3.477 \(\int \frac{x^5}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=515 \[ -\frac{2 x^2 \left (x \left (c d^2-a e^2\right ) \left (-a^2 c d^2 e^4-3 a^3 e^6-11 a c^2 d^4 e^2+7 c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-2 c d e x \left (9 a^2 c d^2 e^4-15 a^3 e^6-61 a c^2 d^4 e^2+35 c^3 d^6\right )+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac{5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac{2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*d*x^4*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2)) - (2*x^2*(a*d*e*(c*d^2 - a*e^2)*(7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(7
*c^3*d^6 - 11*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) - ((105*c^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 45*a^4*
e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3*e^6)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(12*c^3*d^3*e^4*(c*d^2 - a*e^2)^3) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*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^(7/2)*d^(7
/2)*e^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.618946, antiderivative size = 515, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 818, 779, 621, 206} \[ -\frac{2 x^2 \left (x \left (c d^2-a e^2\right ) \left (-a^2 c d^2 e^4-3 a^3 e^6-11 a c^2 d^4 e^2+7 c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-2 c d e x \left (9 a^2 c d^2 e^4-15 a^3 e^6-61 a c^2 d^4 e^2+35 c^3 d^6\right )+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac{5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac{2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-2*d*x^4*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2)) - (2*x^2*(a*d*e*(c*d^2 - a*e^2)*(7*c^2*d^4 - 12*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(7
*c^3*d^6 - 11*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) - ((105*c^4*d^8 - 190*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 30*a^3*c*d^2*e^6 - 45*a^4*
e^8 - 2*c*d*e*(35*c^3*d^6 - 61*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 15*a^3*e^6)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(12*c^3*d^3*e^4*(c*d^2 - a*e^2)^3) + (5*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*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^(7/2)*d^(7
/2)*e^(9/2))

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 621

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{(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 &=\int \frac{x^5 (a e+c d x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx\\ &=-\frac{2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \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 )^{3/2}}+\frac{2 \int \frac{x^3 \left (4 a c d^2 e \left (c d^2-a e^2\right )+\frac{1}{2} c d \left (7 c d^2-3 a e^2\right ) \left (c d^2-a e^2\right ) x\right )}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d e \left (c d^2-a e^2\right )^2}\\ &=-\frac{2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \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 )^{3/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \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}}+\frac{4 \int \frac{x \left (a c d^2 e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\frac{1}{4} c d \left (c d^2-a e^2\right ) \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right )}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2 e^2 \left (c d^2-a e^2\right )^4}\\ &=-\frac{2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \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 )^{3/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \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}}-\frac{\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac{\left (5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^3 d^3 e^4}\\ &=-\frac{2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \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 )^{3/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \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}}-\frac{\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac{\left (5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right )\right ) \operatorname{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 )}{4 c^3 d^3 e^4}\\ &=-\frac{2 d x^4 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \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 )^{3/2}}-\frac{2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (7 c^2 d^4-12 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (7 c^3 d^6-11 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \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}}-\frac{\left (105 c^4 d^8-190 a c^3 d^6 e^2+36 a^2 c^2 d^4 e^4+30 a^3 c d^2 e^6-45 a^4 e^8-2 c d e \left (35 c^3 d^6-61 a c^2 d^4 e^2+9 a^2 c d^2 e^4-15 a^3 e^6\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}+\frac{5 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \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 )}{8 c^{7/2} d^{7/2} e^{9/2}}\\ \end{align*}

Mathematica [C]  time = 8.66389, size = 2132, normalized size = 4.14 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^5/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(x^4*(a*e + c*d*x))/(2*c*d*e*((a*e + c*d*x)*(d + e*x))^(3/2)) + ((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*(((-7*c*d
^2 - 5*a*e^2)*((x^2*Sqrt[a*e + c*d*x])/(c*d*e*(d + e*x)^(3/2)) + ((4*Sqrt[a*e + c*d*x]*(2*a^2*d^2*e^4 - (3*c*d
^4*(-5*c*d^2 - a*e^2))/4 + (5*a*d^2*e^2*(-5*c*d^2 - a*e^2))/4 + e*(3*a^2*d*e^4 - c*d^3*(-5*c*d^2 - a*e^2) - d*
e*(a*c*d^2*e - (3*a*e*(-5*c*d^2 - a*e^2))/2))*x))/(3*e^2*(-(c*d^2) + a*e^2)^2*(d + e*x)^(3/2)) + ((-5*c*d^2 -
a*e^2)*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2))]*Sqrt[(c^2*d^3
)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]*ArcSinh[(Sqrt[c]*Sqrt[d
]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d^2 - a*e^2]*Sqrt[(c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)
])])/(c^(3/2)*d^(3/2)*e^(5/2)*Sqrt[d + e*x]))/(c*d*e)))/(2*c*d) + (a^3*e^3*(-4*a*c*d^2*e - (a*e*(-7*c*d^2 - 5*
a*e^2))/2)*((c*d)/((c^2*d^3)/(c*d^2 - a*e^2) - (a*c*d*e^2)/(c*d^2 - a*e^2)))^(5/2)*Sqrt[(c*d*(d + e*x))/(c*d^2
- a*e^2)]*((2520*c*d*(d + e*x))/(a*e^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (1330*c*d*(d + e*x))/(e*(a*e
+ c*d*x)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (1050*c*d*(a*e + c*d*x)*(d + e*x))/(a^2*e^3*Sqrt[(c*d*(d + e
*x))/(c*d^2 - a*e^2)]) + (196*c*d*(a*e + c*d*x)^2*(d + e*x))/(a^3*e^4*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) +
1568*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] + (1575*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(
a^2*e^4) + (1995*(c*d^2 - a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(e^2*(a*e + c*d*x)^2) - (3780*(c*d^2
- a*e^2)^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a*e^3*(a*e + c*d*x)) - (294*(c*d^2 - a*e^2)^2*(a*e + c*d*x
)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])/(a^3*e^5) - 504*(1 + (c*d*x)/(a*e))*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e
^2)] + 336*(1 + (c*d*x)/(a*e))^2*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)] - 56*(1 + (c*d*x)/(a*e))^3*Sqrt[(c*d*(d
+ e*x))/(c*d^2 - a*e^2)] - (1995*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/((e*(a*e + c*d*x))/(-(c*
d^2) + a*e^2))^(5/2) + (3780*(a*e + c*d*x)*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]])/(a*e*((e*(a*e +
c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) - (1575*(a*e + c*d*x)^2*ArcSin[Sqrt[(e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)]]
)/(a^2*e^2*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) + (294*(a*e + c*d*x)^3*ArcSin[Sqrt[(e*(a*e + c*d*x))/
(-(c*d^2) + a*e^2)]])/(a^3*e^3*((e*(a*e + c*d*x))/(-(c*d^2) + a*e^2))^(5/2)) - (168*e^2*(a*e + c*d*x)^2*Hyperg
eometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d^2 - a*e^2)^2 + (392*e*(a*e + c*d*x)^3*
Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a*(c*d^2 - a*e^2)^2) - (280*(a*e + c
*d*x)^4*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^2*(c*d^2 - a*e^2)^2) + (56
*(a*e + c*d*x)^5*Hypergeometric2F1[3/2, 9/2, 11/2, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^3*e*(c*d^2 - a*e^
2)^2) - (96*e*(a*e + c*d*x)*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e
^2)])/(-(c*d^2) + a*e^2) + (288*(a*e + c*d*x)^2*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d
*x))/(-(c*d^2) + a*e^2)])/(a*(-(c*d^2) + a*e^2)) - (288*(a*e + c*d*x)^3*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1
, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^2*(-(c*d^2*e) + a*e^3)) + (96*(a*e + c*d*x)^4*Hypergeomet
ricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(a^3*e^2*(-(c*d^2) + a*e^2))))/(2
52*c^3*d^3*(c*d^2 - a*e^2)^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(2*c*d*e*((a*e + c*d*x)*(d + e*x))^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.069, size = 1680, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

15/8/d^3/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(
1/2)*a^2+2*d^4/e^5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)+2/3*d^5/e^6/(a*e^2-c*d^2)/(d/e+x)/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-8/3*d^8/e^5*c^2/(a*e^2-c*
d^2)^3/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/8/d^3/c^3*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^
2+1/2/e^2*x^3/d/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/8/e^4*d/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c
)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+9/8*e/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+95/16/e^3*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)*a+51/16/e^5*d^6*c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35
/8/e^4*d/c*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/16*e/d^4/c^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
^3+5/16/e/d^2/c^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-9/4/e^3/c*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)-7/16/e^3/c^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+51/16/e^5*d^2/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)+15/8*e^4/d^3/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+5/2*e^2
/d/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+51/8/e^4*d^5*c/(-a^2*e^4
+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+15/16*e^5/d^4/c^4/(-a^2*e^4+2*a*c*d^2*e^2-c^
2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+35/16*e^3/d^2/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4-16/3*d^7/e^4*c^2/(a*e^2-c*d^2)^3/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(
1/2)*x-8/3*d^6/e^3*c/(a*e^2-c*d^2)^3/(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*a-1/4*d/c/(-a^2*e^4+2*a*c*d
^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+21/8/e*d^2/c/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/4/e/d^2/c^2*x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-15/4/e^2
/d/c^2*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+11/2/e^2*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+15/4/e^2/d/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 50.9616, size = 4361, normalized size = 8.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(15*(7*a*c^5*d^12*e - 15*a^2*c^4*d^10*e^3 + 6*a^3*c^3*d^8*e^5 + 2*a^4*c^2*d^6*e^7 + 3*a^5*c*d^4*e^9 - 3*
a^6*d^2*e^11 + (7*c^6*d^11*e^2 - 15*a*c^5*d^9*e^4 + 6*a^2*c^4*d^7*e^6 + 2*a^3*c^3*d^5*e^8 + 3*a^4*c^2*d^3*e^10
- 3*a^5*c*d*e^12)*x^3 + (14*c^6*d^12*e - 23*a*c^5*d^10*e^3 - 3*a^2*c^4*d^8*e^5 + 10*a^3*c^3*d^6*e^7 + 8*a^4*c
^2*d^4*e^9 - 3*a^5*c*d^2*e^11 - 3*a^6*e^13)*x^2 + (7*c^6*d^13 - a*c^5*d^11*e^2 - 24*a^2*c^4*d^9*e^4 + 14*a^3*c
^3*d^7*e^6 + 7*a^4*c^2*d^5*e^8 + 3*a^5*c*d^3*e^10 - 6*a^6*d*e^12)*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*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*(105*a*c^5*d^11*e^2 - 190*a^2*c^4*d^9*e^4 + 36*a^3*c^3*d^7*e^6 + 30*
a^4*c^2*d^5*e^8 - 45*a^5*c*d^3*e^10 - 6*(c^6*d^9*e^4 - 3*a*c^5*d^7*e^6 + 3*a^2*c^4*d^5*e^8 - a^3*c^3*d^3*e^10)
*x^4 + 3*(7*c^6*d^10*e^3 - 16*a*c^5*d^8*e^5 + 6*a^2*c^4*d^6*e^7 + 8*a^3*c^3*d^4*e^9 - 5*a^4*c^2*d^2*e^11)*x^3
+ (140*c^6*d^11*e^2 - 237*a*c^5*d^9*e^4 + 12*a^2*c^4*d^7*e^6 + 66*a^3*c^3*d^5*e^8 - 45*a^5*c*d*e^12)*x^2 + (10
5*c^6*d^12*e - 50*a*c^5*d^10*e^3 - 222*a^2*c^4*d^8*e^5 + 84*a^3*c^3*d^6*e^7 + 45*a^4*c^2*d^4*e^9 - 90*a^5*c*d^
2*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*c^7*d^12*e^6 - 3*a^2*c^6*d^10*e^8 + 3*a^3*c^5*d^8*e
^10 - a^4*c^4*d^6*e^12 + (c^8*d^11*e^7 - 3*a*c^7*d^9*e^9 + 3*a^2*c^6*d^7*e^11 - a^3*c^5*d^5*e^13)*x^3 + (2*c^8
*d^12*e^6 - 5*a*c^7*d^10*e^8 + 3*a^2*c^6*d^8*e^10 + a^3*c^5*d^6*e^12 - a^4*c^4*d^4*e^14)*x^2 + (c^8*d^13*e^5 -
a*c^7*d^11*e^7 - 3*a^2*c^6*d^9*e^9 + 5*a^3*c^5*d^7*e^11 - 2*a^4*c^4*d^5*e^13)*x), -1/24*(15*(7*a*c^5*d^12*e -
15*a^2*c^4*d^10*e^3 + 6*a^3*c^3*d^8*e^5 + 2*a^4*c^2*d^6*e^7 + 3*a^5*c*d^4*e^9 - 3*a^6*d^2*e^11 + (7*c^6*d^11*
e^2 - 15*a*c^5*d^9*e^4 + 6*a^2*c^4*d^7*e^6 + 2*a^3*c^3*d^5*e^8 + 3*a^4*c^2*d^3*e^10 - 3*a^5*c*d*e^12)*x^3 + (1
4*c^6*d^12*e - 23*a*c^5*d^10*e^3 - 3*a^2*c^4*d^8*e^5 + 10*a^3*c^3*d^6*e^7 + 8*a^4*c^2*d^4*e^9 - 3*a^5*c*d^2*e^
11 - 3*a^6*e^13)*x^2 + (7*c^6*d^13 - a*c^5*d^11*e^2 - 24*a^2*c^4*d^9*e^4 + 14*a^3*c^3*d^7*e^6 + 7*a^4*c^2*d^5*
e^8 + 3*a^5*c*d^3*e^10 - 6*a^6*d*e^12)*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^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(105
*a*c^5*d^11*e^2 - 190*a^2*c^4*d^9*e^4 + 36*a^3*c^3*d^7*e^6 + 30*a^4*c^2*d^5*e^8 - 45*a^5*c*d^3*e^10 - 6*(c^6*d
^9*e^4 - 3*a*c^5*d^7*e^6 + 3*a^2*c^4*d^5*e^8 - a^3*c^3*d^3*e^10)*x^4 + 3*(7*c^6*d^10*e^3 - 16*a*c^5*d^8*e^5 +
6*a^2*c^4*d^6*e^7 + 8*a^3*c^3*d^4*e^9 - 5*a^4*c^2*d^2*e^11)*x^3 + (140*c^6*d^11*e^2 - 237*a*c^5*d^9*e^4 + 12*a
^2*c^4*d^7*e^6 + 66*a^3*c^3*d^5*e^8 - 45*a^5*c*d*e^12)*x^2 + (105*c^6*d^12*e - 50*a*c^5*d^10*e^3 - 222*a^2*c^4
*d^8*e^5 + 84*a^3*c^3*d^6*e^7 + 45*a^4*c^2*d^4*e^9 - 90*a^5*c*d^2*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a
*e^2)*x))/(a*c^7*d^12*e^6 - 3*a^2*c^6*d^10*e^8 + 3*a^3*c^5*d^8*e^10 - a^4*c^4*d^6*e^12 + (c^8*d^11*e^7 - 3*a*c
^7*d^9*e^9 + 3*a^2*c^6*d^7*e^11 - a^3*c^5*d^5*e^13)*x^3 + (2*c^8*d^12*e^6 - 5*a*c^7*d^10*e^8 + 3*a^2*c^6*d^8*e
^10 + a^3*c^5*d^6*e^12 - a^4*c^4*d^4*e^14)*x^2 + (c^8*d^13*e^5 - a*c^7*d^11*e^7 - 3*a^2*c^6*d^9*e^9 + 5*a^3*c^
5*d^7*e^11 - 2*a^4*c^4*d^5*e^13)*x)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(x**5/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, 1]