3.5.78 \(\int \frac {x^4}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [478]

Optimal. Leaf size=438 \[ -\frac {2 d x^3 \left (a e \left (c d^2-a e^2\right )+c d \left (c d^2-a e^2\right ) x\right )}{3 e \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 x \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 a c^2 d^4 e^2-a^2 c d^2 e^4-3 a^3 e^6\right ) x\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{2 c^{5/2} d^{5/2} e^{7/2}} \]

[Out]

-2/3*d*x^3*(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
)-1/2*(3*a*e^2+5*c*d^2)*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^(5/2)/d^(5/2)/e^(7/2)-2/3*x*(a*d*e*(-a*e^2+c*d^2)*(-3*a^2*e^4-10*a*c*d^2*e^2+5*c^2*d^4)+(-a*e
^2+c*d^2)*(-3*a^3*e^6-a^2*c*d^2*e^4-9*a*c^2*d^4*e^2+5*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)+1/3*(-9*a^3*e^6+9*a^2*c*d^2*e^4-31*a*c^2*d^4*e^2+15*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)/c^2/d^2/e^3/(-a*e^2+c*d^2)^3

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Rubi [A]
time = 0.36, antiderivative size = 438, 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, 654, 635, 212} \begin {gather*} -\frac {2 x \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-10 a c d^2 e^2+5 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^4-9 a c^2 d^4 e^2+5 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 (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (3 a e^2+5 c d^2\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 )}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac {2 d x^3 \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^4/((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^3*(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*(a*d*e*(c*d^2 - a*e^2)*(5*c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(5*c
^3*d^6 - 9*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]) + ((15*c^3*d^6 - 31*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 9*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2])/(3*c^2*d^2*e^3*(c*d^2 - a*e^2)^3) - ((5*c*d^2 + 3*a*e^2)*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])])/(2*c^(5/2)*d^(5/2)*e^(7/2))

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 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[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^4}{(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^4 (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^3 \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^2 \left (3 a c d^2 e \left (c d^2-a e^2\right )+\frac {1}{2} c d \left (5 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^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 {\frac {1}{2} a c d^2 e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) x}{\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^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^2 d^2 e^3}\\ &=-\frac {2 d x^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\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 )}{c^2 d^2 e^3}\\ &=-\frac {2 d x^3 \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 \left (a d e \left (c d^2-a e^2\right ) \left (5 c^2 d^4-10 a c d^2 e^2-3 a^2 e^4\right )+\left (c d^2-a e^2\right ) \left (5 c^3 d^6-9 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 (15 c^3 d^6-31 a c^2 d^4 e^2+9 a^2 c d^2 e^4-9 a^3 e^6\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac {\left (5 c d^2+3 a e^2\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 )}{2 c^{5/2} d^{5/2} e^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.55, size = 300, normalized size = 0.68 \begin {gather*} \frac {\frac {\sqrt {c} \sqrt {d} \sqrt {e} (a e+c d x) \left (-9 a^4 e^7 (d+e x)^2+3 a^3 c d e^5 (3 d-e x) (d+e x)^2+c^4 d^7 x \left (15 d^2+20 d e x+3 e^2 x^2\right )+a c^3 d^5 e \left (15 d^3-11 d^2 e x-39 d e^2 x^2-9 e^3 x^3\right )+a^2 c^2 d^3 e^3 \left (-31 d^3-33 d^2 e x+9 d e^2 x^2+9 e^3 x^3\right )\right )}{\left (c d^2-a e^2\right )^3}-3 \left (5 c d^2+3 a e^2\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 )}{3 c^{5/2} d^{5/2} e^{7/2} ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((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)*(-9*a^4*e^7*(d + e*x)^2 + 3*a^3*c*d*e^5*(3*d - e*x)*(d + e*x)^2 + c^4*
d^7*x*(15*d^2 + 20*d*e*x + 3*e^2*x^2) + a*c^3*d^5*e*(15*d^3 - 11*d^2*e*x - 39*d*e^2*x^2 - 9*e^3*x^3) + a^2*c^2
*d^3*e^3*(-31*d^3 - 33*d^2*e*x + 9*d*e^2*x^2 + 9*e^3*x^3)))/(c*d^2 - a*e^2)^3 - 3*(5*c*d^2 + 3*a*e^2)*(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])])/(3*c^(5/2)*d
^(5/2)*e^(7/2)*((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. \(1111\) vs. \(2(408)=816\).
time = 0.09, size = 1112, normalized size = 2.54

method	result	size
default	\(\frac {\frac {x^{2}}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {x}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2 c d e}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{c d e \sqrt {c d e}}\right )}{2 c d e}-\frac {2 a \left (-\frac {1}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{c}}{e}-\frac {d \left (-\frac {x}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{2 c d e}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d e x}{\sqrt {c d e}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}\right )}{c d e \sqrt {c d e}}\right )}{e^{2}}+\frac {d^{2} \left (-\frac {1}{c d e \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{c d e \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{e^{3}}-\frac {2 d^{3} \left (2 c d e x +a \,e^{2}+c \,d^{2}\right )}{e^{4} \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}+\frac {d^{4} \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 c d e \left (2 c d e \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{5}}\)	\(1112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(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*(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/e^2*(-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))+1/e^3*d^2*(-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^3/e^4*(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/e^5*d^4*(-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^4/(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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (393) = 786\).
time = 7.73, size = 1817, normalized size = 4.15 \begin {gather*} \left [\frac {3 \, {\left (5 \, c^{5} d^{11} x + 6 \, a^{3} c^{2} d^{6} e^{5} - 3 \, a^{5} x^{2} e^{11} - 3 \, {\left (a^{4} c d x^{3} + 2 \, a^{5} d x\right )} e^{10} - {\left (2 \, a^{4} c d^{2} x^{2} + 3 \, a^{5} d^{2}\right )} e^{9} + {\left (4 \, a^{3} c^{2} d^{3} x^{3} + 5 \, a^{4} c d^{3} x\right )} e^{8} + 2 \, {\left (7 \, a^{3} c^{2} d^{4} x^{2} + 2 \, a^{4} c d^{4}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{3} + 8 \, a^{3} c^{2} d^{5} x\right )} e^{6} - 6 \, {\left (2 \, a c^{4} d^{7} x^{3} + 3 \, a^{2} c^{3} d^{7} x\right )} e^{4} - {\left (19 \, a c^{4} d^{8} x^{2} + 12 \, a^{2} c^{3} d^{8}\right )} e^{3} + {\left (5 \, c^{5} d^{9} x^{3} - 2 \, a c^{4} d^{9} x\right )} e^{2} + 5 \, {\left (2 \, c^{5} d^{10} x^{2} + a c^{4} d^{10}\right )} e\right )} \sqrt {c d} e^{\frac {1}{2}} \log \left (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} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {c d} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) + 4 \, {\left (15 \, c^{5} d^{10} x e - 9 \, a^{4} c d x^{2} e^{10} - 3 \, {\left (a^{3} c^{2} d^{2} x^{3} + 6 \, a^{4} c d^{2} x\right )} e^{9} + 3 \, {\left (a^{3} c^{2} d^{3} x^{2} - 3 \, a^{4} c d^{3}\right )} e^{8} + 3 \, {\left (3 \, a^{2} c^{3} d^{4} x^{3} + 5 \, a^{3} c^{2} d^{4} x\right )} e^{7} + 9 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{6} - 3 \, {\left (3 \, a c^{4} d^{6} x^{3} + 11 \, a^{2} c^{3} d^{6} x\right )} e^{5} - {\left (39 \, a c^{4} d^{7} x^{2} + 31 \, a^{2} c^{3} d^{7}\right )} e^{4} + {\left (3 \, c^{5} d^{8} x^{3} - 11 \, a c^{4} d^{8} x\right )} e^{3} + 5 \, {\left (4 \, c^{5} d^{9} x^{2} + 3 \, a c^{4} d^{9}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{12 \, {\left (c^{7} d^{12} x e^{4} - a^{4} c^{3} d^{3} x^{2} e^{13} - {\left (a^{3} c^{4} d^{4} x^{3} + 2 \, a^{4} c^{3} d^{4} x\right )} e^{12} + {\left (a^{3} c^{4} d^{5} x^{2} - a^{4} c^{3} d^{5}\right )} e^{11} + {\left (3 \, a^{2} c^{5} d^{6} x^{3} + 5 \, a^{3} c^{4} d^{6} x\right )} e^{10} + 3 \, {\left (a^{2} c^{5} d^{7} x^{2} + a^{3} c^{4} d^{7}\right )} e^{9} - 3 \, {\left (a c^{6} d^{8} x^{3} + a^{2} c^{5} d^{8} x\right )} e^{8} - {\left (5 \, a c^{6} d^{9} x^{2} + 3 \, a^{2} c^{5} d^{9}\right )} e^{7} + {\left (c^{7} d^{10} x^{3} - a c^{6} d^{10} x\right )} e^{6} + {\left (2 \, c^{7} d^{11} x^{2} + a c^{6} d^{11}\right )} e^{5}\right )}}, \frac {3 \, {\left (5 \, c^{5} d^{11} x + 6 \, a^{3} c^{2} d^{6} e^{5} - 3 \, a^{5} x^{2} e^{11} - 3 \, {\left (a^{4} c d x^{3} + 2 \, a^{5} d x\right )} e^{10} - {\left (2 \, a^{4} c d^{2} x^{2} + 3 \, a^{5} d^{2}\right )} e^{9} + {\left (4 \, a^{3} c^{2} d^{3} x^{3} + 5 \, a^{4} c d^{3} x\right )} e^{8} + 2 \, {\left (7 \, a^{3} c^{2} d^{4} x^{2} + 2 \, a^{4} c d^{4}\right )} e^{7} + 2 \, {\left (3 \, a^{2} c^{3} d^{5} x^{3} + 8 \, a^{3} c^{2} d^{5} x\right )} e^{6} - 6 \, {\left (2 \, a c^{4} d^{7} x^{3} + 3 \, a^{2} c^{3} d^{7} x\right )} e^{4} - {\left (19 \, a c^{4} d^{8} x^{2} + 12 \, a^{2} c^{3} d^{8}\right )} e^{3} + {\left (5 \, c^{5} d^{9} x^{3} - 2 \, a c^{4} d^{9} x\right )} e^{2} + 5 \, {\left (2 \, c^{5} d^{10} x^{2} + a c^{4} d^{10}\right )} e\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{3} x e + a c d x e^{3} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (15 \, c^{5} d^{10} x e - 9 \, a^{4} c d x^{2} e^{10} - 3 \, {\left (a^{3} c^{2} d^{2} x^{3} + 6 \, a^{4} c d^{2} x\right )} e^{9} + 3 \, {\left (a^{3} c^{2} d^{3} x^{2} - 3 \, a^{4} c d^{3}\right )} e^{8} + 3 \, {\left (3 \, a^{2} c^{3} d^{4} x^{3} + 5 \, a^{3} c^{2} d^{4} x\right )} e^{7} + 9 \, {\left (a^{2} c^{3} d^{5} x^{2} + a^{3} c^{2} d^{5}\right )} e^{6} - 3 \, {\left (3 \, a c^{4} d^{6} x^{3} + 11 \, a^{2} c^{3} d^{6} x\right )} e^{5} - {\left (39 \, a c^{4} d^{7} x^{2} + 31 \, a^{2} c^{3} d^{7}\right )} e^{4} + {\left (3 \, c^{5} d^{8} x^{3} - 11 \, a c^{4} d^{8} x\right )} e^{3} + 5 \, {\left (4 \, c^{5} d^{9} x^{2} + 3 \, a c^{4} d^{9}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{6 \, {\left (c^{7} d^{12} x e^{4} - a^{4} c^{3} d^{3} x^{2} e^{13} - {\left (a^{3} c^{4} d^{4} x^{3} + 2 \, a^{4} c^{3} d^{4} x\right )} e^{12} + {\left (a^{3} c^{4} d^{5} x^{2} - a^{4} c^{3} d^{5}\right )} e^{11} + {\left (3 \, a^{2} c^{5} d^{6} x^{3} + 5 \, a^{3} c^{4} d^{6} x\right )} e^{10} + 3 \, {\left (a^{2} c^{5} d^{7} x^{2} + a^{3} c^{4} d^{7}\right )} e^{9} - 3 \, {\left (a c^{6} d^{8} x^{3} + a^{2} c^{5} d^{8} x\right )} e^{8} - {\left (5 \, a c^{6} d^{9} x^{2} + 3 \, a^{2} c^{5} d^{9}\right )} e^{7} + {\left (c^{7} d^{10} x^{3} - a c^{6} d^{10} x\right )} e^{6} + {\left (2 \, c^{7} d^{11} x^{2} + a c^{6} d^{11}\right )} e^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(3*(5*c^5*d^11*x + 6*a^3*c^2*d^6*e^5 - 3*a^5*x^2*e^11 - 3*(a^4*c*d*x^3 + 2*a^5*d*x)*e^10 - (2*a^4*c*d^2*
x^2 + 3*a^5*d^2)*e^9 + (4*a^3*c^2*d^3*x^3 + 5*a^4*c*d^3*x)*e^8 + 2*(7*a^3*c^2*d^4*x^2 + 2*a^4*c*d^4)*e^7 + 2*(
3*a^2*c^3*d^5*x^3 + 8*a^3*c^2*d^5*x)*e^6 - 6*(2*a*c^4*d^7*x^3 + 3*a^2*c^3*d^7*x)*e^4 - (19*a*c^4*d^8*x^2 + 12*
a^2*c^3*d^8)*e^3 + (5*c^5*d^9*x^3 - 2*a*c^4*d^9*x)*e^2 + 5*(2*c^5*d^10*x^2 + a*c^4*d^10)*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*(15*c^5*d^10*x*e - 9*a^4*c*d*x^
2*e^10 - 3*(a^3*c^2*d^2*x^3 + 6*a^4*c*d^2*x)*e^9 + 3*(a^3*c^2*d^3*x^2 - 3*a^4*c*d^3)*e^8 + 3*(3*a^2*c^3*d^4*x^
3 + 5*a^3*c^2*d^4*x)*e^7 + 9*(a^2*c^3*d^5*x^2 + a^3*c^2*d^5)*e^6 - 3*(3*a*c^4*d^6*x^3 + 11*a^2*c^3*d^6*x)*e^5
- (39*a*c^4*d^7*x^2 + 31*a^2*c^3*d^7)*e^4 + (3*c^5*d^8*x^3 - 11*a*c^4*d^8*x)*e^3 + 5*(4*c^5*d^9*x^2 + 3*a*c^4*
d^9)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^7*d^12*x*e^4 - a^4*c^3*d^3*x^2*e^13 - (a^3*c^4*d^4*x
^3 + 2*a^4*c^3*d^4*x)*e^12 + (a^3*c^4*d^5*x^2 - a^4*c^3*d^5)*e^11 + (3*a^2*c^5*d^6*x^3 + 5*a^3*c^4*d^6*x)*e^10
+ 3*(a^2*c^5*d^7*x^2 + a^3*c^4*d^7)*e^9 - 3*(a*c^6*d^8*x^3 + a^2*c^5*d^8*x)*e^8 - (5*a*c^6*d^9*x^2 + 3*a^2*c^
5*d^9)*e^7 + (c^7*d^10*x^3 - a*c^6*d^10*x)*e^6 + (2*c^7*d^11*x^2 + a*c^6*d^11)*e^5), 1/6*(3*(5*c^5*d^11*x + 6*
a^3*c^2*d^6*e^5 - 3*a^5*x^2*e^11 - 3*(a^4*c*d*x^3 + 2*a^5*d*x)*e^10 - (2*a^4*c*d^2*x^2 + 3*a^5*d^2)*e^9 + (4*a
^3*c^2*d^3*x^3 + 5*a^4*c*d^3*x)*e^8 + 2*(7*a^3*c^2*d^4*x^2 + 2*a^4*c*d^4)*e^7 + 2*(3*a^2*c^3*d^5*x^3 + 8*a^3*c
^2*d^5*x)*e^6 - 6*(2*a*c^4*d^7*x^3 + 3*a^2*c^3*d^7*x)*e^4 - (19*a*c^4*d^8*x^2 + 12*a^2*c^3*d^8)*e^3 + (5*c^5*d
^9*x^3 - 2*a*c^4*d^9*x)*e^2 + 5*(2*c^5*d^10*x^2 + a*c^4*d^10)*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)*sqrt(-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*(15*c^5*d^10*x*e - 9*a^4*c*d*x^2*e^10 - 3*(a^3*c^2*d^2*x^3 + 6*a^4*c*d^2*x)*e^9 + 3*(a^3*c^2*
d^3*x^2 - 3*a^4*c*d^3)*e^8 + 3*(3*a^2*c^3*d^4*x^3 + 5*a^3*c^2*d^4*x)*e^7 + 9*(a^2*c^3*d^5*x^2 + a^3*c^2*d^5)*e
^6 - 3*(3*a*c^4*d^6*x^3 + 11*a^2*c^3*d^6*x)*e^5 - (39*a*c^4*d^7*x^2 + 31*a^2*c^3*d^7)*e^4 + (3*c^5*d^8*x^3 - 1
1*a*c^4*d^8*x)*e^3 + 5*(4*c^5*d^9*x^2 + 3*a*c^4*d^9)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^7*d^
12*x*e^4 - a^4*c^3*d^3*x^2*e^13 - (a^3*c^4*d^4*x^3 + 2*a^4*c^3*d^4*x)*e^12 + (a^3*c^4*d^5*x^2 - a^4*c^3*d^5)*e
^11 + (3*a^2*c^5*d^6*x^3 + 5*a^3*c^4*d^6*x)*e^10 + 3*(a^2*c^5*d^7*x^2 + a^3*c^4*d^7)*e^9 - 3*(a*c^6*d^8*x^3 +
a^2*c^5*d^8*x)*e^8 - (5*a*c^6*d^9*x^2 + 3*a^2*c^5*d^9)*e^7 + (c^7*d^10*x^3 - a*c^6*d^10*x)*e^6 + (2*c^7*d^11*x
^2 + a*c^6*d^11)*e^5)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\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**4/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Integral(x**4/(((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^4/(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^4/((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^4}{\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^4/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

int(x^4/((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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