\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [1934]

Optimal result

Integrand size = 35, antiderivative size = 356 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {5 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {5 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}} \]

[Out]

-5/384*(-a*e^2+c*d^2)^3*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/e^2+1/24*(-a*e
^2+c*d^2)*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2/e+1/7*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(7/2)/c/d-5/2048*(-a*e^2+c*d^2)^7*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)/e^(7/2)+5/1024*(-a*e^2+c*d^2)^5*(2*c*d*e*x+a*e^2+c*d^2)*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^3

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {654, 626, 635, 212} \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=-\frac {5 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}}+\frac {5 \left (c d^2-a e^2\right )^5 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d} \]

[In]

Int[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(5*(c*d^2 - a*e^2)^5*(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])/(1024*c^4*d^4*e^
3) - (5*(c*d^2 - a*e^2)^3*(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))/(384*c^3*
d^3*e^2) + ((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)^(5/2))/(24*c^2
*d^2*e) + (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(7*c*d) - (5*(c*d^2 - a*e^2)^7*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])])/(2048*c^(9/2)*d^(9/2)*e
^(7/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

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] - Dist[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]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 654

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{2 d} \\ & = \frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^3\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{48 c^2 d^2 e} \\ & = -\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (5 \left (c d^2-a e^2\right )^5\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^2} \\ & = \frac {5 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2048 c^4 d^4 e^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {\left (5 \left (c d^2-a e^2\right )^7\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^4 d^4 e^3} \\ & = \frac {5 \left (c d^2-a e^2\right )^5 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^2}+\frac {\left (c d^2-a e^2\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 c^2 d^2 e}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}-\frac {5 \left (c d^2-a e^2\right )^7 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.23 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^6 e^{12}+70 a^5 c d e^{10} (10 d+e x)-7 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )+4 a^3 c^3 d^3 e^6 \left (768 d^3+323 d^2 e x+92 d e^2 x^2+12 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (1981 d^4+17140 d^3 e x+27648 d^2 e^2 x^2+18800 d e^3 x^3+4736 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (-350 d^5+231 d^4 e x+9032 d^3 e^2 x^2+18248 d^2 e^3 x^3+13824 d e^4 x^4+3712 e^5 x^5\right )+c^6 d^6 \left (105 d^6-70 d^5 e x+56 d^4 e^2 x^2+6096 d^3 e^3 x^3+13696 d^2 e^4 x^4+11008 d e^5 x^5+3072 e^6 x^6\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {105 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{21504 c^{9/2} d^{9/2} e^{7/2}} \]

[In]

Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^6*e^12 + 70*a^5*c*d*e^10*(10*d + e*x) - 7*a
^4*c^2*d^2*e^8*(283*d^2 + 66*d*e*x + 8*e^2*x^2) + 4*a^3*c^3*d^3*e^6*(768*d^3 + 323*d^2*e*x + 92*d*e^2*x^2 + 12
*e^3*x^3) + a^2*c^4*d^4*e^4*(1981*d^4 + 17140*d^3*e*x + 27648*d^2*e^2*x^2 + 18800*d*e^3*x^3 + 4736*e^4*x^4) +
2*a*c^5*d^5*e^2*(-350*d^5 + 231*d^4*e*x + 9032*d^3*e^2*x^2 + 18248*d^2*e^3*x^3 + 13824*d*e^4*x^4 + 3712*e^5*x^
5) + c^6*d^6*(105*d^6 - 70*d^5*e*x + 56*d^4*e^2*x^2 + 6096*d^3*e^3*x^3 + 13696*d^2*e^4*x^4 + 11008*d*e^5*x^5 +
3072*e^6*x^6)))/((a*e + c*d*x)^2*(d + e*x)^2) - (105*(c*d^2 - a*e^2)^7*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]
)/(Sqrt[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(21504*c^(9/2)*d^(9/2)*e^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(744\) vs. \(2(322)=644\).

Time = 2.44 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.09

method	result	size
default	\(d \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )+e \left (\frac {{\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {7}{2}}}{7 c d e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) {\left (a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 x c d e +e^{2} a +c \,d^{2}\right ) \sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (e^{2} a +c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} e^{2} a +\frac {1}{2} c \,d^{2}+x c d e}{\sqrt {c d e}}+\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 c d e}\right )\)	\(745\)

[In]

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

d*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2
)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e
^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e
^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2))/(c*d*e)^(1/2))))+e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/12*(2*c*d
*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/
8*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c
/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^
2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+x*c*d*e)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1
/2)))))

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 1270, normalized size of antiderivative = 3.57 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/86016*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 2
1*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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*(3072*c^7*d^7*e^7*x^6 + 105*c^7*d^13*e - 700*a*c^6*d^11*e^3 + 1981*a^2*c^5*d^9*e^5 + 30
72*a^3*c^4*d^7*e^7 - 1981*a^4*c^3*d^5*e^9 + 700*a^5*c^2*d^3*e^11 - 105*a^6*c*d*e^13 + 256*(43*c^7*d^8*e^6 + 29
*a*c^6*d^6*e^8)*x^5 + 128*(107*c^7*d^9*e^5 + 216*a*c^6*d^7*e^7 + 37*a^2*c^5*d^5*e^9)*x^4 + 16*(381*c^7*d^10*e^
4 + 2281*a*c^6*d^8*e^6 + 1175*a^2*c^5*d^6*e^8 + 3*a^3*c^4*d^4*e^10)*x^3 + 8*(7*c^7*d^11*e^3 + 2258*a*c^6*d^9*e
^5 + 3456*a^2*c^5*d^7*e^7 + 46*a^3*c^4*d^5*e^9 - 7*a^4*c^3*d^3*e^11)*x^2 - 2*(35*c^7*d^12*e^2 - 231*a*c^6*d^10
*e^4 - 8570*a^2*c^5*d^8*e^6 - 646*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5*c^2*d^2*e^12)*x)*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^4), 1/43008*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e
^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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*(3072*c^7*d^7*e^7*x^6 + 105*c^7*d^13*e - 700*a*c^6*d^11*e^
3 + 1981*a^2*c^5*d^9*e^5 + 3072*a^3*c^4*d^7*e^7 - 1981*a^4*c^3*d^5*e^9 + 700*a^5*c^2*d^3*e^11 - 105*a^6*c*d*e^
13 + 256*(43*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(107*c^7*d^9*e^5 + 216*a*c^6*d^7*e^7 + 37*a^2*c^5*d^5*e
^9)*x^4 + 16*(381*c^7*d^10*e^4 + 2281*a*c^6*d^8*e^6 + 1175*a^2*c^5*d^6*e^8 + 3*a^3*c^4*d^4*e^10)*x^3 + 8*(7*c^
7*d^11*e^3 + 2258*a*c^6*d^9*e^5 + 3456*a^2*c^5*d^7*e^7 + 46*a^3*c^4*d^5*e^9 - 7*a^4*c^3*d^3*e^11)*x^2 - 2*(35*
c^7*d^12*e^2 - 231*a*c^6*d^10*e^4 - 8570*a^2*c^5*d^8*e^6 - 646*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5
*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^4)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5268 vs. \(2 (345) = 690\).

Time = 6.15 (sec) , antiderivative size = 5268, normalized size of antiderivative = 14.80 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**3*x**6/7 + x**5*(3*a*c**2*d**2*e**5 +
4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e) + x**4*(3*a**2*c*d*e**6 + 78*a*c**2
*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d*
*2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e) + x**3*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**
2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c)
+ 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11
*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7
)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e) + x**2*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2 - 4*a*(3*a
**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*
c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*e**2/2 + 7*
c*d**2/2)*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3
- c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*
e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**
4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e) + x*(6*a**3*
d**2*e**5 + 12*a**2*c*d**4*e**3 + 3*a*c**2*d**6*e - 3*a*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3
- 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d
**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 +
11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e)
)/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2 - 4*
a*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**
5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*e**2/
2 + 7*c*d**2/2)*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4
*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**
2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c*
*3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c*
d*e) + (4*a**3*d**3*e**4 + 3*a**2*c*d**5*e**2 - 2*a*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2
- 4*a*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**
2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*
e**2/2 + 7*c*d**2/2)*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3
*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(
3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 +
4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c) - (3*
a*e**2/2 + 3*c*d**2/2)*(6*a**3*d**2*e**5 + 12*a**2*c*d**4*e**3 + 3*a*c**2*d**6*e - 3*a*(a**3*e**7 + 12*a**2*c*
d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 1
3*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*
c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**
2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(4*a**3*d*e**6 + 18*a**2*c*d**3*
e**4 + 12*a*c**2*d**5*e**2 - 4*a*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 +
11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))
/(5*c) + c**3*d**7 - (7*a*e**2/2 + 7*c*d**2/2)*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3
*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (
9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**
2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*
e))/(4*c*d*e))/(3*c*d*e))/(2*c*d*e))/(c*d*e)) + (a**3*d**4*e**3 - a*(6*a**3*d**2*e**5 + 12*a**2*c*d**4*e**3 +
3*a*c**2*d**6*e - 3*a*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**
3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*
(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5
+ 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 +
5*c*d**2/2)*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2 - 4*a*(3*a**2*c*d*e**6 + 78*a*c**2*d**
3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e
**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*e**2/2 + 7*c*d**2/2)*(a**3*e**7 + 12*a*
*2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/
2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7
+ 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*
a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*d*e))/(2*c) - (a*e**2 + c*d**2)*(4*a**3*d**3*
e**4 + 3*a**2*c*d**5*e**2 - 2*a*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2 - 4*a*(3*a**2*c*d*e
**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4
*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*e**2/2 + 7*c*d**2/2)
*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d*
*2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78
*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 -
c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c) - (3*a*e**2/2 + 3*c*d**2/
2)*(6*a**3*d**2*e**5 + 12*a**2*c*d**4*e**3 + 3*a*c**2*d**6*e - 3*a*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**
2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c)
+ 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11
*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7
)/(6*c*d*e))/(5*c*d*e))/(4*c) - (5*a*e**2/2 + 5*c*d**2/2)*(4*a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**
5*e**2 - 4*a*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c*
*2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 -
(7*a*e**2/2 + 7*c*d**2/2)*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d**2*e**5 +
4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**
2/2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*
e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e))/(3*c*
d*e))/(2*c*d*e))/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c*d*e)*sqrt(a*d*e + c*d*e*x**2
+ x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)
/(2*c*d*e))*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2)/(2*c*d*e))**2), True)), N
e(c*d*e, 0)), (2*(c*d**3*(a*d*e + x*(a*e**2 + c*d**2))**(7/2)/(7*(a*e**2 + c*d**2)) + e*(a*d*e + x*(a*e**2 + c
*d**2))**(9/2)/(9*(a*e**2 + c*d**2)))/(a*e**2 + c*d**2), Ne(a*e**2 + c*d**2, 0)), ((a*d*e)**(5/2)*(d*x + e*x**
2/2), True))

Maxima [F(-2)]

Exception generated. \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (322) = 644\).

Time = 0.34 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.83 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} d^{2} e^{3} x + \frac {43 \, c^{8} d^{9} e^{8} + 29 \, a c^{7} d^{7} e^{10}}{c^{6} d^{6} e^{6}}\right )} x + \frac {107 \, c^{8} d^{10} e^{7} + 216 \, a c^{7} d^{8} e^{9} + 37 \, a^{2} c^{6} d^{6} e^{11}}{c^{6} d^{6} e^{6}}\right )} x + \frac {381 \, c^{8} d^{11} e^{6} + 2281 \, a c^{7} d^{9} e^{8} + 1175 \, a^{2} c^{6} d^{7} e^{10} + 3 \, a^{3} c^{5} d^{5} e^{12}}{c^{6} d^{6} e^{6}}\right )} x + \frac {7 \, c^{8} d^{12} e^{5} + 2258 \, a c^{7} d^{10} e^{7} + 3456 \, a^{2} c^{6} d^{8} e^{9} + 46 \, a^{3} c^{5} d^{6} e^{11} - 7 \, a^{4} c^{4} d^{4} e^{13}}{c^{6} d^{6} e^{6}}\right )} x - \frac {35 \, c^{8} d^{13} e^{4} - 231 \, a c^{7} d^{11} e^{6} - 8570 \, a^{2} c^{6} d^{9} e^{8} - 646 \, a^{3} c^{5} d^{7} e^{10} + 231 \, a^{4} c^{4} d^{5} e^{12} - 35 \, a^{5} c^{3} d^{3} e^{14}}{c^{6} d^{6} e^{6}}\right )} x + \frac {105 \, c^{8} d^{14} e^{3} - 700 \, a c^{7} d^{12} e^{5} + 1981 \, a^{2} c^{6} d^{10} e^{7} + 3072 \, a^{3} c^{5} d^{8} e^{9} - 1981 \, a^{4} c^{4} d^{6} e^{11} + 700 \, a^{5} c^{3} d^{4} e^{13} - 105 \, a^{6} c^{2} d^{2} e^{15}}{c^{6} d^{6} e^{6}}\right )} + \frac {5 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2048 \, \sqrt {c d e} c^{4} d^{4} e^{3}} \]

[In]

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/21504*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(12*c^2*d^2*e^3*x + (43*c^8*d^9*e^8 + 29*a*
c^7*d^7*e^10)/(c^6*d^6*e^6))*x + (107*c^8*d^10*e^7 + 216*a*c^7*d^8*e^9 + 37*a^2*c^6*d^6*e^11)/(c^6*d^6*e^6))*x
+ (381*c^8*d^11*e^6 + 2281*a*c^7*d^9*e^8 + 1175*a^2*c^6*d^7*e^10 + 3*a^3*c^5*d^5*e^12)/(c^6*d^6*e^6))*x + (7*
c^8*d^12*e^5 + 2258*a*c^7*d^10*e^7 + 3456*a^2*c^6*d^8*e^9 + 46*a^3*c^5*d^6*e^11 - 7*a^4*c^4*d^4*e^13)/(c^6*d^6
*e^6))*x - (35*c^8*d^13*e^4 - 231*a*c^7*d^11*e^6 - 8570*a^2*c^6*d^9*e^8 - 646*a^3*c^5*d^7*e^10 + 231*a^4*c^4*d
^5*e^12 - 35*a^5*c^3*d^3*e^14)/(c^6*d^6*e^6))*x + (105*c^8*d^14*e^3 - 700*a*c^7*d^12*e^5 + 1981*a^2*c^6*d^10*e
^7 + 3072*a^3*c^5*d^8*e^9 - 1981*a^4*c^4*d^6*e^11 + 700*a^5*c^3*d^4*e^13 - 105*a^6*c^2*d^2*e^15)/(c^6*d^6*e^6)
) + 5/2048*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a
^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*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^4*d^4*e^3)

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \]

[In]

int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)