 + 40*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(e^6*x 
^2 + 2*d*e^5*x + d^2*e^4)]
 

Sympy [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{6}}\, dx \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**6,x)
 

Output:

Integral(((d + e*x)*(a*e + c*d*x))**(7/2)/(d + e*x)**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,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. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,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. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{{\left (d+e\,x\right )}^6} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^6,x)
                                                                                    
                                                                                    
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^6, x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 991, normalized size of antiderivative = 3.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^6,x)
 

Output:

( - 64*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*e**7 - 448*sqrt(d + e*x)*sqrt( 
a*e + c*d*x)*a**2*c*d**2*e**5 - 640*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c 
*d*e**6*x + 1400*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**3 + 1904*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x + 312*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a*c**2*d**2*e**5*x**2 - 840*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
c**3*d**6*e - 1120*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**5*e**2*x - 168* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**4*e**3*x**2 + 48*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*c**3*d**3*e**4*x**3 + 840*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*d**3*e**4 + 1680*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*d* 
*2*e**5*x + 840*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d*e**6*x**2 - 
1680*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 - 3360*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 - 1680*sqrt(e)*sqrt(d)*sqr 
t(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 + 840*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...