3.5.68 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^8 (d+e x)} \, dx\) [468]

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

[Out]

-1/384*(-a*e^2+c*d^2)*(9*a^2*e^4+10*a*c*d^2*e^2+5*c^2*d^4)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)/a^3/d^4/e^3/x^4-1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^7-1/84*(5*c/a/e-9*e/d^2)*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^6+1/840*(-63*a^2*e^4+20*a*c*d^2*e^2+35*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(5/2)/a^2/d^3/e^2/x^5-1/2048*(-a*e^2+c*d^2)^5*(9*a^2*e^4+10*a*c*d^2*e^2+5*c^2*d^4)*arctanh(1/2*(2*a*d*
e+(a*e^2+c*d^2)*x)/a^(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))/a^(9/2)/d^(11/2)/e^(9/2)+1
/1024*(-a*e^2+c*d^2)^3*(9*a^2*e^4+10*a*c*d^2*e^2+5*c^2*d^4)*(2*a*d*e+(a*e^2+c*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)/a^4/d^5/e^4/x^2

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Rubi [A]
time = 0.41, 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 = {863, 848, 820, 734, 738, 212} \begin {gather*} \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} \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 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 734

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] /; Free
Q[{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 738

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 820

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)/(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[S
implify[m + 2*p + 3], 0]

Rule 848

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 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 {\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 ) \text {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 = 1.30, size = 497, normalized size = 0.99 \begin {gather*} \frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (-525 c^6 d^{12} x^6+350 a c^5 d^{10} e x^5 (d+4 e x)-35 a^2 c^4 d^8 e^2 x^4 \left (8 d^2+26 d e x+15 e^2 x^2\right )+60 a^3 c^3 d^6 e^3 x^3 \left (4 d^3+12 d^2 e x+5 d e^2 x^2-10 e^3 x^3\right )+a^4 c^2 d^4 e^4 x^2 \left (23680 d^4+33520 d^3 e x+1824 d^2 e^2 x^2-2332 d e^3 x^3+3689 e^4 x^4\right )+2 a^5 c d^2 e^5 x \left (18560 d^5+24320 d^4 e x+744 d^3 e^2 x^2-872 d^2 e^3 x^3+1099 d e^4 x^4-1680 e^5 x^5\right )+3 a^6 e^6 \left (5120 d^6+6400 d^5 e x+128 d^4 e^2 x^2-144 d^3 e^3 x^3+168 d^2 e^4 x^4-210 d e^5 x^5+315 e^6 x^6\right )\right )}{\left (c d^2-a e^2\right )^5 x^7 (a e+c d x) (d+e x)}+\frac {105 \left (5 c^2 d^4+10 a c d^2 e^2+9 a^2 e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{107520 a^{9/2} d^{11/2} e^{9/2}} \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]

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(-525*c^6*d^12*x^6 + 350*a*c^5
*d^10*e*x^5*(d + 4*e*x) - 35*a^2*c^4*d^8*e^2*x^4*(8*d^2 + 26*d*e*x + 15*e^2*x^2) + 60*a^3*c^3*d^6*e^3*x^3*(4*d
^3 + 12*d^2*e*x + 5*d*e^2*x^2 - 10*e^3*x^3) + a^4*c^2*d^4*e^4*x^2*(23680*d^4 + 33520*d^3*e*x + 1824*d^2*e^2*x^
2 - 2332*d*e^3*x^3 + 3689*e^4*x^4) + 2*a^5*c*d^2*e^5*x*(18560*d^5 + 24320*d^4*e*x + 744*d^3*e^2*x^2 - 872*d^2*
e^3*x^3 + 1099*d*e^4*x^4 - 1680*e^5*x^5) + 3*a^6*e^6*(5120*d^6 + 6400*d^5*e*x + 128*d^4*e^2*x^2 - 144*d^3*e^3*
x^3 + 168*d^2*e^4*x^4 - 210*d*e^5*x^5 + 315*e^6*x^6)))/((c*d^2 - a*e^2)^5*x^7*(a*e + c*d*x)*(d + e*x)) + (105*
(5*c^2*d^4 + 10*a*c*d^2*e^2 + 9*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])
/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(107520*a^(9/2)*d^(11/2)*e^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(45105\) vs. \(2(462)=924\).
time = 0.08, size = 45106, normalized size = 90.21

method	result	size
default	\(\text {Expression too large to display}\)	\(45106\)

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^8/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((x*e + d)*x^8), x)

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Fricas [A]
time = 176.66, size = 1355, normalized size = 2.71 \begin {gather*} \left [-\frac {{\left (105 \, {\left (5 \, c^{7} d^{14} x^{7} - 15 \, a c^{6} d^{12} x^{7} e^{2} + 9 \, a^{2} c^{5} d^{10} x^{7} e^{4} + 5 \, a^{3} c^{4} d^{8} x^{7} e^{6} + 15 \, a^{4} c^{3} d^{6} x^{7} e^{8} - 45 \, a^{5} c^{2} d^{4} x^{7} e^{10} + 35 \, a^{6} c d^{2} x^{7} e^{12} - 9 \, a^{7} x^{7} e^{14}\right )} \sqrt {a d} e^{\frac {1}{2}} \log \left (\frac {c^{2} d^{4} x^{2} + 8 \, a c d^{3} x e + a^{2} x^{2} e^{4} + 8 \, a^{2} d x e^{3} + 4 \, {\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {a d} e^{\frac {1}{2}} + 2 \, {\left (3 \, a c d^{2} x^{2} + 4 \, a^{2} d^{2}\right )} e^{2}}{x^{2}}\right ) - 4 \, {\left (525 \, a c^{6} d^{13} x^{6} e - 350 \, a^{2} c^{5} d^{12} x^{5} e^{2} - 945 \, a^{7} d x^{6} e^{13} + 630 \, a^{7} d^{2} x^{5} e^{12} + 168 \, {\left (20 \, a^{6} c d^{3} x^{6} - 3 \, a^{7} d^{3} x^{4}\right )} e^{11} - 2 \, {\left (1099 \, a^{6} c d^{4} x^{5} - 216 \, a^{7} d^{4} x^{3}\right )} e^{10} - {\left (3689 \, a^{5} c^{2} d^{5} x^{6} - 1744 \, a^{6} c d^{5} x^{4} + 384 \, a^{7} d^{5} x^{2}\right )} e^{9} + 4 \, {\left (583 \, a^{5} c^{2} d^{6} x^{5} - 372 \, a^{6} c d^{6} x^{3} - 4800 \, a^{7} d^{6} x\right )} e^{8} + 8 \, {\left (75 \, a^{4} c^{3} d^{7} x^{6} - 228 \, a^{5} c^{2} d^{7} x^{4} - 6080 \, a^{6} c d^{7} x^{2} - 1920 \, a^{7} d^{7}\right )} e^{7} - 20 \, {\left (15 \, a^{4} c^{3} d^{8} x^{5} + 1676 \, a^{5} c^{2} d^{8} x^{3} + 1856 \, a^{6} c d^{8} x\right )} e^{6} + 5 \, {\left (105 \, a^{3} c^{4} d^{9} x^{6} - 144 \, a^{4} c^{3} d^{9} x^{4} - 4736 \, a^{5} c^{2} d^{9} x^{2}\right )} e^{5} + 10 \, {\left (91 \, a^{3} c^{4} d^{10} x^{5} - 24 \, a^{4} c^{3} d^{10} x^{3}\right )} e^{4} - 280 \, {\left (5 \, a^{2} c^{5} d^{11} x^{6} - a^{3} c^{4} d^{11} x^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{430080 \, a^{5} d^{6} x^{7}}, \frac {{\left (105 \, {\left (5 \, c^{7} d^{14} x^{7} - 15 \, a c^{6} d^{12} x^{7} e^{2} + 9 \, a^{2} c^{5} d^{10} x^{7} e^{4} + 5 \, a^{3} c^{4} d^{8} x^{7} e^{6} + 15 \, a^{4} c^{3} d^{6} x^{7} e^{8} - 45 \, a^{5} c^{2} d^{4} x^{7} e^{10} + 35 \, a^{6} c d^{2} x^{7} e^{12} - 9 \, a^{7} x^{7} e^{14}\right )} \sqrt {-a d e} \arctan \left (\frac {{\left (c d^{2} x + a x e^{2} + 2 \, a d e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {-a d e}}{2 \, {\left (a c d^{3} x e + a^{2} d x e^{3} + {\left (a c d^{2} x^{2} + a^{2} d^{2}\right )} e^{2}\right )}}\right ) + 2 \, {\left (525 \, a c^{6} d^{13} x^{6} e - 350 \, a^{2} c^{5} d^{12} x^{5} e^{2} - 945 \, a^{7} d x^{6} e^{13} + 630 \, a^{7} d^{2} x^{5} e^{12} + 168 \, {\left (20 \, a^{6} c d^{3} x^{6} - 3 \, a^{7} d^{3} x^{4}\right )} e^{11} - 2 \, {\left (1099 \, a^{6} c d^{4} x^{5} - 216 \, a^{7} d^{4} x^{3}\right )} e^{10} - {\left (3689 \, a^{5} c^{2} d^{5} x^{6} - 1744 \, a^{6} c d^{5} x^{4} + 384 \, a^{7} d^{5} x^{2}\right )} e^{9} + 4 \, {\left (583 \, a^{5} c^{2} d^{6} x^{5} - 372 \, a^{6} c d^{6} x^{3} - 4800 \, a^{7} d^{6} x\right )} e^{8} + 8 \, {\left (75 \, a^{4} c^{3} d^{7} x^{6} - 228 \, a^{5} c^{2} d^{7} x^{4} - 6080 \, a^{6} c d^{7} x^{2} - 1920 \, a^{7} d^{7}\right )} e^{7} - 20 \, {\left (15 \, a^{4} c^{3} d^{8} x^{5} + 1676 \, a^{5} c^{2} d^{8} x^{3} + 1856 \, a^{6} c d^{8} x\right )} e^{6} + 5 \, {\left (105 \, a^{3} c^{4} d^{9} x^{6} - 144 \, a^{4} c^{3} d^{9} x^{4} - 4736 \, a^{5} c^{2} d^{9} x^{2}\right )} e^{5} + 10 \, {\left (91 \, a^{3} c^{4} d^{10} x^{5} - 24 \, a^{4} c^{3} d^{10} x^{3}\right )} e^{4} - 280 \, {\left (5 \, a^{2} c^{5} d^{11} x^{6} - a^{3} c^{4} d^{11} x^{4}\right )} e^{3}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}\right )} e^{\left (-5\right )}}{215040 \, a^{5} d^{6} x^{7}}\right ] \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="fricas")

[Out]

[-1/430080*(105*(5*c^7*d^14*x^7 - 15*a*c^6*d^12*x^7*e^2 + 9*a^2*c^5*d^10*x^7*e^4 + 5*a^3*c^4*d^8*x^7*e^6 + 15*
a^4*c^3*d^6*x^7*e^8 - 45*a^5*c^2*d^4*x^7*e^10 + 35*a^6*c*d^2*x^7*e^12 - 9*a^7*x^7*e^14)*sqrt(a*d)*e^(1/2)*log(
(c^2*d^4*x^2 + 8*a*c*d^3*x*e + a^2*x^2*e^4 + 8*a^2*d*x*e^3 + 4*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*
x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(a*d)*e^(1/2) + 2*(3*a*c*d^2*x^2 + 4*a^2*d^2)*e^2)/x^2) - 4*(525*a*c^6*d^13*x^6
*e - 350*a^2*c^5*d^12*x^5*e^2 - 945*a^7*d*x^6*e^13 + 630*a^7*d^2*x^5*e^12 + 168*(20*a^6*c*d^3*x^6 - 3*a^7*d^3*
x^4)*e^11 - 2*(1099*a^6*c*d^4*x^5 - 216*a^7*d^4*x^3)*e^10 - (3689*a^5*c^2*d^5*x^6 - 1744*a^6*c*d^5*x^4 + 384*a
^7*d^5*x^2)*e^9 + 4*(583*a^5*c^2*d^6*x^5 - 372*a^6*c*d^6*x^3 - 4800*a^7*d^6*x)*e^8 + 8*(75*a^4*c^3*d^7*x^6 - 2
28*a^5*c^2*d^7*x^4 - 6080*a^6*c*d^7*x^2 - 1920*a^7*d^7)*e^7 - 20*(15*a^4*c^3*d^8*x^5 + 1676*a^5*c^2*d^8*x^3 +
1856*a^6*c*d^8*x)*e^6 + 5*(105*a^3*c^4*d^9*x^6 - 144*a^4*c^3*d^9*x^4 - 4736*a^5*c^2*d^9*x^2)*e^5 + 10*(91*a^3*
c^4*d^10*x^5 - 24*a^4*c^3*d^10*x^3)*e^4 - 280*(5*a^2*c^5*d^11*x^6 - a^3*c^4*d^11*x^4)*e^3)*sqrt(c*d^2*x + a*x*
e^2 + (c*d*x^2 + a*d)*e))*e^(-5)/(a^5*d^6*x^7), 1/215040*(105*(5*c^7*d^14*x^7 - 15*a*c^6*d^12*x^7*e^2 + 9*a^2*
c^5*d^10*x^7*e^4 + 5*a^3*c^4*d^8*x^7*e^6 + 15*a^4*c^3*d^6*x^7*e^8 - 45*a^5*c^2*d^4*x^7*e^10 + 35*a^6*c*d^2*x^7
*e^12 - 9*a^7*x^7*e^14)*sqrt(-a*d*e)*arctan(1/2*(c*d^2*x + a*x*e^2 + 2*a*d*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^
2 + a*d)*e)*sqrt(-a*d*e)/(a*c*d^3*x*e + a^2*d*x*e^3 + (a*c*d^2*x^2 + a^2*d^2)*e^2)) + 2*(525*a*c^6*d^13*x^6*e
- 350*a^2*c^5*d^12*x^5*e^2 - 945*a^7*d*x^6*e^13 + 630*a^7*d^2*x^5*e^12 + 168*(20*a^6*c*d^3*x^6 - 3*a^7*d^3*x^4
)*e^11 - 2*(1099*a^6*c*d^4*x^5 - 216*a^7*d^4*x^3)*e^10 - (3689*a^5*c^2*d^5*x^6 - 1744*a^6*c*d^5*x^4 + 384*a^7*
d^5*x^2)*e^9 + 4*(583*a^5*c^2*d^6*x^5 - 372*a^6*c*d^6*x^3 - 4800*a^7*d^6*x)*e^8 + 8*(75*a^4*c^3*d^7*x^6 - 228*
a^5*c^2*d^7*x^4 - 6080*a^6*c*d^7*x^2 - 1920*a^7*d^7)*e^7 - 20*(15*a^4*c^3*d^8*x^5 + 1676*a^5*c^2*d^8*x^3 + 185
6*a^6*c*d^8*x)*e^6 + 5*(105*a^3*c^4*d^9*x^6 - 144*a^4*c^3*d^9*x^4 - 4736*a^5*c^2*d^9*x^2)*e^5 + 10*(91*a^3*c^4
*d^10*x^5 - 24*a^4*c^3*d^10*x^3)*e^4 - 280*(5*a^2*c^5*d^11*x^6 - a^3*c^4*d^11*x^4)*e^3)*sqrt(c*d^2*x + a*x*e^2
+ (c*d*x^2 + a*d)*e))*e^(-5)/(a^5*d^6*x^7)]

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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 4505 vs. \(2 (452) = 904\).
time = 1.81, size = 4505, normalized size = 9.01 \begin {gather*} \text {Too large to display} \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]

1/1024*(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)*arctan(-(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e
^2 + a*d*e))/sqrt(-a*d*e))*e^(-4)/(sqrt(-a*d*e)*a^4*d^5) - 1/107520*(525*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e
+ c*d^2*x + a*x*e^2 + a*d*e))*a^6*c^7*d^20*e^6 - 3500*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e
^2 + a*d*e))^3*a^5*c^7*d^19*e^5 + 9905*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a
^4*c^7*d^18*e^4 + 15360*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^3*c^7*d^17*e^3
- 9905*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^2*c^7*d^16*e^2 + 3500*(sqrt(c*
d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a*c^7*d^15*e - 525*(sqrt(c*d)*x*e^(1/2) - sqrt(
c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^13*c^7*d^14 + 215040*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x +
a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^4*c^6*d^16*e^(9/2) - 1575*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a
*x*e^2 + a*d*e))*a^7*c^6*d^18*e^8 + 10500*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^
3*a^6*c^6*d^17*e^7 + 615405*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^5*c^6*d^16
*e^6 + 752640*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^4*c^6*d^15*e^5 + 29715*(
sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^3*c^6*d^14*e^4 - 10500*(sqrt(c*d)*x*e^(
1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^2*c^6*d^13*e^3 + 1575*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d
*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^13*a*c^6*d^12*e^2 + 1075200*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^6*c^5*d^15*e^(15/2) + 2938880*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d
^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^5*c^5*d^14*e^(13/2) + 1576960*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c
*d^2*x + a*x*e^2 + a*d*e))^8*sqrt(c*d)*a^4*c^5*d^13*e^(11/2) + 945*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d
^2*x + a*x*e^2 + a*d*e))*a^8*c^5*d^16*e^10 + 1068900*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e))^3*a^7*c^5*d^15*e^9 + 5823909*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*
a^6*c^5*d^14*e^8 + 6730752*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^5*c^5*d^13*
e^7 + 1745499*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9*a^4*c^5*d^12*e^6 + 6300*(s
qrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^3*c^5*d^11*e^5 - 945*(sqrt(c*d)*x*e^(1/
2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^13*a^2*c^5*d^10*e^4 + 645120*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d
*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sqrt(c*d)*a^8*c^4*d^14*e^(21/2) + 6666240*(sqrt(c*d)*x*e^(1/2) - sqrt(c
*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4*sqrt(c*d)*a^7*c^4*d^13*e^(19/2) + 14192640*(sqrt(c*d)*x*e^(1/2) - sqr
t(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^6*sqrt(c*d)*a^6*c^4*d^12*e^(17/2) + 8171520*(sqrt(c*d)*x*e^(1/2) - s
qrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^8*sqrt(c*d)*a^5*c^4*d^11*e^(15/2) + 1075200*(sqrt(c*d)*x*e^(1/2) -
sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^10*sqrt(c*d)*a^4*c^4*d^10*e^(13/2) + 215565*(sqrt(c*d)*x*e^(1/2)
- sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*a^9*c^4*d^14*e^12 + 4655700*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^
2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^8*c^4*d^13*e^11 + 17499825*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2
*x + a*x*e^2 + a*d*e))^5*a^7*c^4*d^12*e^10 + 18170880*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^
2 + a*d*e))^7*a^6*c^4*d^11*e^9 + 5294415*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^9
*a^5*c^4*d^10*e^8 + 290220*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^11*a^4*c^4*d^9*
e^7 - 525*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^13*a^3*c^4*d^8*e^6 + 30720*sqrt(
c*d)*a^10*c^3*d^13*e^(27/2) + 1935360*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*sq
rt(c*d)*a^9*c^3*d^12*e^(25/2) + 13332480*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^4
*sqrt(c*d)*a^8*c^3*d^11*e^(23/2) + 23654400*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)
)^6*sqrt(c*d)*a^7*c^3*d^10*e^(21/2) + 12257280*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d
*e))^8*sqrt(c*d)*a^6*c^3*d^9*e^(19/2) + 1505280*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*
d*e))^10*sqrt(c*d)*a^5*c^3*d^8*e^(17/2) + 431655*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a
*d*e))*a^10*c^3*d^12*e^14 + 6225660*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^3*a^9*
c^3*d^11*e^13 + 19168275*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^5*a^8*c^3*d^10*e^
12 + 16665600*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^7*a^7*c^3*d^9*e^11 + 3625965
*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*...

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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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