3.20.18 \(\int (d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2} \, dx\) [1918]

Optimal. Leaf size=461 \[ -\frac {99 \left (c d^2-a e^2\right )^6 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^5 d^5 e}+\frac {33 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {99 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{13/2} d^{13/2} e^{5/2}} \]

[Out]

33/2048*(-a*e^2+c*d^2)^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)^(3/2)/c^5/d^5/e+33/640*(-a*
e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^4/d^4+33/448*(-a*e^2+c*d^2)^2*(e*x+d)*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3+11/112*(-a*e^2+c*d^2)*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2
+1/8*(e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d+99/32768*(-a*e^2+c*d^2)^8*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^(13/2)/d^(13/2)/e^(5/2)-99/16384
*(-a*e^2+c*d^2)^6*(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^6/d^6/e^2

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 461, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {684, 654, 626, 635, 212} \begin {gather*} \frac {99 \left (c d^2-a e^2\right )^8 \tanh ^{-1}\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 )}{32768 c^{13/2} d^{13/2} e^{5/2}}-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \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}}{2048 c^5 d^5 e}+\frac {33 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{640 c^4 d^4}+\frac {33 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

(-99*(c*d^2 - a*e^2)^6*(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])/(16384*c^6*d^6
*e^2) + (33*(c*d^2 - a*e^2)^4*(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))/(2048
*c^5*d^5*e) + (33*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(640*c^4*d^4) + (33*(c*d^2
- a*e^2)^2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(448*c^3*d^3) + (11*(c*d^2 - a*e^2)*(d + e
*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(112*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x +
c*d*e*x^2)^(5/2))/(8*c*d) + (99*(c*d^2 - a*e^2)^8*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])])/(32768*c^(13/2)*d^(13/2)*e^(5/2))

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]

Rule 684

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int (d+e x)^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx &=\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (11 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{16 d}\\ &=\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (99 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{224 d^2}\\ &=\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (33 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{128 d^3}\\ &=\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (33 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{256 d^4}\\ &=\frac {33 \left (c d^2-a e^2\right )^4 \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}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}-\frac {\left (99 \left (c d^2-a e^2\right )^6\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^5 d^5 e}\\ &=-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \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}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (99 \left (c d^2-a e^2\right )^8\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32768 c^6 d^6 e^2}\\ &=-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \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}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {\left (99 \left (c d^2-a e^2\right )^8\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 )}{16384 c^6 d^6 e^2}\\ &=-\frac {99 \left (c d^2-a e^2\right )^6 \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}}{16384 c^6 d^6 e^2}+\frac {33 \left (c d^2-a e^2\right )^4 \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}}{2048 c^5 d^5 e}+\frac {33 \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 )^{5/2}}{640 c^4 d^4}+\frac {33 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 c^3 d^3}+\frac {11 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 c d}+\frac {99 \left (c d^2-a e^2\right )^8 \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 )}{32768 c^{13/2} d^{13/2} e^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.50, size = 352, normalized size = 0.76 \begin {gather*} \frac {\left (c d^2-a e^2\right )^8 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x)^6 \left (3465 e^7-\frac {26565 c d e^6 (d+e x)}{a e+c d x}+\frac {88473 c^2 d^2 e^5 (d+e x)^2}{(a e+c d x)^2}-\frac {166749 c^3 d^3 e^4 (d+e x)^3}{(a e+c d x)^3}+\frac {193699 c^4 d^4 e^3 (d+e x)^4}{(a e+c d x)^4}-\frac {140903 c^5 d^5 e^2 (d+e x)^5}{(a e+c d x)^5}-\frac {26565 c^6 d^6 e (d+e x)^6}{(a e+c d x)^6}+\frac {3465 c^7 d^7 (d+e x)^7}{(a e+c d x)^7}\right )}{\left (c d^2-a e^2\right )^8 (d+e x)}+\frac {3465 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{573440 c^{13/2} d^{13/2} e^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]

[Out]

((c*d^2 - a*e^2)^8*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^6*(3465*e^7 - (26
565*c*d*e^6*(d + e*x))/(a*e + c*d*x) + (88473*c^2*d^2*e^5*(d + e*x)^2)/(a*e + c*d*x)^2 - (166749*c^3*d^3*e^4*(
d + e*x)^3)/(a*e + c*d*x)^3 + (193699*c^4*d^4*e^3*(d + e*x)^4)/(a*e + c*d*x)^4 - (140903*c^5*d^5*e^2*(d + e*x)
^5)/(a*e + c*d*x)^5 - (26565*c^6*d^6*e*(d + e*x)^6)/(a*e + c*d*x)^6 + (3465*c^7*d^7*(d + e*x)^7)/(a*e + c*d*x)
^7))/((c*d^2 - a*e^2)^8*(d + e*x))) + (3465*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x]
)])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(573440*c^(13/2)*d^(13/2)*e^(5/2))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3890\) vs. \(2(419)=838\).
time = 0.80, size = 3891, normalized size = 8.44

method	result	size
default	\(\text {Expression too large to display}\)	\(3891\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

e^4*(1/8*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-11/16*(a*e^2+c*d^2)/c/d/e*(1/7*x^2*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-9/14*(a*e^2+c*d^2)/c/d/e*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-7/
12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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))))-1/
6*a/c*(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+c*d*e*x)/(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))))-2/7*a/c*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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)))))-
3/8*a/c*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-7/12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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))))-1/6*a/c*(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+c*
d*e*x)/(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)))))+4*d*e^3*(1/7*x^2*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-9/14*(a*e^2+c*d^2)/c/d/e*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e
-7/12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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))))
-1/6*a/c*(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+c*d*e*x)/(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))))-2/7*a/c*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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)))
))+6*d^2*e^2*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-7/12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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))))-1/6*a/c*(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+c*d*e*x)/(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))))+4*d^3*e*(1/5*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^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+c*d*e*x)/(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))))+d^4*(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...

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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(c*d^2-%e^2*a>0)', see `assume?
` for more d

________________________________________________________________________________________

Fricas [A]
time = 3.19, size = 1477, normalized size = 3.20 \begin {gather*} \left [\frac {{\left (3465 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (2310 \, c^{8} d^{14} x e^{2} - 3465 \, c^{8} d^{15} e + 2310 \, a^{6} c^{2} d^{2} x e^{14} - 3465 \, a^{7} c d e^{15} - 231 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 132 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - 11 \, {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} + 2 \, {\left (640 \, a^{2} c^{6} d^{6} x^{5} - 5984 \, a^{3} c^{5} d^{6} x^{3} + 28941 \, a^{4} c^{4} d^{6} x\right )} e^{10} + {\left (87040 \, a c^{7} d^{7} x^{6} + 10624 \, a^{2} c^{6} d^{7} x^{4} - 45936 \, a^{3} c^{5} d^{7} x^{2} + 166749 \, a^{4} c^{4} d^{7}\right )} e^{9} + 280 \, {\left (256 \, c^{8} d^{8} x^{7} + 1856 \, a c^{7} d^{8} x^{5} + 140 \, a^{2} c^{6} d^{8} x^{3} - 385 \, a^{3} c^{5} d^{8} x\right )} e^{8} + {\left (414720 \, c^{8} d^{9} x^{6} + 1288576 \, a c^{7} d^{9} x^{4} + 85136 \, a^{2} c^{6} d^{9} x^{2} - 193699 \, a^{3} c^{5} d^{9}\right )} e^{7} + 2 \, {\left (492160 \, c^{8} d^{10} x^{5} + 845984 \, a c^{7} d^{10} x^{3} + 61709 \, a^{2} c^{6} d^{10} x\right )} e^{6} + {\left (1211008 \, c^{8} d^{11} x^{4} + 1226408 \, a c^{7} d^{11} x^{2} + 140903 \, a^{2} c^{6} d^{11}\right )} e^{5} + 4 \, {\left (197004 \, c^{8} d^{12} x^{3} + 110299 \, a c^{7} d^{12} x\right )} e^{4} + 7 \, {\left (32504 \, c^{8} d^{13} x^{2} + 3795 \, a c^{7} d^{13}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{2293760 \, c^{7} d^{7}}, -\frac {{\left (3465 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) - 2 \, {\left (2310 \, c^{8} d^{14} x e^{2} - 3465 \, c^{8} d^{15} e + 2310 \, a^{6} c^{2} d^{2} x e^{14} - 3465 \, a^{7} c d e^{15} - 231 \, {\left (8 \, a^{5} c^{3} d^{3} x^{2} - 115 \, a^{6} c^{2} d^{3}\right )} e^{13} + 132 \, {\left (12 \, a^{4} c^{4} d^{4} x^{3} - 133 \, a^{5} c^{3} d^{4} x\right )} e^{12} - 11 \, {\left (128 \, a^{3} c^{5} d^{5} x^{4} - 1272 \, a^{4} c^{4} d^{5} x^{2} + 8043 \, a^{5} c^{3} d^{5}\right )} e^{11} + 2 \, {\left (640 \, a^{2} c^{6} d^{6} x^{5} - 5984 \, a^{3} c^{5} d^{6} x^{3} + 28941 \, a^{4} c^{4} d^{6} x\right )} e^{10} + {\left (87040 \, a c^{7} d^{7} x^{6} + 10624 \, a^{2} c^{6} d^{7} x^{4} - 45936 \, a^{3} c^{5} d^{7} x^{2} + 166749 \, a^{4} c^{4} d^{7}\right )} e^{9} + 280 \, {\left (256 \, c^{8} d^{8} x^{7} + 1856 \, a c^{7} d^{8} x^{5} + 140 \, a^{2} c^{6} d^{8} x^{3} - 385 \, a^{3} c^{5} d^{8} x\right )} e^{8} + {\left (414720 \, c^{8} d^{9} x^{6} + 1288576 \, a c^{7} d^{9} x^{4} + 85136 \, a^{2} c^{6} d^{9} x^{2} - 193699 \, a^{3} c^{5} d^{9}\right )} e^{7} + 2 \, {\left (492160 \, c^{8} d^{10} x^{5} + 845984 \, a c^{7} d^{10} x^{3} + 61709 \, a^{2} c^{6} d^{10} x\right )} e^{6} + {\left (1211008 \, c^{8} d^{11} x^{4} + 1226408 \, a c^{7} d^{11} x^{2} + 140903 \, a^{2} c^{6} d^{11}\right )} e^{5} + 4 \, {\left (197004 \, c^{8} d^{12} x^{3} + 110299 \, a c^{7} d^{12} x\right )} e^{4} + 7 \, {\left (32504 \, c^{8} d^{13} x^{2} + 3795 \, a c^{7} d^{13}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-3\right )}}{1146880 \, c^{7} d^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2293760*(3465*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8
- 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*sqrt(c*d)*e^(1/2)*log(8*c^2*d^3*x*
e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 + 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e
^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) + 4*(2310*c^8*d^14*x*e^2 - 3465*c^8*d^15*e + 2310*a
^6*c^2*d^2*x*e^14 - 3465*a^7*c*d*e^15 - 231*(8*a^5*c^3*d^3*x^2 - 115*a^6*c^2*d^3)*e^13 + 132*(12*a^4*c^4*d^4*x
^3 - 133*a^5*c^3*d^4*x)*e^12 - 11*(128*a^3*c^5*d^5*x^4 - 1272*a^4*c^4*d^5*x^2 + 8043*a^5*c^3*d^5)*e^11 + 2*(64
0*a^2*c^6*d^6*x^5 - 5984*a^3*c^5*d^6*x^3 + 28941*a^4*c^4*d^6*x)*e^10 + (87040*a*c^7*d^7*x^6 + 10624*a^2*c^6*d^
7*x^4 - 45936*a^3*c^5*d^7*x^2 + 166749*a^4*c^4*d^7)*e^9 + 280*(256*c^8*d^8*x^7 + 1856*a*c^7*d^8*x^5 + 140*a^2*
c^6*d^8*x^3 - 385*a^3*c^5*d^8*x)*e^8 + (414720*c^8*d^9*x^6 + 1288576*a*c^7*d^9*x^4 + 85136*a^2*c^6*d^9*x^2 - 1
93699*a^3*c^5*d^9)*e^7 + 2*(492160*c^8*d^10*x^5 + 845984*a*c^7*d^10*x^3 + 61709*a^2*c^6*d^10*x)*e^6 + (1211008
*c^8*d^11*x^4 + 1226408*a*c^7*d^11*x^2 + 140903*a^2*c^6*d^11)*e^5 + 4*(197004*c^8*d^12*x^3 + 110299*a*c^7*d^12
*x)*e^4 + 7*(32504*c^8*d^13*x^2 + 3795*a*c^7*d^13)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-3)/(c
^7*d^7), -1/1146880*(3465*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4 - 56*a^3*c^5*d^10*e^6 + 70*a^4*c^
4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^8*e^16)*sqrt(-c*d*e)*arctan(1/2*s
qrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3
+ (c^2*d^2*x^2 + a*c*d^2)*e^2)) - 2*(2310*c^8*d^14*x*e^2 - 3465*c^8*d^15*e + 2310*a^6*c^2*d^2*x*e^14 - 3465*a
^7*c*d*e^15 - 231*(8*a^5*c^3*d^3*x^2 - 115*a^6*c^2*d^3)*e^13 + 132*(12*a^4*c^4*d^4*x^3 - 133*a^5*c^3*d^4*x)*e^
12 - 11*(128*a^3*c^5*d^5*x^4 - 1272*a^4*c^4*d^5*x^2 + 8043*a^5*c^3*d^5)*e^11 + 2*(640*a^2*c^6*d^6*x^5 - 5984*a
^3*c^5*d^6*x^3 + 28941*a^4*c^4*d^6*x)*e^10 + (87040*a*c^7*d^7*x^6 + 10624*a^2*c^6*d^7*x^4 - 45936*a^3*c^5*d^7*
x^2 + 166749*a^4*c^4*d^7)*e^9 + 280*(256*c^8*d^8*x^7 + 1856*a*c^7*d^8*x^5 + 140*a^2*c^6*d^8*x^3 - 385*a^3*c^5*
d^8*x)*e^8 + (414720*c^8*d^9*x^6 + 1288576*a*c^7*d^9*x^4 + 85136*a^2*c^6*d^9*x^2 - 193699*a^3*c^5*d^9)*e^7 + 2
*(492160*c^8*d^10*x^5 + 845984*a*c^7*d^10*x^3 + 61709*a^2*c^6*d^10*x)*e^6 + (1211008*c^8*d^11*x^4 + 1226408*a*
c^7*d^11*x^2 + 140903*a^2*c^6*d^11)*e^5 + 4*(197004*c^8*d^12*x^3 + 110299*a*c^7*d^12*x)*e^4 + 7*(32504*c^8*d^1
3*x^2 + 3795*a*c^7*d^13)*e^3)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))*e^(-3)/(c^7*d^7)]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**4, x)

________________________________________________________________________________________

Giac [A]
time = 0.81, size = 726, normalized size = 1.57 \begin {gather*} \frac {1}{573440} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, c d x e^{5} + \frac {{\left (81 \, c^{8} d^{9} e^{11} + 17 \, a c^{7} d^{7} e^{13}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (769 \, c^{8} d^{10} e^{10} + 406 \, a c^{7} d^{8} e^{12} + a^{2} c^{6} d^{6} e^{14}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (9461 \, c^{8} d^{11} e^{9} + 10067 \, a c^{7} d^{9} e^{11} + 83 \, a^{2} c^{6} d^{7} e^{13} - 11 \, a^{3} c^{5} d^{5} e^{15}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (49251 \, c^{8} d^{12} e^{8} + 105748 \, a c^{7} d^{10} e^{10} + 2450 \, a^{2} c^{6} d^{8} e^{12} - 748 \, a^{3} c^{5} d^{6} e^{14} + 99 \, a^{4} c^{4} d^{4} e^{16}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (28441 \, c^{8} d^{13} e^{7} + 153301 \, a c^{7} d^{11} e^{9} + 10642 \, a^{2} c^{6} d^{9} e^{11} - 5742 \, a^{3} c^{5} d^{7} e^{13} + 1749 \, a^{4} c^{4} d^{5} e^{15} - 231 \, a^{5} c^{3} d^{3} e^{17}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x + \frac {{\left (1155 \, c^{8} d^{14} e^{6} + 220598 \, a c^{7} d^{12} e^{8} + 61709 \, a^{2} c^{6} d^{10} e^{10} - 53900 \, a^{3} c^{5} d^{8} e^{12} + 28941 \, a^{4} c^{4} d^{6} e^{14} - 8778 \, a^{5} c^{3} d^{4} e^{16} + 1155 \, a^{6} c^{2} d^{2} e^{18}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} x - \frac {{\left (3465 \, c^{8} d^{15} e^{5} - 26565 \, a c^{7} d^{13} e^{7} - 140903 \, a^{2} c^{6} d^{11} e^{9} + 193699 \, a^{3} c^{5} d^{9} e^{11} - 166749 \, a^{4} c^{4} d^{7} e^{13} + 88473 \, a^{5} c^{3} d^{5} e^{15} - 26565 \, a^{6} c^{2} d^{3} e^{17} + 3465 \, a^{7} c d e^{19}\right )} e^{\left (-7\right )}}{c^{7} d^{7}}\right )} - \frac {99 \, {\left (c^{8} d^{16} - 8 \, a c^{7} d^{14} e^{2} + 28 \, a^{2} c^{6} d^{12} e^{4} - 56 \, a^{3} c^{5} d^{10} e^{6} + 70 \, a^{4} c^{4} d^{8} e^{8} - 56 \, a^{5} c^{3} d^{6} e^{10} + 28 \, a^{6} c^{2} d^{4} e^{12} - 8 \, a^{7} c d^{2} e^{14} + a^{8} e^{16}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{32768 \, \sqrt {c d} c^{6} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

1/573440*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*(4*(14*c*d*x*e^5 + (81*c^8*d^9*e^11 + 17*
a*c^7*d^7*e^13)*e^(-7)/(c^7*d^7))*x + (769*c^8*d^10*e^10 + 406*a*c^7*d^8*e^12 + a^2*c^6*d^6*e^14)*e^(-7)/(c^7*
d^7))*x + (9461*c^8*d^11*e^9 + 10067*a*c^7*d^9*e^11 + 83*a^2*c^6*d^7*e^13 - 11*a^3*c^5*d^5*e^15)*e^(-7)/(c^7*d
^7))*x + (49251*c^8*d^12*e^8 + 105748*a*c^7*d^10*e^10 + 2450*a^2*c^6*d^8*e^12 - 748*a^3*c^5*d^6*e^14 + 99*a^4*
c^4*d^4*e^16)*e^(-7)/(c^7*d^7))*x + (28441*c^8*d^13*e^7 + 153301*a*c^7*d^11*e^9 + 10642*a^2*c^6*d^9*e^11 - 574
2*a^3*c^5*d^7*e^13 + 1749*a^4*c^4*d^5*e^15 - 231*a^5*c^3*d^3*e^17)*e^(-7)/(c^7*d^7))*x + (1155*c^8*d^14*e^6 +
220598*a*c^7*d^12*e^8 + 61709*a^2*c^6*d^10*e^10 - 53900*a^3*c^5*d^8*e^12 + 28941*a^4*c^4*d^6*e^14 - 8778*a^5*c
^3*d^4*e^16 + 1155*a^6*c^2*d^2*e^18)*e^(-7)/(c^7*d^7))*x - (3465*c^8*d^15*e^5 - 26565*a*c^7*d^13*e^7 - 140903*
a^2*c^6*d^11*e^9 + 193699*a^3*c^5*d^9*e^11 - 166749*a^4*c^4*d^7*e^13 + 88473*a^5*c^3*d^5*e^15 - 26565*a^6*c^2*
d^3*e^17 + 3465*a^7*c*d*e^19)*e^(-7)/(c^7*d^7)) - 99/32768*(c^8*d^16 - 8*a*c^7*d^14*e^2 + 28*a^2*c^6*d^12*e^4
- 56*a^3*c^5*d^10*e^6 + 70*a^4*c^4*d^8*e^8 - 56*a^5*c^3*d^6*e^10 + 28*a^6*c^2*d^4*e^12 - 8*a^7*c*d^2*e^14 + a^
8*e^16)*e^(-5/2)*log(abs(-c*d^2 - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c
*d)*e^(1/2) - a*e^2))/(sqrt(c*d)*c^6*d^6)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^4\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)

[Out]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), x)

________________________________________________________________________________________