\(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [457]

Optimal result

Integrand size = 40, antiderivative size = 574 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 \left (c d^2-a e^2\right )^3 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 c^5 d^5 e^6}+\frac {\left (c d^2-a e^2\right ) \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}+\frac {1}{112} \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 e}-\frac {\left (231 c^3 d^6-15 a c^2 d^4 e^2-95 a^2 c d^2 e^4-105 a^3 e^6-10 c d e \left (33 c^2 d^4-10 a c d^2 e^2-15 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}+\frac {3 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}} \]

[Out]

1/2048*(-a*e^2+c*d^2)*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^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)^(3/2)/c^4/d^4/e^5+1/112*(5*a/c/d-11*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
5/2)+1/8*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e-1/4480*(231*c^3*d^6-15*a*c^2*d^4*e^2-95*a^2*c*d^2*e^4-1
05*a^3*e^6-10*c*d*e*(-15*a^2*e^4-10*a*c*d^2*e^2+33*c^2*d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^3/d^3
/e^4+3/32768*(-a*e^2+c*d^2)^5*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^6)*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^(11/2)/d^(11/2)/e^(13/2)-3/16
384*(-a*e^2+c*d^2)^3*(15*a^3*e^6+35*a^2*c*d^2*e^4+45*a*c^2*d^4*e^2+33*c^3*d^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^5/d^5/e^6

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 846, 793, 626, 635, 212} \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{32768 c^{11/2} d^{11/2} e^{13/2}}-\frac {\left (-105 a^3 e^6-10 c d e x \left (-15 a^2 e^4-10 a c d^2 e^2+33 c^2 d^4\right )-95 a^2 c d^2 e^4-15 a c^2 d^4 e^2+231 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 c^3 d^3 e^4}-\frac {3 \left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{16384 c^5 d^5 e^6}+\frac {\left (15 a^3 e^6+35 a^2 c d^2 e^4+45 a c^2 d^4 e^2+33 c^3 d^6\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2048 c^4 d^4 e^5}+\frac {1}{112} x^2 \left (\frac {5 a}{c d}-\frac {11 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 e} \]

[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 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 793

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 846

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 863

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
]))

Rubi steps \begin{align*} \text {integral}& = \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 ) \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^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] (verified)

Time = 1.40 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (1575 a^7 e^{14}-525 a^6 c d e^{12} (7 d+2 e x)+35 a^5 c^2 d^2 e^{10} \left (29 d^2+68 d e x+24 e^2 x^2\right )+5 a^4 c^3 d^3 e^8 \left (185 d^3-110 d^2 e x-376 d e^2 x^2-144 e^3 x^3\right )+5 a^3 c^4 d^4 e^6 \left (265 d^4-120 d^3 e x+80 d^2 e^2 x^2+320 d e^3 x^3+128 e^4 x^4\right )+a^2 c^5 d^5 e^4 \left (-11193 d^5+7034 d^4 e x-5488 d^3 e^2 x^2+4640 d^2 e^3 x^3+137600 d e^4 x^4+103680 e^5 x^5\right )+a c^6 d^6 e^2 \left (11445 d^6-7476 d^5 e x+5928 d^4 e^2 x^2-5056 d^3 e^3 x^3+4480 d^2 e^4 x^4+212480 d e^5 x^5+168960 e^6 x^6\right )+c^7 d^7 \left (-3465 d^7+2310 d^6 e x-1848 d^5 e^2 x^2+1584 d^4 e^3 x^3-1408 d^3 e^4 x^4+1280 d^2 e^5 x^5+87040 d e^6 x^6+71680 e^7 x^7\right )\right )+\frac {105 \left (c d^2-a e^2\right )^5 \left (33 c^3 d^6+45 a c^2 d^4 e^2+35 a^2 c d^2 e^4+15 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{573440 c^{11/2} d^{11/2} e^{13/2}} \]

[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[c]*Sqrt[d]*Sqrt[e]*(1575*a^7*e^14 - 525*a^6*c*d*e^12*(7*d + 2*e*x) + 35*a
^5*c^2*d^2*e^10*(29*d^2 + 68*d*e*x + 24*e^2*x^2) + 5*a^4*c^3*d^3*e^8*(185*d^3 - 110*d^2*e*x - 376*d*e^2*x^2 -
144*e^3*x^3) + 5*a^3*c^4*d^4*e^6*(265*d^4 - 120*d^3*e*x + 80*d^2*e^2*x^2 + 320*d*e^3*x^3 + 128*e^4*x^4) + a^2*
c^5*d^5*e^4*(-11193*d^5 + 7034*d^4*e*x - 5488*d^3*e^2*x^2 + 4640*d^2*e^3*x^3 + 137600*d*e^4*x^4 + 103680*e^5*x
^5) + a*c^6*d^6*e^2*(11445*d^6 - 7476*d^5*e*x + 5928*d^4*e^2*x^2 - 5056*d^3*e^3*x^3 + 4480*d^2*e^4*x^4 + 21248
0*d*e^5*x^5 + 168960*e^6*x^6) + c^7*d^7*(-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 + 1280*d^2*e^5*x^5 + 87040*d*e^6*x^6 + 71680*e^7*x^7)) + (105*(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[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e
*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(573440*c^(11/2)*d^(11/2)*e^(13/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1894\) vs. \(2(536)=1072\).

Time = 0.66 (sec) , antiderivative size = 1895, normalized size of antiderivative = 3.30

method	result	size
default	\(\text {Expression too large to display}\)	\(1895\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 1524, normalized size of antiderivative = 2.66 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \]

[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)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \]

[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

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]

[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 >> 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 [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.37 \[ \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=\frac {1}{573440} \, \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 (10 \, {\left (4 \, {\left (14 \, c^{2} d^{2} e x + \frac {17 \, c^{9} d^{10} e^{7} + 33 \, a c^{8} d^{8} e^{9}}{c^{7} d^{7} e^{7}}\right )} x + \frac {c^{9} d^{11} e^{6} + 166 \, a c^{8} d^{9} e^{8} + 81 \, a^{2} c^{7} d^{7} e^{10}}{c^{7} d^{7} e^{7}}\right )} x - \frac {11 \, c^{9} d^{12} e^{5} - 35 \, a c^{8} d^{10} e^{7} - 1075 \, a^{2} c^{7} d^{8} e^{9} - 5 \, a^{3} c^{6} d^{6} e^{11}}{c^{7} d^{7} e^{7}}\right )} x + \frac {99 \, c^{9} d^{13} e^{4} - 316 \, a c^{8} d^{11} e^{6} + 290 \, a^{2} c^{7} d^{9} e^{8} + 100 \, a^{3} c^{6} d^{7} e^{10} - 45 \, a^{4} c^{5} d^{5} e^{12}}{c^{7} d^{7} e^{7}}\right )} x - \frac {231 \, c^{9} d^{14} e^{3} - 741 \, a c^{8} d^{12} e^{5} + 686 \, a^{2} c^{7} d^{10} e^{7} - 50 \, a^{3} c^{6} d^{8} e^{9} + 235 \, a^{4} c^{5} d^{6} e^{11} - 105 \, a^{5} c^{4} d^{4} e^{13}}{c^{7} d^{7} e^{7}}\right )} x + \frac {1155 \, c^{9} d^{15} e^{2} - 3738 \, a c^{8} d^{13} e^{4} + 3517 \, a^{2} c^{7} d^{11} e^{6} - 300 \, a^{3} c^{6} d^{9} e^{8} - 275 \, a^{4} c^{5} d^{7} e^{10} + 1190 \, a^{5} c^{4} d^{5} e^{12} - 525 \, a^{6} c^{3} d^{3} e^{14}}{c^{7} d^{7} e^{7}}\right )} x - \frac {3465 \, c^{9} d^{16} e - 11445 \, a c^{8} d^{14} e^{3} + 11193 \, a^{2} c^{7} d^{12} e^{5} - 1325 \, a^{3} c^{6} d^{10} e^{7} - 925 \, a^{4} c^{5} d^{8} e^{9} - 1015 \, a^{5} c^{4} d^{6} e^{11} + 3675 \, a^{6} c^{3} d^{4} e^{13} - 1575 \, a^{7} c^{2} d^{2} e^{15}}{c^{7} d^{7} e^{7}}\right )} - \frac {3 \, {\left (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}\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 )}{32768 \, \sqrt {c d e} c^{5} d^{5} e^{6}} \]

[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]

1/573440*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*(4*(14*c^2*d^2*e*x + (17*c^9*d^10*e^7 + 3
3*a*c^8*d^8*e^9)/(c^7*d^7*e^7))*x + (c^9*d^11*e^6 + 166*a*c^8*d^9*e^8 + 81*a^2*c^7*d^7*e^10)/(c^7*d^7*e^7))*x
- (11*c^9*d^12*e^5 - 35*a*c^8*d^10*e^7 - 1075*a^2*c^7*d^8*e^9 - 5*a^3*c^6*d^6*e^11)/(c^7*d^7*e^7))*x + (99*c^9
*d^13*e^4 - 316*a*c^8*d^11*e^6 + 290*a^2*c^7*d^9*e^8 + 100*a^3*c^6*d^7*e^10 - 45*a^4*c^5*d^5*e^12)/(c^7*d^7*e^
7))*x - (231*c^9*d^14*e^3 - 741*a*c^8*d^12*e^5 + 686*a^2*c^7*d^10*e^7 - 50*a^3*c^6*d^8*e^9 + 235*a^4*c^5*d^6*e
^11 - 105*a^5*c^4*d^4*e^13)/(c^7*d^7*e^7))*x + (1155*c^9*d^15*e^2 - 3738*a*c^8*d^13*e^4 + 3517*a^2*c^7*d^11*e^
6 - 300*a^3*c^6*d^9*e^8 - 275*a^4*c^5*d^7*e^10 + 1190*a^5*c^4*d^5*e^12 - 525*a^6*c^3*d^3*e^14)/(c^7*d^7*e^7))*
x - (3465*c^9*d^16*e - 11445*a*c^8*d^14*e^3 + 11193*a^2*c^7*d^12*e^5 - 1325*a^3*c^6*d^10*e^7 - 925*a^4*c^5*d^8
*e^9 - 1015*a^5*c^4*d^6*e^11 + 3675*a^6*c^3*d^4*e^13 - 1575*a^7*c^2*d^2*e^15)/(c^7*d^7*e^7)) - 3/32768*(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)*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^5*d^5*e^6)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]

[In]

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

[Out]

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