3.5.77 \(\int \frac {x^5}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [477]

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

[Out]

-2/3*d*x^4*(a*e*(-a*e^2+c*d^2)+c*d*(-a*e^2+c*d^2)*x)/e/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2
)+5/8*(3*a^2*e^4+6*a*c*d^2*e^2+7*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^(7/2)/d^(7/2)/e^(9/2)-2/3*x^2*(a*d*e*(-a*e^2+c*d^2)*(-3*a^2*e^4-12*a*c*d^2*
e^2+7*c^2*d^4)+(-a*e^2+c*d^2)*(-3*a^3*e^6-a^2*c*d^2*e^4-11*a*c^2*d^4*e^2+7*c^3*d^6)*x)/c/d/e^2/(-a*e^2+c*d^2)^
4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/12*(105*c^4*d^8-190*a*c^3*d^6*e^2+36*a^2*c^2*d^4*e^4+30*a^3*c*d^2*
e^6-45*a^4*e^8-2*c*d*e*(-15*a^3*e^6+9*a^2*c*d^2*e^4-61*a*c^2*d^4*e^2+35*c^3*d^6)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)/c^3/d^3/e^4/(-a*e^2+c*d^2)^3

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Rubi [A]
time = 0.41, antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 832, 793, 635, 212} \begin {gather*} \frac {5 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{7/2} d^{7/2} e^{9/2}}-\frac {2 x^2 \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-12 a c d^2 e^2+7 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-11 a c^2 d^4 e^2+7 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {\left (-45 a^4 e^8+30 a^3 c d^2 e^6+36 a^2 c^2 d^4 e^4-2 c d e x \left (-15 a^3 e^6+9 a^2 c d^2 e^4-61 a c^2 d^4 e^2+35 c^3 d^6\right )-190 a c^3 d^6 e^2+105 c^4 d^8\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 c^3 d^3 e^4 \left (c d^2-a e^2\right )^3}-\frac {2 d x^4 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 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 832

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

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

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Mathematica [A]
time = 0.83, size = 388, normalized size = 0.75 \begin {gather*} \frac {-\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (-45 a^5 e^9 (d+e x)^2+15 a^4 c d e^7 (2 d-e x) (d+e x)^2+6 a^3 c^2 d^2 e^5 (d+e x)^2 \left (6 d^2+2 d e x+e^2 x^2\right )+c^5 d^8 x \left (105 d^3+140 d^2 e x+21 d e^2 x^2-6 e^3 x^3\right )-2 a^2 c^3 d^4 e^3 \left (95 d^4+111 d^3 e x-6 d^2 e^2 x^2-9 d e^3 x^3+9 e^4 x^4\right )+a c^4 d^6 e \left (105 d^4-50 d^3 e x-237 d^2 e^2 x^2-48 d e^3 x^3+18 e^4 x^4\right )\right )}{\left (c d^2-a e^2\right )^3}+15 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{12 c^{7/2} d^{7/2} e^{9/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)*(-45*a^5*e^9*(d + e*x)^2 + 15*a^4*c*d*e^7*(2*d - e*x)*(d + e*x)^2 +
6*a^3*c^2*d^2*e^5*(d + e*x)^2*(6*d^2 + 2*d*e*x + e^2*x^2) + c^5*d^8*x*(105*d^3 + 140*d^2*e*x + 21*d*e^2*x^2 -
6*e^3*x^3) - 2*a^2*c^3*d^4*e^3*(95*d^4 + 111*d^3*e*x - 6*d^2*e^2*x^2 - 9*d*e^3*x^3 + 9*e^4*x^4) + a*c^4*d^6*e*
(105*d^4 - 50*d^3*e*x - 237*d^2*e^2*x^2 - 48*d*e^3*x^3 + 18*e^4*x^4)))/(c*d^2 - a*e^2)^3) + 15*(7*c^2*d^4 + 6*
a*c*d^2*e^2 + 3*a^2*e^4)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*
Sqrt[a*e + c*d*x])])/(12*c^(7/2)*d^(7/2)*e^(9/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1922\) vs. \(2(485)=970\).
time = 0.11, size = 1923, normalized size = 3.73

method	result	size
default	\(\text {Expression too large to display}\)	\(1923\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(1/2*x^3/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-5/4*(a*e^2+c*d^2)/c/d/e*(x^2/c/d/e/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)-3/2*(a*e^2+c*d^2)/c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c
*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a
*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*a*e^2+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))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)))-3/2*a/c*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/
c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^
2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*a*e^2+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*(x^2/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)-3/2*(a*e^2+c*d^2)/c/d/e*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^
2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*a*e^2+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))-2*a/c*(-1/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2
+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))
)+d^2/e^3*(-x/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/2*(a*e^2+c*d^2)/c/d/e*(-1/c/d/e/(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+1/c/d/e*ln((1/2*a*e^2+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/c/d/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-(a*e^2+c*d^2)/c/d/e*
(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))+2*d^4/e^5*(2*
c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-d^5/e^6*(-2/3/(a*
e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+8/3*c*d*e/(a*e^2-c*d^2)^3*(2*c*d*e*(x+d/e)+a*
e^2-c*d^2)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1073 vs. \(2 (466) = 932\).
time = 16.74, size = 2161, normalized size = 4.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*(7*c^6*d^13*x - 3*a^6*x^2*e^13 - 3*(a^5*c*d*x^3 + 2*a^6*d*x)*e^12 - 3*(a^5*c*d^2*x^2 + a^6*d^2)*e^11
+ 3*(a^4*c^2*d^3*x^3 + a^5*c*d^3*x)*e^10 + (8*a^4*c^2*d^4*x^2 + 3*a^5*c*d^4)*e^9 + (2*a^3*c^3*d^5*x^3 + 7*a^4
*c^2*d^5*x)*e^8 + 2*(5*a^3*c^3*d^6*x^2 + a^4*c^2*d^6)*e^7 + 2*(3*a^2*c^4*d^7*x^3 + 7*a^3*c^3*d^7*x)*e^6 - 3*(a
^2*c^4*d^8*x^2 - 2*a^3*c^3*d^8)*e^5 - 3*(5*a*c^5*d^9*x^3 + 8*a^2*c^4*d^9*x)*e^4 - (23*a*c^5*d^10*x^2 + 15*a^2*
c^4*d^10)*e^3 + (7*c^6*d^11*x^3 - a*c^5*d^11*x)*e^2 + 7*(2*c^6*d^12*x^2 + a*c^5*d^12)*e)*sqrt(c*d)*e^(1/2)*log
(8*c^2*d^3*x*e + c^2*d^4 + 8*a*c*d*x*e^3 + a^2*e^4 + 4*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e
+ c*d^2 + a*e^2)*sqrt(c*d)*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(105*c^6*d^12*x*e - 45*a^5*c*d*x^2
*e^12 - 15*(a^4*c^2*d^2*x^3 + 6*a^5*c*d^2*x)*e^11 + 3*(2*a^3*c^3*d^3*x^4 - 15*a^5*c*d^3)*e^10 + 3*(8*a^3*c^3*d
^4*x^3 + 15*a^4*c^2*d^4*x)*e^9 - 6*(3*a^2*c^4*d^5*x^4 - 11*a^3*c^3*d^5*x^2 - 5*a^4*c^2*d^5)*e^8 + 6*(3*a^2*c^4
*d^6*x^3 + 14*a^3*c^3*d^6*x)*e^7 + 6*(3*a*c^5*d^7*x^4 + 2*a^2*c^4*d^7*x^2 + 6*a^3*c^3*d^7)*e^6 - 6*(8*a*c^5*d^
8*x^3 + 37*a^2*c^4*d^8*x)*e^5 - (6*c^6*d^9*x^4 + 237*a*c^5*d^9*x^2 + 190*a^2*c^4*d^9)*e^4 + (21*c^6*d^10*x^3 -
50*a*c^5*d^10*x)*e^3 + 35*(4*c^6*d^11*x^2 + 3*a*c^5*d^11)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(
c^8*d^13*x*e^5 - a^4*c^4*d^4*x^2*e^14 - (a^3*c^5*d^5*x^3 + 2*a^4*c^4*d^5*x)*e^13 + (a^3*c^5*d^6*x^2 - a^4*c^4*
d^6)*e^12 + (3*a^2*c^6*d^7*x^3 + 5*a^3*c^5*d^7*x)*e^11 + 3*(a^2*c^6*d^8*x^2 + a^3*c^5*d^8)*e^10 - 3*(a*c^7*d^9
*x^3 + a^2*c^6*d^9*x)*e^9 - (5*a*c^7*d^10*x^2 + 3*a^2*c^6*d^10)*e^8 + (c^8*d^11*x^3 - a*c^7*d^11*x)*e^7 + (2*c
^8*d^12*x^2 + a*c^7*d^12)*e^6), -1/24*(15*(7*c^6*d^13*x - 3*a^6*x^2*e^13 - 3*(a^5*c*d*x^3 + 2*a^6*d*x)*e^12 -
3*(a^5*c*d^2*x^2 + a^6*d^2)*e^11 + 3*(a^4*c^2*d^3*x^3 + a^5*c*d^3*x)*e^10 + (8*a^4*c^2*d^4*x^2 + 3*a^5*c*d^4)*
e^9 + (2*a^3*c^3*d^5*x^3 + 7*a^4*c^2*d^5*x)*e^8 + 2*(5*a^3*c^3*d^6*x^2 + a^4*c^2*d^6)*e^7 + 2*(3*a^2*c^4*d^7*x
^3 + 7*a^3*c^3*d^7*x)*e^6 - 3*(a^2*c^4*d^8*x^2 - 2*a^3*c^3*d^8)*e^5 - 3*(5*a*c^5*d^9*x^3 + 8*a^2*c^4*d^9*x)*e^
4 - (23*a*c^5*d^10*x^2 + 15*a^2*c^4*d^10)*e^3 + (7*c^6*d^11*x^3 - a*c^5*d^11*x)*e^2 + 7*(2*c^6*d^12*x^2 + a*c^
5*d^12)*e)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqr
t(-c*d*e)/(c^2*d^3*x*e + a*c*d*x*e^3 + (c^2*d^2*x^2 + a*c*d^2)*e^2)) + 2*(105*c^6*d^12*x*e - 45*a^5*c*d*x^2*e^
12 - 15*(a^4*c^2*d^2*x^3 + 6*a^5*c*d^2*x)*e^11 + 3*(2*a^3*c^3*d^3*x^4 - 15*a^5*c*d^3)*e^10 + 3*(8*a^3*c^3*d^4*
x^3 + 15*a^4*c^2*d^4*x)*e^9 - 6*(3*a^2*c^4*d^5*x^4 - 11*a^3*c^3*d^5*x^2 - 5*a^4*c^2*d^5)*e^8 + 6*(3*a^2*c^4*d^
6*x^3 + 14*a^3*c^3*d^6*x)*e^7 + 6*(3*a*c^5*d^7*x^4 + 2*a^2*c^4*d^7*x^2 + 6*a^3*c^3*d^7)*e^6 - 6*(8*a*c^5*d^8*x
^3 + 37*a^2*c^4*d^8*x)*e^5 - (6*c^6*d^9*x^4 + 237*a*c^5*d^9*x^2 + 190*a^2*c^4*d^9)*e^4 + (21*c^6*d^10*x^3 - 50
*a*c^5*d^10*x)*e^3 + 35*(4*c^6*d^11*x^2 + 3*a*c^5*d^11)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^8
*d^13*x*e^5 - a^4*c^4*d^4*x^2*e^14 - (a^3*c^5*d^5*x^3 + 2*a^4*c^4*d^5*x)*e^13 + (a^3*c^5*d^6*x^2 - a^4*c^4*d^6
)*e^12 + (3*a^2*c^6*d^7*x^3 + 5*a^3*c^5*d^7*x)*e^11 + 3*(a^2*c^6*d^8*x^2 + a^3*c^5*d^8)*e^10 - 3*(a*c^7*d^9*x^
3 + a^2*c^6*d^9*x)*e^9 - (5*a*c^7*d^10*x^2 + 3*a^2*c^6*d^10)*e^8 + (c^8*d^11*x^3 - a*c^7*d^11*x)*e^7 + (2*c^8*
d^12*x^2 + a*c^7*d^12)*e^6)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{\left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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