\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^3} \, dx\) [238]

Optimal result

Integrand size = 37, antiderivative size = 371 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=-\frac {7 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c d e^4}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 c d e^3 (d+e x)}-\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{240 c d e^2 (d+e x)^2}+\frac {\left (\frac {d}{e}-\frac {a e}{c d}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{40 (d+e x)^3}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{5 c d (d+e x)^4}+\frac {7 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{128 c^{3/2} d^{3/2} e^{9/2}} \] Output:

-7/128*(-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/d/e^4+7/ 
192*(-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/d/e^3/(e*x+ 
d)-7/240*(-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/d/e^2/ 
(e*x+d)^2+1/40*(d/e-a*e/c/d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+ 
d)^3+1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/c/d/(e*x+d)^4+7/128*(-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^(3/2)/d^(3/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (105 a^4 e^8+10 a^3 c d e^6 (79 d+121 e x)+2 a^2 c^2 d^2 e^4 \left (-448 d^2+289 d e x+1052 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (245 d^3-161 d^2 e x+128 d e^2 x^2+744 e^3 x^3\right )+c^4 d^4 \left (-105 d^4+70 d^3 e x-56 d^2 e^2 x^2+48 d e^3 x^3+384 e^4 x^4\right )\right )+\frac {105 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 c^{3/2} d^{3/2} e^{9/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^3,x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(105*a^4*e^8 + 10* 
a^3*c*d*e^6*(79*d + 121*e*x) + 2*a^2*c^2*d^2*e^4*(-448*d^2 + 289*d*e*x + 1 
052*e^2*x^2) + 2*a*c^3*d^3*e^2*(245*d^3 - 161*d^2*e*x + 128*d*e^2*x^2 + 74 
4*e^3*x^3) + c^4*d^4*(-105*d^4 + 70*d^3*e*x - 56*d^2*e^2*x^2 + 48*d*e^3*x^ 
3 + 384*e^4*x^4)) + (105*(c*d^2 - a*e^2)^5*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]))) 
/(1920*c^(3/2)*d^(3/2)*e^(9/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1127, 1134, 1134, 1160, 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.

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

\(\Big \downarrow \) 1127

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

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \int (a e+c d x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{10 e}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \int (a e+c d x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{8 e}\right )}{10 e}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{2 e}\right )}{8 e}\right )}{10 e}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{2 e}\right )}{8 e}\right )}{10 e}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{2 e}\right )}{8 e}\right )}{10 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(a e+c d x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e}-\frac {7 \left (c d^2-a e^2\right ) \left (\frac {(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e}-\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{2 e}\right )}{8 e}\right )}{10 e}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^3,x]
 

Output:

((a*e + c*d*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) - (7 
*(c*d^2 - a*e^2)*(((a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
3/2))/(4*e) - (5*(c*d^2 - a*e^2)*((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^ 
(3/2)/(3*e) - ((c*d^2 - a*e^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))))/(2*e)))/(8*e)))/(10 
*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 = 2.16 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.58

method	result	size
default	\(\frac {\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {14 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 d e c}-\frac {5 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 d e c}-\frac {3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )}{2}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{a \,e^{2}-c \,d^{2}}}{e^{3}}\)	\(586\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^3,x,method=_RETURNVERB 
OSE)
 

Output:

1/e^3*(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 = 0.11 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^3,x, algorithm=" 
fricas")
                                                                                    
                                                                                    
 

Output:

[-1/7680*(105*(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*(384*c^5*d^5*e^5*x^4 - 105*c^5*d^9*e + 490*a*c^4*d^7*e^3 - 896* 
a^2*c^3*d^5*e^5 + 790*a^3*c^2*d^3*e^7 + 105*a^4*c*d*e^9 + 48*(c^5*d^6*e^4 
+ 31*a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 32*a*c^4*d^5*e^5 - 263*a^2*c^ 
3*d^3*e^7)*x^2 + 2*(35*c^5*d^8*e^2 - 161*a*c^4*d^6*e^4 + 289*a^2*c^3*d^4*e 
^6 + 605*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^2*d^2*e^5), -1/3840*(105*(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*(384*c^5*d^5*e^5*x^4 - 105*c^5*d^9*e + 490*a*c^4*d^7*e^3 - 896*a^2*c 
^3*d^5*e^5 + 790*a^3*c^2*d^3*e^7 + 105*a^4*c*d*e^9 + 48*(c^5*d^6*e^4 + 31* 
a*c^4*d^4*e^6)*x^3 - 8*(7*c^5*d^7*e^3 - 32*a*c^4*d^5*e^5 - 263*a^2*c^3*d^3 
*e^7)*x^2 + 2*(35*c^5*d^8*e^2 - 161*a*c^4*d^6*e^4 + 289*a^2*c^3*d^4*e^6 + 
605*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^2* 
d^2*e^5)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^3,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 [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=\frac {1}{1920} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c^{3} d^{3} x + \frac {c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}}{c^{4} d^{4} e^{4}}\right )} x - \frac {7 \, c^{7} d^{9} e^{2} - 32 \, a c^{6} d^{7} e^{4} - 263 \, a^{2} c^{5} d^{5} e^{6}}{c^{4} d^{4} e^{4}}\right )} x + \frac {35 \, c^{7} d^{10} e - 161 \, a c^{6} d^{8} e^{3} + 289 \, a^{2} c^{5} d^{6} e^{5} + 605 \, a^{3} c^{4} d^{4} e^{7}}{c^{4} d^{4} e^{4}}\right )} x - \frac {105 \, c^{7} d^{11} - 490 \, a c^{6} d^{9} e^{2} + 896 \, a^{2} c^{5} d^{7} e^{4} - 790 \, a^{3} c^{4} d^{5} e^{6} - 105 \, a^{4} c^{3} d^{3} e^{8}}{c^{4} d^{4} e^{4}}\right )} - \frac {7 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{256 \, \sqrt {c d e} c d e^{4}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^3,x, algorithm=" 
giac")
 

Output:

1/1920*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(6*(8*c^3*d^3*x + 
 (c^7*d^8*e^3 + 31*a*c^6*d^6*e^5)/(c^4*d^4*e^4))*x - (7*c^7*d^9*e^2 - 32*a 
*c^6*d^7*e^4 - 263*a^2*c^5*d^5*e^6)/(c^4*d^4*e^4))*x + (35*c^7*d^10*e - 16 
1*a*c^6*d^8*e^3 + 289*a^2*c^5*d^6*e^5 + 605*a^3*c^4*d^4*e^7)/(c^4*d^4*e^4) 
)*x - (105*c^7*d^11 - 490*a*c^6*d^9*e^2 + 896*a^2*c^5*d^7*e^4 - 790*a^3*c^ 
4*d^5*e^6 - 105*a^4*c^3*d^3*e^8)/(c^4*d^4*e^4)) - 7/256*(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*d*e^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{7/2}}{{\left (d+e\,x\right )}^3} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^3,x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.13 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^3,x)
 

Output:

(105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c*d*e**9 + 790*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**3*c**2*d**3*e**7 + 1210*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**3*c**2*d**2*e**8*x - 896*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**5 
*e**5 + 578*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**4*e**6*x + 2104*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**7*x**2 + 490*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a*c**4*d**7*e**3 - 322*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a*c**4*d**6*e**4*x + 256*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**5*e**5 
*x**2 + 1488*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e**6*x**3 - 105*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**9*e + 70*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*c**5*d**8*e**2*x - 56*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**7*e**3 
*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**6*e**4*x**3 + 384*sqrt( 
d + e*x)*sqrt(a*e + c*d*x)*c**5*d**5*e**5*x**4 - 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))*a**5*e**10 + 525*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 - 1050*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 + 1050*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**3*d**6*e**4 - 525*s 
qrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)...