3.4.2 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^8 (d+e x)} \, dx\)

Optimal. Leaf size=500 \[ \frac {\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2048 a^{9/2} d^{11/2} e^{9/2}}+\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{384 a^3 d^4 e^3 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6} \]

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Rubi [A]  time = 0.64, antiderivative size = 500, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 834, 806, 720, 724, 206} \begin {gather*} \frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (c d^2-a e^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}}{384 a^3 d^4 e^3 x^4}+\frac {\left (-63 a^2 e^4+20 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \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 )}{2048 a^{9/2} d^{11/2} e^{9/2}}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{84 x^6}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 d x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]

[Out]

((c*d^2 - a*e^2)^3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(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])/(1024*a^4*d^5*e^4*x^2) - ((c*d^2 - a*e^2)*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(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))/(384*a^3*d^4*e^3*x^4) - (a*d*e + (c
*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(7*d*x^7) - (((5*c)/(a*e) - (9*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2))/(84*x^6) + ((35*c^2*d^4 + 20*a*c*d^2*e^2 - 63*a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2
))/(840*a^2*d^3*e^2*x^5) - ((c*d^2 - a*e^2)^5*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*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])])/(2048*a^(9/2)*d^(11/2
)*e^(9/2))

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 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 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 834

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

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
]))

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^8 (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^8} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-9 a e^2\right )+2 a c d e^2 x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^7} \, dx}{7 a d e}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\int \frac {\left (-\frac {1}{4} a e \left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right )-\frac {1}{2} a c d e^2 \left (5 c d^2-9 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}}{x^6} \, dx}{42 a^2 d^2 e^2}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}{48 a^2 d^3 e^2}\\ &=-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right )\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{256 a^3 d^4 e^3}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}+\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}{2048 a^4 d^5 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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 )}{1024 a^4 d^5 e^4}\\ &=\frac {\left (c d^2-a e^2\right )^3 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \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}}{1024 a^4 d^5 e^4 x^2}-\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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}}{384 a^3 d^4 e^3 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 d x^7}-\frac {\left (\frac {5 c}{a e}-\frac {9 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{84 x^6}+\frac {\left (35 c^2 d^4+20 a c d^2 e^2-63 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{840 a^2 d^3 e^2 x^5}-\frac {\left (c d^2-a e^2\right )^5 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\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 )}{2048 a^{9/2} d^{11/2} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.75, size = 408, normalized size = 0.82 \begin {gather*} \frac {((d+e x) (a e+c d x))^{3/2} \left (\frac {7 \left (9 a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \left (5 x \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 )-128 d (d+e x)^{5/2} (a e+c d x)^{5/2}\right )}{15360 a d^3 e x^5 (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac {(d+e x) \left (9 a e^2+7 c d^2\right ) (a e+c d x)^2}{12 a d e x^6}-\frac {(d+e x) (a e+c d x)^2}{x^7}\right )}{7 a d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]

[Out]

(((a*e + c*d*x)*(d + e*x))^(3/2)*(-(((a*e + c*d*x)^2*(d + e*x))/x^7) + ((7*c*d^2 + 9*a*e^2)*(a*e + c*d*x)^2*(d
+ e*x))/(12*a*d*e*x^6) + (7*(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*(-128*d*(a*e + c*d*x)^(5/2)*(d + e*x)^(5
/2) + 5*(c*d^2 - a*e^2)*x*(-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*ArcTanh[(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)))/(15360*a*d^3*e*x^5*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7*a*d*e)

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IntegrateAlgebraic [F]  time = 184.30, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x]

[Out]

$Aborted

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fricas [A]  time = 124.86, size = 1300, normalized size = 2.60

result too large to display

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^8/(e*x+d),x, algorithm="fricas")

[Out]

[-1/430080*(105*(5*c^7*d^14 - 15*a*c^6*d^12*e^2 + 9*a^2*c^5*d^10*e^4 + 5*a^3*c^4*d^8*e^6 + 15*a^4*c^3*d^6*e^8
- 45*a^5*c^2*d^4*e^10 + 35*a^6*c*d^2*e^12 - 9*a^7*e^14)*sqrt(a*d*e)*x^7*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*(15360*a^7*d^7*e^7 - (525*a*c^6*d^13*e - 1400*a^2*c^5*d^11*e^3 + 52
5*a^3*c^4*d^9*e^5 + 600*a^4*c^3*d^7*e^7 - 3689*a^5*c^2*d^5*e^9 + 3360*a^6*c*d^3*e^11 - 945*a^7*d*e^13)*x^6 + 2
*(175*a^2*c^5*d^12*e^2 - 455*a^3*c^4*d^10*e^4 + 150*a^4*c^3*d^8*e^6 - 1166*a^5*c^2*d^6*e^8 + 1099*a^6*c*d^4*e^
10 - 315*a^7*d^2*e^12)*x^5 - 8*(35*a^3*c^4*d^11*e^3 - 90*a^4*c^3*d^9*e^5 - 228*a^5*c^2*d^7*e^7 + 218*a^6*c*d^5
*e^9 - 63*a^7*d^3*e^11)*x^4 + 16*(15*a^4*c^3*d^10*e^4 + 2095*a^5*c^2*d^8*e^6 + 93*a^6*c*d^6*e^8 - 27*a^7*d^4*e
^10)*x^3 + 128*(185*a^5*c^2*d^9*e^5 + 380*a^6*c*d^7*e^7 + 3*a^7*d^5*e^9)*x^2 + 1280*(29*a^6*c*d^8*e^6 + 15*a^7
*d^6*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5*d^6*e^5*x^7), 1/215040*(105*(5*c^7*d^14 - 15*a*
c^6*d^12*e^2 + 9*a^2*c^5*d^10*e^4 + 5*a^3*c^4*d^8*e^6 + 15*a^4*c^3*d^6*e^8 - 45*a^5*c^2*d^4*e^10 + 35*a^6*c*d^
2*e^12 - 9*a^7*e^14)*sqrt(-a*d*e)*x^7*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)) - 2*(15360*a^7*d^7*e^7
- (525*a*c^6*d^13*e - 1400*a^2*c^5*d^11*e^3 + 525*a^3*c^4*d^9*e^5 + 600*a^4*c^3*d^7*e^7 - 3689*a^5*c^2*d^5*e^9
+ 3360*a^6*c*d^3*e^11 - 945*a^7*d*e^13)*x^6 + 2*(175*a^2*c^5*d^12*e^2 - 455*a^3*c^4*d^10*e^4 + 150*a^4*c^3*d^
8*e^6 - 1166*a^5*c^2*d^6*e^8 + 1099*a^6*c*d^4*e^10 - 315*a^7*d^2*e^12)*x^5 - 8*(35*a^3*c^4*d^11*e^3 - 90*a^4*c
^3*d^9*e^5 - 228*a^5*c^2*d^7*e^7 + 218*a^6*c*d^5*e^9 - 63*a^7*d^3*e^11)*x^4 + 16*(15*a^4*c^3*d^10*e^4 + 2095*a
^5*c^2*d^8*e^6 + 93*a^6*c*d^6*e^8 - 27*a^7*d^4*e^10)*x^3 + 128*(185*a^5*c^2*d^9*e^5 + 380*a^6*c*d^7*e^7 + 3*a^
7*d^5*e^9)*x^2 + 1280*(29*a^6*c*d^8*e^6 + 15*a^7*d^6*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5
*d^6*e^5*x^7)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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^8/(e*x+d),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.09, size = 5353, normalized size = 10.71 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)/x^8/(e*x+d),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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^{8}}\,{d x} \end {gather*}

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^8/(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^8), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^8\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^8*(d + e*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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**8/(e*x+d),x)

[Out]

Timed out

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