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

Optimal. Leaf size=574 \[ -\frac{3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac{\left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}-\frac{\left (-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-105 a^3 e^6-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}}+\frac{x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}+\frac{1}{112} x^2 \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

[Out]

(-3*(c*d^2 - a*e^2)^3*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^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^5*d^5*e^6) + ((c*d^2 - a*e^2)*(33*c^3*d^6 + 45*a*c^2
*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(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^4*d^4*e^5) + (((5*a)/(c*d) - (11*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/112
+ (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(8*e) - ((231*c^3*d^6 - 15*a*c^2*d^4*e^2 - 95*a^2*c*d^2
*e^4 - 105*a^3*e^6 - 10*c*d*e*(33*c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(5/2))/(4480*c^3*d^3*e^4) + (3*(c*d^2 - a*e^2)^5*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*
a^3*e^6)*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^(11/2)*d^(11/2)*e^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.694762, antiderivative size = 574, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {849, 832, 779, 612, 621, 206} \[ -\frac{3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac{\left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}-\frac{\left (-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-105 a^3 e^6-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{3 \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}}+\frac{x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e}+\frac{1}{112} x^2 \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(c*d^2 - a*e^2)^3*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^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^5*d^5*e^6) + ((c*d^2 - a*e^2)*(33*c^3*d^6 + 45*a*c^2
*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*(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^4*d^4*e^5) + (((5*a)/(c*d) - (11*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/112
+ (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(8*e) - ((231*c^3*d^6 - 15*a*c^2*d^4*e^2 - 95*a^2*c*d^2
*e^4 - 105*a^3*e^6 - 10*c*d*e*(33*c^2*d^4 - 10*a*c*d^2*e^2 - 15*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(5/2))/(4480*c^3*d^3*e^4) + (3*(c*d^2 - a*e^2)^5*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*
a^3*e^6)*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^(11/2)*d^(11/2)*e^(13/2))

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 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 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 612

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 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^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx &=\int x^3 (a e+c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx\\ &=\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}+\frac{\int x^2 \left (-3 a c d^2 e-\frac{1}{2} c d \left (11 c d^2-5 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}{8 c d e}\\ &=\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}+\frac{\int x \left (a c d^2 e \left (11 c d^2-5 a e^2\right )+\frac{3}{4} c d \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\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}{56 c^2 d^2 e^2}\\ &=\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac{\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{\left (\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right )\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 c^3 d^3 e^4}\\ &=\frac{\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )^{3/2}}{2048 c^4 d^4 e^5}+\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac{\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}-\frac{\left (3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right )\right ) \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4096 c^4 d^4 e^5}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \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^5 d^5 e^6}+\frac{\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )^{3/2}}{2048 c^4 d^4 e^5}+\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac{\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{\left (3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right )\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^5 d^5 e^6}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \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^5 d^5 e^6}+\frac{\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )^{3/2}}{2048 c^4 d^4 e^5}+\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac{\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{\left (3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )}{16384 c^5 d^5 e^6}\\ &=-\frac{3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \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^5 d^5 e^6}+\frac{\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )^{3/2}}{2048 c^4 d^4 e^5}+\frac{1}{112} \left (\frac{5 a}{c d}-\frac{11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac{\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac{3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\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 )}{32768 c^{11/2} d^{11/2} e^{13/2}}\\ \end{align*}

Mathematica [A]  time = 4.2967, size = 681, normalized size = 1.19 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{e} \left (35 a^6 c^2 d^2 e^{11} \left (29 d^2-37 d e x-6 e^2 x^2\right )+5 a^5 c^3 d^3 e^9 \left (93 d^2 e x+185 d^3+100 d e^2 x^2+24 e^3 x^3\right )+5 a^4 c^4 d^4 e^7 \left (-30 d^2 e^2 x^2+65 d^3 e x+265 d^4-56 d e^3 x^3-16 e^4 x^4\right )+a^3 c^5 d^5 e^5 \left (-6088 d^3 e^2 x^2+5040 d^2 e^3 x^3+8359 d^4 e x-11193 d^5+139200 d e^4 x^4+104320 e^5 x^5\right )+a^2 c^6 d^6 e^3 \left (12962 d^4 e^2 x^2-10544 d^3 e^3 x^3+9120 d^2 e^4 x^4-18669 d^5 e x+11445 d^6+350080 d e^5 x^5+272640 e^6 x^6\right )-525 a^7 c d e^{13} (7 d-e x)+1575 a^8 e^{15}+a c^7 d^7 e \left (-9324 d^5 e^2 x^2+7512 d^4 e^3 x^3-6464 d^3 e^4 x^4+5760 d^2 e^5 x^5+13755 d^6 e x-3465 d^7+299520 d e^6 x^6+240640 e^7 x^7\right )+c^8 d^8 x \left (-1848 d^5 e^2 x^2+1584 d^4 e^3 x^3-1408 d^3 e^4 x^4+1280 d^2 e^5 x^5+2310 d^6 e x-3465 d^7+87040 d e^6 x^6+71680 e^7 x^7\right )\right )}{a e+c d x}+\frac{105 \sqrt{c} \sqrt{d} \left (35 a^2 c d^2 e^4+15 a^3 e^6+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^{9/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{c d} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}\right )}{573440 c^5 d^5 e^{13/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*(1575*a^8*e^15 - 525*a^7*c*d*e^13*(7*d - e*x) + 35*a^6*c^2*d^2*e^11*(
29*d^2 - 37*d*e*x - 6*e^2*x^2) + 5*a^5*c^3*d^3*e^9*(185*d^3 + 93*d^2*e*x + 100*d*e^2*x^2 + 24*e^3*x^3) + 5*a^4
*c^4*d^4*e^7*(265*d^4 + 65*d^3*e*x - 30*d^2*e^2*x^2 - 56*d*e^3*x^3 - 16*e^4*x^4) + a^3*c^5*d^5*e^5*(-11193*d^5
+ 8359*d^4*e*x - 6088*d^3*e^2*x^2 + 5040*d^2*e^3*x^3 + 139200*d*e^4*x^4 + 104320*e^5*x^5) + a^2*c^6*d^6*e^3*(
11445*d^6 - 18669*d^5*e*x + 12962*d^4*e^2*x^2 - 10544*d^3*e^3*x^3 + 9120*d^2*e^4*x^4 + 350080*d*e^5*x^5 + 2726
40*e^6*x^6) + c^8*d^8*x*(-3465*d^7 + 2310*d^6*e*x - 1848*d^5*e^2*x^2 + 1584*d^4*e^3*x^3 - 1408*d^3*e^4*x^4 + 1
280*d^2*e^5*x^5 + 87040*d*e^6*x^6 + 71680*e^7*x^7) + a*c^7*d^7*e*(-3465*d^7 + 13755*d^6*e*x - 9324*d^5*e^2*x^2
+ 7512*d^4*e^3*x^3 - 6464*d^3*e^4*x^4 + 5760*d^2*e^5*x^5 + 299520*d*e^6*x^6 + 240640*e^7*x^7)))/(a*e + c*d*x)
+ (105*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(9/2)*(33*c^3*d^6 + 45*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 + 15*a^3*e^6)*
ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(Sqrt[c*d]*Sqrt[a*e + c*
d*x]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(573440*c^5*d^5*e^(13/2))

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Maple [B]  time = 0.074, size = 3178, normalized size = 5.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-95/1024/e^4*d^5*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+285/8192/e^5*d^8*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*x+45/16384*e^8/d^5/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^7-9/112/e/d^2/c^2*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(7/2)*a-285/32768/e^6*d^11*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)-705/16384/e^4*d^7*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-15/204
8*e^5/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^5-1/16*d^2/e*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e
+x))^(3/2)-9/64*d^4/e*a^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/128*d*e^2*a^4/c^2*(c*d*e*(d/e+x)^2
+(a*e^2-c*d^2)*(d/e+x))^(1/2)+3/64*d^7/e^4*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-3/64*d^8/e^5*c^2*
(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-1/8*d^3/e^2*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+
1/8*d^5/e^4*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+3/256*d^11/e^6*c^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+
x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-75/16384*e^4/d/c^3*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^5+65/1024/e*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2-465/16384*e^
2*d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+29/128/e^2*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a-5
/512*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a^2+1155/8192/e*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*x*a^2-45/8192*e^3/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+35/256/e^2*a*d^3*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)*x+5/32/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a+1/8/e^2*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(7/2)/d/c+3/128*e^2/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^3-1/5*d^3/e^4*(c*d*e*(d/e+x)^2+(a*e
^2-c*d^2)*(d/e+x))^(5/2)+19/128/e^4*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-25/112/e^3/c*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(7/2)-35/2048*e^3/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^4+15/16384*e^6/d^3/c^4*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^6-15/128*d^5*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+
(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-3/64*d^3*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/
e+x))^(1/2)+1/16*d^6/e^5*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)-3/128*d^9/e^6*c^2*(c*d*e*(d/e+x)^2+(a
*e^2-c*d^2)*(d/e+x))^(1/2)+465/4096*d^5*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^3+735/16384*d^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+13/128/d/c^2*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a^2+19/64/e^3*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+45/2048/e^3
*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a-15/1024*e/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+165/1
6384/e^2*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-95/2048/e^5*d^6*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)+285/16384/e^6*d^9*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-975/16384*e^2*d^3/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)*a^4-855/8192/e^2*d^7*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)*a^2-45/32768*e^1
0/d^5/c^5*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^8+15/4096*e^8/d^3/c^4*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^7-15/8192*e^6/d/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^6-105/2048*e*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+3/64*e/d^2/
c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x*a^2-15/1024*e^4/d^3/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*
x*a^4+15/256*d^3*e^2*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)
*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/64*d^2*e*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+9/64*d^6/e^3*a
*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256*d*e^4*a^5/c^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/
(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+15/128*d^7/e^2*a^2*c*ln((1/2*a*e^2-
1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)-15/256*d^9
/e^4*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))
/(d*e*c)^(1/2)+45/8192*e^7/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^6-15/4096*e^5/d^2/c^3*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^5-5/256*e^2*a^3/c^2/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x+195/4096/e^
4*d^9*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-495/4096/e^3*d^6*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a+45/4096*e^4*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^5

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 3.52865, size = 3413, normalized size = 5.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2293760*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^2 + 140*a^2*c^6*d^12*e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^
8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a^6*c^2*d^4*e^12 + 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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*(71680*c^8*d^8*e^8*x^7 - 3465*c^8*d^15*e + 11445*a*c^
7*d^13*e^3 - 11193*a^2*c^6*d^11*e^5 + 1325*a^3*c^5*d^9*e^7 + 925*a^4*c^4*d^7*e^9 + 1015*a^5*c^3*d^5*e^11 - 367
5*a^6*c^2*d^3*e^13 + 1575*a^7*c*d*e^15 + 5120*(17*c^8*d^9*e^7 + 33*a*c^7*d^7*e^9)*x^6 + 1280*(c^8*d^10*e^6 + 1
66*a*c^7*d^8*e^8 + 81*a^2*c^6*d^6*e^10)*x^5 - 128*(11*c^8*d^11*e^5 - 35*a*c^7*d^9*e^7 - 1075*a^2*c^6*d^7*e^9 -
5*a^3*c^5*d^5*e^11)*x^4 + 16*(99*c^8*d^12*e^4 - 316*a*c^7*d^10*e^6 + 290*a^2*c^6*d^8*e^8 + 100*a^3*c^5*d^6*e^
10 - 45*a^4*c^4*d^4*e^12)*x^3 - 8*(231*c^8*d^13*e^3 - 741*a*c^7*d^11*e^5 + 686*a^2*c^6*d^9*e^7 - 50*a^3*c^5*d^
7*e^9 + 235*a^4*c^4*d^5*e^11 - 105*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*c^8*d^14*e^2 - 3738*a*c^7*d^12*e^4 + 3517*a
^2*c^6*d^10*e^6 - 300*a^3*c^5*d^8*e^8 - 275*a^4*c^4*d^6*e^10 + 1190*a^5*c^3*d^4*e^12 - 525*a^6*c^2*d^2*e^14)*x
)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^7), -1/1146880*(105*(33*c^8*d^16 - 120*a*c^7*d^14*e^
2 + 140*a^2*c^6*d^12*e^4 - 40*a^3*c^5*d^10*e^6 - 10*a^4*c^4*d^8*e^8 - 8*a^5*c^3*d^6*e^10 - 20*a^6*c^2*d^4*e^12
+ 40*a^7*c*d^2*e^14 - 15*a^8*e^16)*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*(71680*c^8
*d^8*e^8*x^7 - 3465*c^8*d^15*e + 11445*a*c^7*d^13*e^3 - 11193*a^2*c^6*d^11*e^5 + 1325*a^3*c^5*d^9*e^7 + 925*a^
4*c^4*d^7*e^9 + 1015*a^5*c^3*d^5*e^11 - 3675*a^6*c^2*d^3*e^13 + 1575*a^7*c*d*e^15 + 5120*(17*c^8*d^9*e^7 + 33*
a*c^7*d^7*e^9)*x^6 + 1280*(c^8*d^10*e^6 + 166*a*c^7*d^8*e^8 + 81*a^2*c^6*d^6*e^10)*x^5 - 128*(11*c^8*d^11*e^5
- 35*a*c^7*d^9*e^7 - 1075*a^2*c^6*d^7*e^9 - 5*a^3*c^5*d^5*e^11)*x^4 + 16*(99*c^8*d^12*e^4 - 316*a*c^7*d^10*e^6
+ 290*a^2*c^6*d^8*e^8 + 100*a^3*c^5*d^6*e^10 - 45*a^4*c^4*d^4*e^12)*x^3 - 8*(231*c^8*d^13*e^3 - 741*a*c^7*d^1
1*e^5 + 686*a^2*c^6*d^9*e^7 - 50*a^3*c^5*d^7*e^9 + 235*a^4*c^4*d^5*e^11 - 105*a^5*c^3*d^3*e^13)*x^2 + 2*(1155*
c^8*d^14*e^2 - 3738*a*c^7*d^12*e^4 + 3517*a^2*c^6*d^10*e^6 - 300*a^3*c^5*d^8*e^8 - 275*a^4*c^4*d^6*e^10 + 1190
*a^5*c^3*d^4*e^12 - 525*a^6*c^2*d^2*e^14)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^7)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError