\(\int \frac {x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{d+e x} \, dx\) [446]

Optimal result

Integrand size = 40, antiderivative size = 449 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) x^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}} \]

[Out]

1/20*(a/c/d-3*d/e^2)*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/6*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)/e-1/960*(105*c^3*d^6-21*a*c^2*d^4*e^2-33*a^2*c*d^2*e^4-35*a^3*e^6-6*c*d*e*(-7*a^2*e^4-6*a*c*d^2*e^2+21*c^2*
d^4)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/e^4-1/1024*(-a*e^2+c*d^2)^3*(7*a^3*e^6+15*a^2*c*d^2*e^
4+21*a*c^2*d^4*e^2+21*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^(9/2)/d^(9/2)/e^(11/2)+1/512*(-7*a^4*e^8-8*a^3*c*d^2*e^6-6*a^2*c^2*d^4*e^4+21*c^4*d^8)
*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^5

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 449, 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 = {863, 846, 793, 626, 635, 212} \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}}-\frac {\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}+\frac {\left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^5}+\frac {1}{20} x^2 \left (\frac {a}{c d}-\frac {3 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac {x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e} \]

[In]

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

[Out]

((21*c^4*d^8 - 6*a^2*c^2*d^4*e^4 - 8*a^3*c*d^2*e^6 - 7*a^4*e^8)*(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])/(512*c^4*d^4*e^5) + ((a/(c*d) - (3*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(3/2))/20 + (x^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - ((105*c^3*d^6 - 21*a*c^2*d^4*e^2 -
33*a^2*c*d^2*e^4 - 35*a^3*e^6 - 6*c*d*e*(21*c^2*d^4 - 6*a*c*d^2*e^2 - 7*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*
x + c*d*e*x^2)^(3/2))/(960*c^3*d^3*e^4) - ((c*d^2 - a*e^2)^3*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4
+ 7*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])])/(1024*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 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) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx \\ & = \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}+\frac {\int x^2 \left (-3 a c d^2 e-\frac {3}{2} c d \left (3 c d^2-a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{6 c d e} \\ & = \frac {1}{20} \left (\frac {a}{c d}-\frac {3 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 )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}+\frac {\int x \left (3 a c d^2 e \left (3 c d^2-a e^2\right )+\frac {3}{4} c d \left (21 c^2 d^4-6 a c d^2 e^2-7 a^2 e^4\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{30 c^2 d^2 e^2} \\ & = \frac {1}{20} \left (\frac {a}{c d}-\frac {3 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 )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 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}}{960 c^3 d^3 e^4}+\frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 c^3 d^3 e^4} \\ & = \frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\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}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 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 )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 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}}{960 c^3 d^3 e^4}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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}{1024 c^4 d^4 e^5} \\ & = \frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\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}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 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 )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 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}}{960 c^3 d^3 e^4}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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 )}{512 c^4 d^4 e^5} \\ & = \frac {\left (21 c^4 d^8-6 a^2 c^2 d^4 e^4-8 a^3 c d^2 e^6-7 a^4 e^8\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}}{512 c^4 d^4 e^5}+\frac {1}{20} \left (\frac {a}{c d}-\frac {3 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 )^{3/2}+\frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 e}-\frac {\left (105 c^3 d^6-21 a c^2 d^4 e^2-33 a^2 c d^2 e^4-35 a^3 e^6-6 c d e \left (21 c^2 d^4-6 a c d^2 e^2-7 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}}{960 c^3 d^3 e^4}-\frac {\left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 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 )}{1024 c^{9/2} d^{9/2} e^{11/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^5 e^{10}+5 a^4 c d e^8 (11 d+14 e x)+2 a^3 c^2 d^2 e^6 \left (27 d^2-16 d e x-28 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (13 d^3-6 d^2 e x+4 d e^2 x^2+8 e^3 x^3\right )+a c^4 d^4 e^2 \left (-525 d^4+336 d^3 e x-264 d^2 e^2 x^2+224 d e^3 x^3+1664 e^4 x^4\right )+c^5 d^5 \left (315 d^5-210 d^4 e x+168 d^3 e^2 x^2-144 d^2 e^3 x^3+128 d e^4 x^4+1280 e^5 x^5\right )\right )-\frac {15 \left (c d^2-a e^2\right )^3 \left (21 c^3 d^6+21 a c^2 d^4 e^2+15 a^2 c d^2 e^4+7 a^3 e^6\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{7680 c^{9/2} d^{9/2} e^{11/2}} \]

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^5*e^10 + 5*a^4*c*d*e^8*(11*d + 14*e*x) + 2*a^3
*c^2*d^2*e^6*(27*d^2 - 16*d*e*x - 28*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(13*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 + 8*e^3*x^
3) + a*c^4*d^4*e^2*(-525*d^4 + 336*d^3*e*x - 264*d^2*e^2*x^2 + 224*d*e^3*x^3 + 1664*e^4*x^4) + c^5*d^5*(315*d^
5 - 210*d^4*e*x + 168*d^3*e^2*x^2 - 144*d^2*e^3*x^3 + 128*d*e^4*x^4 + 1280*e^5*x^5)) - (15*(c*d^2 - a*e^2)^3*(
21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]
*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(7680*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. \(1425\) vs. \(2(415)=830\).

Time = 0.64 (sec) , antiderivative size = 1426, normalized size of antiderivative = 3.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1426\)

[In]

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

[Out]

1/e*(1/6*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-7/12*(a*e^2+c*d^2)/c/d/e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/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))))-1/6*a/c*(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*(1/8*(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*(1/5*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d/e-1/2*(a*e^2+c*d^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/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*(1/3*(c*d*e*(x+d/e)^2+(a*e^2-
c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(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 = 0.34 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.33 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\left [-\frac {15 \, {\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \sqrt {c d e} \log \left (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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e - 525 \, a c^{5} d^{9} e^{3} + 78 \, a^{2} c^{4} d^{7} e^{5} + 54 \, a^{3} c^{3} d^{5} e^{7} + 55 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} - 16 \, {\left (9 \, c^{6} d^{8} e^{4} - 14 \, a c^{5} d^{6} e^{6} - 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (21 \, c^{6} d^{9} e^{3} - 33 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (105 \, c^{6} d^{10} e^{2} - 168 \, a c^{5} d^{8} e^{4} + 18 \, a^{2} c^{4} d^{6} e^{6} + 16 \, a^{3} c^{3} d^{4} e^{8} - 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{5} d^{5} e^{6}}, \frac {15 \, {\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e - 525 \, a c^{5} d^{9} e^{3} + 78 \, a^{2} c^{4} d^{7} e^{5} + 54 \, a^{3} c^{3} d^{5} e^{7} + 55 \, a^{4} c^{2} d^{3} e^{9} - 105 \, a^{5} c d e^{11} + 128 \, {\left (c^{6} d^{7} e^{5} + 13 \, a c^{5} d^{5} e^{7}\right )} x^{4} - 16 \, {\left (9 \, c^{6} d^{8} e^{4} - 14 \, a c^{5} d^{6} e^{6} - 3 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (21 \, c^{6} d^{9} e^{3} - 33 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - 7 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} - 2 \, {\left (105 \, c^{6} d^{10} e^{2} - 168 \, a c^{5} d^{8} e^{4} + 18 \, a^{2} c^{4} d^{6} e^{6} + 16 \, a^{3} c^{3} d^{4} e^{8} - 35 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{5} d^{5} e^{6}}\right ] \]

[In]

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

[Out]

[-1/30720*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 +
6*a^5*c*d^2*e^10 - 7*a^6*e^12)*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*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e - 525*a*c^5*d^9*e^3 + 78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*
a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*d^5*e^7)*x^4 - 16*(9*c^6*d^8*e^4 - 14*a*c^5*d
^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c^6*d^9*e^3 - 33*a*c^5*d^7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9
)*x^2 - 2*(105*c^6*d^10*e^2 - 168*a*c^5*d^8*e^4 + 18*a^2*c^4*d^6*e^6 + 16*a^3*c^3*d^4*e^8 - 35*a^4*c^2*d^2*e^1
0)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^6), 1/15360*(15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2
+ 15*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt(-c*d*e)*ar
ctan(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*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e - 525*a*c^5*d^9*e^3 +
78*a^2*c^4*d^7*e^5 + 54*a^3*c^3*d^5*e^7 + 55*a^4*c^2*d^3*e^9 - 105*a^5*c*d*e^11 + 128*(c^6*d^7*e^5 + 13*a*c^5*
d^5*e^7)*x^4 - 16*(9*c^6*d^8*e^4 - 14*a*c^5*d^6*e^6 - 3*a^2*c^4*d^4*e^8)*x^3 + 8*(21*c^6*d^9*e^3 - 33*a*c^5*d^
7*e^5 + 3*a^2*c^4*d^5*e^7 - 7*a^3*c^3*d^3*e^9)*x^2 - 2*(105*c^6*d^10*e^2 - 168*a*c^5*d^8*e^4 + 18*a^2*c^4*d^6*
e^6 + 16*a^3*c^3*d^4*e^8 - 35*a^4*c^2*d^2*e^10)*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. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(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 = 0.37 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.14 \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{7680} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c d x + \frac {c^{6} d^{7} e^{4} + 13 \, a c^{5} d^{5} e^{6}}{c^{5} d^{5} e^{5}}\right )} x - \frac {9 \, c^{6} d^{8} e^{3} - 14 \, a c^{5} d^{6} e^{5} - 3 \, a^{2} c^{4} d^{4} e^{7}}{c^{5} d^{5} e^{5}}\right )} x + \frac {21 \, c^{6} d^{9} e^{2} - 33 \, a c^{5} d^{7} e^{4} + 3 \, a^{2} c^{4} d^{5} e^{6} - 7 \, a^{3} c^{3} d^{3} e^{8}}{c^{5} d^{5} e^{5}}\right )} x - \frac {105 \, c^{6} d^{10} e - 168 \, a c^{5} d^{8} e^{3} + 18 \, a^{2} c^{4} d^{6} e^{5} + 16 \, a^{3} c^{3} d^{4} e^{7} - 35 \, a^{4} c^{2} d^{2} e^{9}}{c^{5} d^{5} e^{5}}\right )} x + \frac {315 \, c^{6} d^{11} - 525 \, a c^{5} d^{9} e^{2} + 78 \, a^{2} c^{4} d^{7} e^{4} + 54 \, a^{3} c^{3} d^{5} e^{6} + 55 \, a^{4} c^{2} d^{3} e^{8} - 105 \, a^{5} c d e^{10}}{c^{5} d^{5} e^{5}}\right )} + \frac {{\left (21 \, c^{6} d^{12} - 42 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} + 4 \, a^{3} c^{3} d^{6} e^{6} + 3 \, a^{4} c^{2} d^{4} e^{8} + 6 \, a^{5} c d^{2} e^{10} - 7 \, a^{6} e^{12}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{1024 \, \sqrt {c d e} c^{4} d^{4} e^{5}} \]

[In]

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

[Out]

1/7680*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(10*c*d*x + (c^6*d^7*e^4 + 13*a*c^5*d^5*e^6)/(c
^5*d^5*e^5))*x - (9*c^6*d^8*e^3 - 14*a*c^5*d^6*e^5 - 3*a^2*c^4*d^4*e^7)/(c^5*d^5*e^5))*x + (21*c^6*d^9*e^2 - 3
3*a*c^5*d^7*e^4 + 3*a^2*c^4*d^5*e^6 - 7*a^3*c^3*d^3*e^8)/(c^5*d^5*e^5))*x - (105*c^6*d^10*e - 168*a*c^5*d^8*e^
3 + 18*a^2*c^4*d^6*e^5 + 16*a^3*c^3*d^4*e^7 - 35*a^4*c^2*d^2*e^9)/(c^5*d^5*e^5))*x + (315*c^6*d^11 - 525*a*c^5
*d^9*e^2 + 78*a^2*c^4*d^7*e^4 + 54*a^3*c^3*d^5*e^6 + 55*a^4*c^2*d^3*e^8 - 105*a^5*c*d*e^10)/(c^5*d^5*e^5)) + 1
/1024*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*
d^2*e^10 - 7*a^6*e^12)*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. \[ \int \frac {x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {x^3\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]

[In]

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

[Out]

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