\(\int \frac {x^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{d+e x} \, dx\) [458]

Optimal result

Integrand size = 40, antiderivative size = 452 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 c d^2-5 a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}} \]

[Out]

-1/384*(-a*e^2+c*d^2)*(5*a^2*e^4+10*a*c*d^2*e^2+9*c^2*d^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^3/d^3/e^4+1/7*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e+1/840*(63*c^2*d^4-20*a*c*d^2*e^2-35
*a^2*e^4-10*c*d*e*(-5*a*e^2+9*c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c^2/d^2/e^3-1/2048*(-a*e^2+c*d
^2)^5*(5*a^2*e^4+10*a*c*d^2*e^2+9*c^2*d^4)*arctanh(1/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)/e^(11/2)+1/1024*(-a*e^2+c*d^2)^3*(5*a^2*e^4+10*a*c*d^2*e^2+9
*c^2*d^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^4/d^4/e^5

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {865, 846, 793, 626, 635, 212} \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 c^{9/2} d^{9/2} e^{11/2}}+\frac {\left (-35 a^2 e^4-10 c d e x \left (9 c d^2-5 a e^2\right )-20 a c d^2 e^2+63 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 c^2 d^2 e^3}+\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 c^4 d^4 e^5}-\frac {\left (5 a^2 e^4+10 a c d^2 e^2+9 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 c^3 d^3 e^4}+\frac {x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 e} \]

[In]

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

[Out]

((c*d^2 - a*e^2)^3*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])/(1024*c^4*d^4*e^5) - ((c*d^2 - a*e^2)*(9*c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^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))/(384*c^3*d^3*e^4) + (x^2*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(5/2))/(7*e) + ((63*c^2*d^4 - 20*a*c*d^2*e^2 - 35*a^2*e^4 - 10*c*d*e*(9*c*d^2 - 5*a*e^2)
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(840*c^2*d^2*e^3) - ((c*d^2 - a*e^2)^5*(9*c^2*d^4 + 10*a*c*
d^2*e^2 + 5*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])])/(2048*c^(9/2)*d^(9/2)*e^(11/2))

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 865

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :>
Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + p))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] &&
NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[m, 0] && In
tegerQ[n] && (LtQ[n, 0] || GtQ[p, 0])

Rubi steps \begin{align*} \text {integral}& = \int x^2 (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^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\int x \left (-2 a c d^2 e-\frac {1}{2} c d \left (9 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}{7 c d e} \\ & = \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 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 )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}{48 c^2 d^2 e^3} \\ & = -\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 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 )^{5/2}}{840 c^2 d^2 e^3}+\frac {\left (\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{256 c^3 d^3 e^4} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 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 )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 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}{2048 c^4 d^4 e^5} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 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 )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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 )}{1024 c^4 d^4 e^5} \\ & = \frac {\left (c d^2-a e^2\right )^3 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{1024 c^4 d^4 e^5}-\frac {\left (c d^2-a e^2\right ) \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\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}}{384 c^3 d^3 e^4}+\frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 e}+\frac {\left (63 c^2 d^4-20 a c d^2 e^2-35 a^2 e^4-10 c d e \left (9 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 )^{5/2}}{840 c^2 d^2 e^3}-\frac {\left (c d^2-a e^2\right )^5 \left (9 c^2 d^4+10 a c d^2 e^2+5 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 )}{2048 c^{9/2} d^{9/2} e^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.06 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-525 a^6 e^{12}+350 a^5 c d e^{10} (4 d+e x)-35 a^4 c^2 d^2 e^8 \left (15 d^2+26 d e x+8 e^2 x^2\right )-60 a^3 c^3 d^3 e^6 \left (10 d^3-5 d^2 e x-12 d e^2 x^2-4 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (3689 d^4-2332 d^3 e x+1824 d^2 e^2 x^2+33520 d e^3 x^3+23680 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (-1680 d^5+1099 d^4 e x-872 d^3 e^2 x^2+744 d^2 e^3 x^3+24320 d e^4 x^4+18560 e^5 x^5\right )+3 c^6 d^6 \left (315 d^6-210 d^5 e x+168 d^4 e^2 x^2-144 d^3 e^3 x^3+128 d^2 e^4 x^4+6400 d e^5 x^5+5120 e^6 x^6\right )\right )}{\left (c d^2-a e^2\right )^5 (a e+c d x) (d+e x)}-\frac {105 \left (9 c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{107520 c^{9/2} d^{9/2} e^{11/2}} \]

[In]

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

[Out]

((c*d^2 - a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-525*a^6*e^12 + 350*a^5*c*d*e^10
*(4*d + e*x) - 35*a^4*c^2*d^2*e^8*(15*d^2 + 26*d*e*x + 8*e^2*x^2) - 60*a^3*c^3*d^3*e^6*(10*d^3 - 5*d^2*e*x - 1
2*d*e^2*x^2 - 4*e^3*x^3) + a^2*c^4*d^4*e^4*(3689*d^4 - 2332*d^3*e*x + 1824*d^2*e^2*x^2 + 33520*d*e^3*x^3 + 236
80*e^4*x^4) + 2*a*c^5*d^5*e^2*(-1680*d^5 + 1099*d^4*e*x - 872*d^3*e^2*x^2 + 744*d^2*e^3*x^3 + 24320*d*e^4*x^4
+ 18560*e^5*x^5) + 3*c^6*d^6*(315*d^6 - 210*d^5*e*x + 168*d^4*e^2*x^2 - 144*d^3*e^3*x^3 + 128*d^2*e^4*x^4 + 64
00*d*e^5*x^5 + 5120*e^6*x^6)))/((c*d^2 - a*e^2)^5*(a*e + c*d*x)*(d + e*x)) - (105*(9*c^2*d^4 + 10*a*c*d^2*e^2
+ 5*a^2*e^4)*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/((a*e + c*d*x)^(3/2)*(d + e
*x)^(3/2))))/(107520*c^(9/2)*d^(9/2)*e^(11/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1079\) vs. \(2(418)=836\).

Time = 0.66 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.39

method	result	size
default	\(\text {Expression too large to display}\)	\(1080\)

[In]

int(x^2*(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*(1/7*(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/e^2*(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*(1/5*(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 = 0.67 (sec) , antiderivative size = 1272, normalized size of antiderivative = 2.81 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/430080*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8
- 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^
2*e^2 + a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c
^2*d^3*e + a*c*d*e^3)*x) - 4*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 3360*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*
e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 - 525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*
e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d
^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*
d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c
^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6), 1/215040*(105*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^
2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8 - 9*a^5*c^2*d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)
*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/
(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(15360*c^7*d^7*e^7*x^6 + 945*c^7*d^13*e - 336
0*a*c^6*d^11*e^3 + 3689*a^2*c^5*d^9*e^5 - 600*a^3*c^4*d^7*e^7 - 525*a^4*c^3*d^5*e^9 + 1400*a^5*c^2*d^3*e^11 -
525*a^6*c*d*e^13 + 1280*(15*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(3*c^7*d^9*e^5 + 380*a*c^6*d^7*e^7 + 185
*a^2*c^5*d^5*e^9)*x^4 - 16*(27*c^7*d^10*e^4 - 93*a*c^6*d^8*e^6 - 2095*a^2*c^5*d^6*e^8 - 15*a^3*c^4*d^4*e^10)*x
^3 + 8*(63*c^7*d^11*e^3 - 218*a*c^6*d^9*e^5 + 228*a^2*c^5*d^7*e^7 + 90*a^3*c^4*d^5*e^9 - 35*a^4*c^3*d^3*e^11)*
x^2 - 2*(315*c^7*d^12*e^2 - 1099*a*c^6*d^10*e^4 + 1166*a^2*c^5*d^8*e^6 - 150*a^3*c^4*d^6*e^8 + 455*a^4*c^3*d^4
*e^10 - 175*a^5*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6)]

Sympy [F(-1)]

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

[In]

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

[In]

integrate(x^2*(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 = 648, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {1}{107520} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, c^{2} d^{2} e x + \frac {15 \, c^{8} d^{9} e^{6} + 29 \, a c^{7} d^{7} e^{8}}{c^{6} d^{6} e^{6}}\right )} x + \frac {3 \, c^{8} d^{10} e^{5} + 380 \, a c^{7} d^{8} e^{7} + 185 \, a^{2} c^{6} d^{6} e^{9}}{c^{6} d^{6} e^{6}}\right )} x - \frac {27 \, c^{8} d^{11} e^{4} - 93 \, a c^{7} d^{9} e^{6} - 2095 \, a^{2} c^{6} d^{7} e^{8} - 15 \, a^{3} c^{5} d^{5} e^{10}}{c^{6} d^{6} e^{6}}\right )} x + \frac {63 \, c^{8} d^{12} e^{3} - 218 \, a c^{7} d^{10} e^{5} + 228 \, a^{2} c^{6} d^{8} e^{7} + 90 \, a^{3} c^{5} d^{6} e^{9} - 35 \, a^{4} c^{4} d^{4} e^{11}}{c^{6} d^{6} e^{6}}\right )} x - \frac {315 \, c^{8} d^{13} e^{2} - 1099 \, a c^{7} d^{11} e^{4} + 1166 \, a^{2} c^{6} d^{9} e^{6} - 150 \, a^{3} c^{5} d^{7} e^{8} + 455 \, a^{4} c^{4} d^{5} e^{10} - 175 \, a^{5} c^{3} d^{3} e^{12}}{c^{6} d^{6} e^{6}}\right )} x + \frac {945 \, c^{8} d^{14} e - 3360 \, a c^{7} d^{12} e^{3} + 3689 \, a^{2} c^{6} d^{10} e^{5} - 600 \, a^{3} c^{5} d^{8} e^{7} - 525 \, a^{4} c^{4} d^{6} e^{9} + 1400 \, a^{5} c^{3} d^{4} e^{11} - 525 \, a^{6} c^{2} d^{2} e^{13}}{c^{6} d^{6} e^{6}}\right )} + \frac {{\left (9 \, c^{7} d^{14} - 35 \, a c^{6} d^{12} e^{2} + 45 \, a^{2} c^{5} d^{10} e^{4} - 15 \, a^{3} c^{4} d^{8} e^{6} - 5 \, a^{4} c^{3} d^{6} e^{8} - 9 \, a^{5} c^{2} d^{4} e^{10} + 15 \, a^{6} c d^{2} e^{12} - 5 \, a^{7} e^{14}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2048 \, \sqrt {c d e} c^{4} d^{4} e^{5}} \]

[In]

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

[Out]

1/107520*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*(12*c^2*d^2*e*x + (15*c^8*d^9*e^6 + 29*a*
c^7*d^7*e^8)/(c^6*d^6*e^6))*x + (3*c^8*d^10*e^5 + 380*a*c^7*d^8*e^7 + 185*a^2*c^6*d^6*e^9)/(c^6*d^6*e^6))*x -
(27*c^8*d^11*e^4 - 93*a*c^7*d^9*e^6 - 2095*a^2*c^6*d^7*e^8 - 15*a^3*c^5*d^5*e^10)/(c^6*d^6*e^6))*x + (63*c^8*d
^12*e^3 - 218*a*c^7*d^10*e^5 + 228*a^2*c^6*d^8*e^7 + 90*a^3*c^5*d^6*e^9 - 35*a^4*c^4*d^4*e^11)/(c^6*d^6*e^6))*
x - (315*c^8*d^13*e^2 - 1099*a*c^7*d^11*e^4 + 1166*a^2*c^6*d^9*e^6 - 150*a^3*c^5*d^7*e^8 + 455*a^4*c^4*d^5*e^1
0 - 175*a^5*c^3*d^3*e^12)/(c^6*d^6*e^6))*x + (945*c^8*d^14*e - 3360*a*c^7*d^12*e^3 + 3689*a^2*c^6*d^10*e^5 - 6
00*a^3*c^5*d^8*e^7 - 525*a^4*c^4*d^6*e^9 + 1400*a^5*c^3*d^4*e^11 - 525*a^6*c^2*d^2*e^13)/(c^6*d^6*e^6)) + 1/20
48*(9*c^7*d^14 - 35*a*c^6*d^12*e^2 + 45*a^2*c^5*d^10*e^4 - 15*a^3*c^4*d^8*e^6 - 5*a^4*c^3*d^6*e^8 - 9*a^5*c^2*
d^4*e^10 + 15*a^6*c*d^2*e^12 - 5*a^7*e^14)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*
x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*c^4*d^4*e^5)

Mupad [F(-1)]

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

[In]

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

[Out]

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