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

Optimal. Leaf size=352 \[ \frac{\left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e}-\frac{\left (-45 a^2 c d^2 e^4-5 a^3 e^6-15 a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x} \]

[Out]

((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2])/(8*e) - ((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*x) - ((c^3*d^6 - 15*a*c^2*d^4
*e^2 - 45*a^2*c*d^2*e^4 - 5*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])])/(16*Sqrt[c]*Sqrt[d]*e^(3/2)) - (a^(3/2)*Sqrt[d]*e^(3/2)*(5*c*d^2 + 3*a*e^
2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])])/2

________________________________________________________________________________________

Rubi [A]  time = 0.432167, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {849, 812, 814, 843, 621, 206, 724} \[ \frac{\left (19 a^2 e^4+2 c d e x \left (7 a e^2+c d^2\right )+28 a c d^2 e^2+c^2 d^4\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e}-\frac{\left (-45 a^2 c d^2 e^4-5 a^3 e^6-15 a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{(3 a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c^2*d^4 + 28*a*c*d^2*e^2 + 19*a^2*e^4 + 2*c*d*e*(c*d^2 + 7*a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*
x^2])/(8*e) - ((3*a*e - c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3*x) - ((c^3*d^6 - 15*a*c^2*d^4
*e^2 - 45*a^2*c*d^2*e^4 - 5*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])])/(16*Sqrt[c]*Sqrt[d]*e^(3/2)) - (a^(3/2)*Sqrt[d]*e^(3/2)*(5*c*d^2 + 3*a*e^
2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
])])/2

Rule 849

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
]))

Rule 812

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

Rule 814

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^2 (d+e x)} \, dx &=\int \frac{(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}}{x^2} \, dx\\ &=-\frac{(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}}{3 x}-\frac{1}{2} \int \frac{\left (-a e \left (5 c d^2+3 a e^2\right )-c d \left (c d^2+7 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x} \, dx\\ &=\frac{\left (c^2 d^4+28 a c d^2 e^2+19 a^2 e^4+2 c d e \left (c d^2+7 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e}-\frac{(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}}{3 x}+\frac{\int \frac{4 a^2 c d^2 e^3 \left (5 c d^2+3 a e^2\right )-\frac{1}{2} c d \left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 a^3 e^6\right ) x}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c d e}\\ &=\frac{\left (c^2 d^4+28 a c d^2 e^2+19 a^2 e^4+2 c d e \left (c d^2+7 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e}-\frac{(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}}{3 x}+\frac{1}{2} \left (a^2 d e^2 \left (5 c d^2+3 a e^2\right )\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx-\frac{\left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 a^3 e^6\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 e}\\ &=\frac{\left (c^2 d^4+28 a c d^2 e^2+19 a^2 e^4+2 c d e \left (c d^2+7 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e}-\frac{(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}}{3 x}-\left (a^2 d e^2 \left (5 c d^2+3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )-\frac{\left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 a^3 e^6\right ) \operatorname{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 )}{8 e}\\ &=\frac{\left (c^2 d^4+28 a c d^2 e^2+19 a^2 e^4+2 c d e \left (c d^2+7 a e^2\right ) x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e}-\frac{(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}}{3 x}-\frac{\left (c^3 d^6-15 a c^2 d^4 e^2-45 a^2 c d^2 e^4-5 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 )}{16 \sqrt{c} \sqrt{d} e^{3/2}}-\frac{1}{2} a^{3/2} \sqrt{d} e^{3/2} \left (5 c d^2+3 a e^2\right ) \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 2.21351, size = 350, normalized size = 0.99 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{e} \sqrt{a e+c d x} \left (3 a^2 e^3 (11 e x-8 d)+2 a c d e^2 x (34 d+13 e x)+c^2 d^2 x \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )}{x}-\frac{3 \sqrt{c} \sqrt{d} \left (-45 a^2 c d^2 e^4-5 a^3 e^6-15 a c^2 d^4 e^2+c^3 d^6\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{24 a^{3/2} \sqrt{d} e^3 \left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{\sqrt{d+e x}}\right )}{24 e^{3/2} \sqrt{a e+c d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[e]*Sqrt[a*e + c*d*x]*(3*a^2*e^3*(-8*d + 11*e*x) + 2*a*c*d*e^2*x*(34*d +
13*e*x) + c^2*d^2*x*(3*d^2 + 14*d*e*x + 8*e^2*x^2)))/x - (3*Sqrt[c]*Sqrt[d]*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a
^2*c*d^2*e^4 - 5*a^3*e^6)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])]
)/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)]) - (24*a^(3/2)*Sqrt[d]*e^3*(5*c*d^2 + 3
*a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(24*e^(3/2)*Sqrt
[a*e + c*d*x])

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Maple [B]  time = 0.066, size = 2364, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/256/e*d^6*c^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^
(1/2))/(d*e*c)^(1/2)+1/16*e^4/d^3*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+9/64*e^4/d*a^2*(c*d*e*(d
/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/128*e^7/d^4*a^4/c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-3/6
4*e*d^2*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+1/8*e^3/d^2*a*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x)
)^(3/2)*x+3/64*e^5/d^2*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/8*e^3/d^2*a*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(3/2)*x-3/2*e^4*d*a^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/x)-13/256/e*d^6*c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)-1/16*e^4/d^3*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-3/64*e^5/d^2*a^
3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+227/64*e*d^2*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/64*e^4/
d*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/128*e^7/d^4*a^4/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+
1/d*c/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-1/d^2/a/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/5*e/d^
2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)-1/16*d*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+4/5*e/d
^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+19/8*e^3*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+67/48*d*c*(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+15/128*e^5*a^3*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e
*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/64*d^3*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1
/2)*x-1/8*e*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+3/128/e*d^4*c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(
d/e+x))^(1/2)+1/a/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+25/128*e^5*a^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/
(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+13/64*d^3*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2)*x+e^2/d*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+9/8*e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*x+13/128/e*d^4*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-15/256*e^7/d^2*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(d/e+
x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/256*e^9/d^4*a^5/c^2*ln(
(1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)
+225/256*e*d^4*a*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(
d*e*c)^(1/2)+15/256*e^7/d^2*a^4/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2))/(d*e*c)^(1/2)+121/64*e^2*d*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/256*e^9/d^4*a^5/c^2*ln(
(1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+375/128*e^3
*d^2*a^2*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/
2)+3/64*e^6/d^3*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-15/128*e^3*d^2*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(
d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+15/256*e*d^4*a*c^2*ln
((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2
)-3/64*e^6/d^3*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-9/64*e^2*d*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*
d^2)*(d/e+x))^(1/2)*x-5/2*d^3*a^2*e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 104.892, size = 3652, normalized size = 10.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(c*d*e)*x*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)*sq
rt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 24*(5*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*sqrt(a*d*e)*x*log((8*a^2*d^2*e
^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2
+ a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) - 4*(8*c^3*d^3*e^3*x^3 - 24*a^2*c*d^2*e^4 + 2*(7*
c^3*d^4*e^2 + 13*a*c^2*d^2*e^4)*x^2 + (3*c^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*x), 1/48*(3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqr
t(-c*d*e)*x*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)) + 12*(5*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*sqrt(a*d*e)*
x*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 2*(8*c^3*d^3*e^3*x^3 - 24*a^2
*c*d^2*e^4 + 2*(7*c^3*d^4*e^2 + 13*a*c^2*d^2*e^4)*x^2 + (3*c^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*s
qrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*x), 1/96*(48*(5*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*sqrt(-a*d*
e)*x*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^
2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) - 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2*e^4 - 5*a
^3*e^6)*sqrt(c*d*e)*x*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*(8*c^3*d^3*e^3*x^
3 - 24*a^2*c*d^2*e^4 + 2*(7*c^3*d^4*e^2 + 13*a*c^2*d^2*e^4)*x^2 + (3*c^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d
*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*x), 1/48*(24*(5*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*
sqrt(-a*d*e)*x*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*
e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)) + 3*(c^3*d^6 - 15*a*c^2*d^4*e^2 - 45*a^2*c*d^2
*e^4 - 5*a^3*e^6)*sqrt(-c*d*e)*x*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*(8*c^3*d^3*e^3*x^3 - 24*a^
2*c*d^2*e^4 + 2*(7*c^3*d^4*e^2 + 13*a*c^2*d^2*e^4)*x^2 + (3*c^3*d^5*e + 68*a*c^2*d^3*e^3 + 33*a^2*c*d*e^5)*x)*
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c*d*e^2*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError