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

Optimal. Leaf size=394 \[ -\frac{\left (-83 a^2 c d^2 e^4-5 a^3 e^6-11 a c^2 d^4 e^2+2 c d e x \left (c d^2-5 a e^2\right ) \left (a e^2+3 c d^2\right )+3 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^2}+\frac{\left (90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8-20 a c^3 d^6 e^2+3 c^4 d^8\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{5/2}}-a^{5/2} d^{3/2} e^{5/2} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )+\frac{\left (11 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e} \]

[Out]

-((3*c^3*d^6 - 11*a*c^2*d^4*e^2 - 83*a^2*c*d^2*e^4 - 5*a^3*e^6 + 2*c*d*e*(c*d^2 - 5*a*e^2)*(3*c*d^2 + a*e^2)*x
)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*e^2) + ((3*c*d^2 + 11*a*e^2 + 6*c*d*e*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*e) + ((3*c^4*d^8 - 20*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e
^6 - 5*a^4*e^8)*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])])/(128*c^(3/2)*d^(3/2)*e^(5/2)) - a^(5/2)*d^(3/2)*e^(5/2)*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])]

________________________________________________________________________________________

Rubi [A]  time = 0.449671, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {849, 814, 843, 621, 206, 724} \[ -\frac{\left (-83 a^2 c d^2 e^4-5 a^3 e^6-11 a c^2 d^4 e^2+2 c d e x \left (c d^2-5 a e^2\right ) \left (a e^2+3 c d^2\right )+3 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c d e^2}+\frac{\left (90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8-20 a c^3 d^6 e^2+3 c^4 d^8\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{3/2} d^{3/2} e^{5/2}}-a^{5/2} d^{3/2} e^{5/2} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )+\frac{\left (11 a e^2+3 c d^2+6 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*c^3*d^6 - 11*a*c^2*d^4*e^2 - 83*a^2*c*d^2*e^4 - 5*a^3*e^6 + 2*c*d*e*(c*d^2 - 5*a*e^2)*(3*c*d^2 + a*e^2)*x
)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*c*d*e^2) + ((3*c*d^2 + 11*a*e^2 + 6*c*d*e*x)*(a*d*e + (c*d^
2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*e) + ((3*c^4*d^8 - 20*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e
^6 - 5*a^4*e^8)*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])])/(128*c^(3/2)*d^(3/2)*e^(5/2)) - a^(5/2)*d^(3/2)*e^(5/2)*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])]

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 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 (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} \, dx\\ &=\frac{\left (3 c d^2+11 a e^2+6 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}}{24 e}-\frac{\int \frac{\left (-8 a^2 c d^2 e^3+\frac{1}{2} c d \left (c d^2-5 a e^2\right ) \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}}{x} \, dx}{8 c d e}\\ &=-\frac{\left (3 c^3 d^6-11 a c^2 d^4 e^2-83 a^2 c d^2 e^4-5 a^3 e^6+2 c d e \left (c d^2-5 a e^2\right ) \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}}{64 c d e^2}+\frac{\left (3 c d^2+11 a e^2+6 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}}{24 e}+\frac{\int \frac{32 a^3 c^2 d^4 e^5+\frac{1}{4} c d \left (3 c^4 d^8-20 a c^3 d^6 e^2+90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8\right ) x}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 c^2 d^2 e^2}\\ &=-\frac{\left (3 c^3 d^6-11 a c^2 d^4 e^2-83 a^2 c d^2 e^4-5 a^3 e^6+2 c d e \left (c d^2-5 a e^2\right ) \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}}{64 c d e^2}+\frac{\left (3 c d^2+11 a e^2+6 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}}{24 e}+\left (a^3 d^2 e^3\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 (3 c^4 d^8-20 a c^3 d^6 e^2+90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 c d e^2}\\ &=-\frac{\left (3 c^3 d^6-11 a c^2 d^4 e^2-83 a^2 c d^2 e^4-5 a^3 e^6+2 c d e \left (c d^2-5 a e^2\right ) \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}}{64 c d e^2}+\frac{\left (3 c d^2+11 a e^2+6 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}}{24 e}-\left (2 a^3 d^2 e^3\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 (3 c^4 d^8-20 a c^3 d^6 e^2+90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8\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 )}{64 c d e^2}\\ &=-\frac{\left (3 c^3 d^6-11 a c^2 d^4 e^2-83 a^2 c d^2 e^4-5 a^3 e^6+2 c d e \left (c d^2-5 a e^2\right ) \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}}{64 c d e^2}+\frac{\left (3 c d^2+11 a e^2+6 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}}{24 e}+\frac{\left (3 c^4 d^8-20 a c^3 d^6 e^2+90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8\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 )}{128 c^{3/2} d^{3/2} e^{5/2}}-a^{5/2} d^{3/2} e^{5/2} \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 = 2.03991, size = 390, normalized size = 0.99 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\sqrt{e} \sqrt{a e+c d x} \left (a^2 c d e^4 (337 d+118 e x)+15 a^3 e^6+a c^2 d^2 e^2 \left (57 d^2+244 d e x+136 e^2 x^2\right )+c^3 \left (72 d^4 e^2 x^2+48 d^3 e^3 x^3+6 d^5 e x-9 d^6\right )\right )+\frac{3 \sqrt{c} \sqrt{d} \left (90 a^2 c^2 d^4 e^4+60 a^3 c d^2 e^6-5 a^4 e^8-20 a c^3 d^6 e^2+3 c^4 d^8\right ) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}-\frac{384 a^{5/2} c d^{5/2} e^5 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{\sqrt{d+e x}}\right )}{192 c d e^{5/2} \sqrt{a e+c d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[e]*Sqrt[a*e + c*d*x]*(15*a^3*e^6 + a^2*c*d*e^4*(337*d + 118*e*x) + a*c^2*
d^2*e^2*(57*d^2 + 244*d*e*x + 136*e^2*x^2) + c^3*(-9*d^6 + 6*d^5*e*x + 72*d^4*e^2*x^2 + 48*d^3*e^3*x^3)) + (3*
Sqrt[c]*Sqrt[d]*(3*c^4*d^8 - 20*a*c^3*d^6*e^2 + 90*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 - 5*a^4*e^8)*ArcSinh[(Sq
rt[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)]) - (384*a^(5/2)*c*d^(5/2)*e^5*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*S
qrt[e]*Sqrt[d + e*x])])/Sqrt[d + e*x]))/(192*c*d*e^(5/2)*Sqrt[a*e + c*d*x])

________________________________________________________________________________________

Maple [B]  time = 0.063, size = 2180, normalized size = 5.5 \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/(e*x+d),x)

[Out]

1/8*d^3*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/128*d^5*c^2/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+
83/64*d*a^2*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*d*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-9/64
*a^2*e^3*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/64*d^3*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^
(1/2)+1/8*d*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x-1/5/d*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5
/2)+1/5/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+15/128*d^3*a^2*e^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)-25/256/d*a^4*e^6/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/d^2*a^3*e^5/c*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+15/256/d*a^4*e^6/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)+3/64/d^2*a^3*e^5/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*
(d/e+x))^(1/2)*x+9/64*d^2*a*e*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256/d^3*a^5*e^8/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)+11/2
4*a*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/8/d*a^3*e^4/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-d^2*a^3*
e^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)+75/1
28*d^3*a^2*e^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)+19/64*d^2*a*e*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/16*d^2*c/e*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)
*(d/e+x))^(3/2)-3/128*d^5*c^2/e^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+1/16*d^2*c/e*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)+19/64*a^2*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-15/128*d*a^3*e^4*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^4*c
^2/e*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/256*d^7*c^3/e^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)-1/8/d*a*e^2*(c*d*e*(d/e+x)^2+(a*e^2
-c*d^2)*(d/e+x))^(3/2)*x-3/64*d^4*c^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/128/d^3*a^4*e^6/c^2*(c*d*e
*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-3/64/d*a^3*e^4/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-15/256*
d^5*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)-1/16/d^2*a^2*e^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+1/8/d*a*e^2*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(3/2)*x+1/16/d^2*a^2*e^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-3/128/d^3*a^4*e^6/c^2*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-25/256*d^5*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)+3/256*d^7*c^3/e^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)+75/128*d*a^3*e^4*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)+3/256/d^3*a^5*e^8/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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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/(e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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/(e*x+d),x, algorithm="fricas")

[Out]

Timed out

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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/(e*x+d),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} 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/(e*x+d),x, algorithm="giac")

[Out]

sage0*x