3.466 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^6 (d+e x)} \, dx\)

Optimal. Leaf size=289 \[ \frac{3 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 x^4} \]

[Out]

(3*(c*d^2 - a*e^2)^3*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*a^2*d^3*e
^2*x^2) - ((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*
x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*d*x^5) - (3*(c*d^2 - a*e^2)^5*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])])/(256*a^(5/2)*d^(7/2)*e
^(5/2))

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Rubi [A]  time = 0.327656, antiderivative size = 289, normalized size of antiderivative = 1., 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 = {849, 806, 720, 724, 206} \[ \frac{3 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac{3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x]

[Out]

(3*(c*d^2 - a*e^2)^3*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(128*a^2*d^3*e
^2*x^2) - ((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(16*
x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(5*d*x^5) - (3*(c*d^2 - a*e^2)^5*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])])/(256*a^(5/2)*d^(7/2)*e
^(5/2))

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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

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

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])

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^6 (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^6} \, dx\\ &=-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{\left (-2 a c d^2 e+a e \left (c d^2+a e^2\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^5} \, dx}{2 a d e}\\ &=-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (2 a d e+\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}}{16 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{\left (3 \left (c d^2-a e^2\right )^3\right ) \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{32 a d^2 e}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (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}}{128 a^2 d^3 e^2 x^2}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (2 a d e+\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}}{16 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}+\frac{\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 a^2 d^3 e^2}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (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}}{128 a^2 d^3 e^2 x^2}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (2 a d e+\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}}{16 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{\left (3 \left (c d^2-a e^2\right )^5\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 )}{128 a^2 d^3 e^2}\\ &=\frac{3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (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}}{128 a^2 d^3 e^2 x^2}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (2 a d e+\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}}{16 x^4}-\frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac{3 \left (c d^2-a e^2\right )^5 \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 )}{256 a^{5/2} d^{7/2} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.960561, size = 295, normalized size = 1.02 \[ \frac{((d+e x) (a e+c d x))^{3/2} \left (\frac{5 \left (c d^2-a e^2\right ) \left (\frac{x \left (c d^2-a e^2\right ) \left (\frac{x \left (a e^2-c d^2\right ) \left (3 x^2 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )+\sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a e (2 d+5 e x)-3 c d^2 x\right )\right )}{a^{5/2} \sqrt{d} e^{5/2}}-8 (d+e x)^{5/2} \sqrt{a e+c d x}\right )}{d}-16 (d+e x)^{5/2} (a e+c d x)^{3/2}\right )}{64 d x^4 (d+e x)^{3/2} (a e+c d x)^{3/2}}-\frac{2 (d+e x) (a e+c d x)}{x^5}\right )}{10 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*((-2*(a*e + c*d*x)*(d + e*x))/x^5 + (5*(c*d^2 - a*e^2)*(-16*(a*e + c*d*x)^(3/
2)*(d + e*x)^(5/2) + ((c*d^2 - a*e^2)*x*(-8*Sqrt[a*e + c*d*x]*(d + e*x)^(5/2) + ((-(c*d^2) + a*e^2)*x*(Sqrt[a]
*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-3*c*d^2*x + a*e*(2*d + 5*e*x)) + 3*(c*d^2 - a*e^2)^2*x^2*Ar
cTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(a^(5/2)*Sqrt[d]*e^(5/2))))/d))/(64*d*x^4
*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(10*d)

________________________________________________________________________________________

Maple [B]  time = 0.102, size = 3991, normalized size = 13.8 \begin{align*} \text{output 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^6/(e*x+d),x)

[Out]

-17/640*d^2/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^4+3/640*d^4/a^5/e^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(5/2)*c^5+25/128*e^5/d^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-15/128*e^3*c^2*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)+1/5*e^5/d^6*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(5/2)+3/128*e^3*c^2*(c*d*e*(d/e+x)^2+(a*e^
2-c*d^2)*(d/e+x))^(1/2)-1/16*e^4/d^3*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+3/32*e*d^2/a*c^3*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-15/128*e^2*d^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)*c^3-31/80*e/d^4/a/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/5/d/a^3
*c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+109/320/d^3/a^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-1
/5/d^2/a/e/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+1/128*d^5/a^4/e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*c^5+3/128*d^6/a^3/e^3*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-9/128*d^4/a^2/e*c^4*(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)-1/80/a^3/e^3/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2+11/320/a^4/e^3/x*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3+1/16*e^8/d^7*a^2/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)+9/64*e^8/d^5*a
^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/128*e^11/d^8*a^4/c^2*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+
x))^(1/2)-3/64*e^5/d^2*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+15/128*e^9/d^4*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*e^4/d*c^2
*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/256*e^3*d^2*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)+1/8*e^7/d^6*a*(c*d*e*(d/e+x)^2+(a*e^2-c
*d^2)*(d/e+x))^(3/2)*x+3/64*e^9/d^6*a^3/c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)-1/8*e^5/d^4*c*(c*d*e*(
d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(3/2)*x+15/256*e^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)-3/64*e^9/d^6*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2)+15/128*e^5/d^2*a*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-15/128*e^9/d^4*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)-1/16*e^8/d^7*a^2/c*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(3/2)-9/64*e^8/d^5*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+3/128*e^11/d^8*a^4/c^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/256*e^3*d^2*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)-15/256*e^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*e^8/d^3*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)-1/8*e^7/d^6*a*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(3/2)*x+15/128*e^5/d^4*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-7/128*d^3/a^3/e^2*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*c^4+15/256*d^5/a/(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^4-3/64*d^3/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^4+273/640*e^3/d^4
/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c+129/320*e^2/d^5/a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+47/
160*e/d^2/a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2-253/640*e^3/d^6/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(7/2)-15/256*e^11/d^6*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)-3/64*e^10/d^7*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^6
/d^3*a*c*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/256*e^13/d^8*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)-15/128*e^7/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)+1
5/256*e^11/d^6*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)+3/64*e^10/d^7*a^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+15/128*e^6/d^3*a*c*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x-3/256*e^13/d^8*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)+15/128*e^7/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)+139/320*e^2/d^3/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(5/2)*x+15/128*e^4*d*a/(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^2-1/5/e/d^2/a^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2-139/320*e/d^4/a^2/x*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-5/64*d^2/a^3/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^4-1/5/d^2/a^2/e/x^
3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c+3/40/d/a^2/e^2/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c-3/256
*d^7/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^5+19/320/d/a^3/e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^2-11/320*d/a^4/e^2*c^4*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(5/2)*x+3/128*d^5/a^3/e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^5-1/320*d/a^4/e^4/x^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^3+3/640*d^3/a^5/e^4*c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x-3
/640*d^2/a^5/e^5/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)*c^4+1/128*d^4/a^4/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*x*c^5-15/256*e^6/d*a^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+5/64*e^3/d^2/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*c^2+253/640*e^4/d^5/a*c*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*x+17/160/e/a^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^3-1/128*e^6/d^
5*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-3/128*e^7/d^4*a^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+15/128*e
^4/d^3*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+13/40/d^3/a/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)

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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^{6}}\,{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^6/(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^6), x)

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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^6/(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**6/(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: NotImplementedError} \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^6/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError