3.5.69 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^9 (d+e x)} \, dx\) [469]

Optimal. Leaf size=628 \[ -\frac {3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \tanh ^{-1}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{32768 a^{11/2} d^{13/2} e^{11/2}} \]

[Out]

1/2048*(-a*e^2+c*d^2)*(33*a^3*e^6+45*a^2*c*d^2*e^4+35*a*c^2*d^4*e^2+15*c^3*d^6)*(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^4/d^5/e^4/x^4-1/8*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^8-1/112*(5
*c/a/e-11*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7+1/448*(-33*a^2*e^4+10*a*c*d^2*e^2+15*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^6-1/4480*(-231*a^3*e^6+15*a^2*c*d^2*e^4+95*a*c^2*d^4*e^2+10
5*c^3*d^6)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/a^3/d^4/e^3/x^5+3/32768*(-a*e^2+c*d^2)^5*(33*a^3*e^6+45*a^2
*c*d^2*e^4+35*a*c^2*d^4*e^2+15*c^3*d^6)*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^(11/2)/d^(13/2)/e^(11/2)-3/16384*(-a*e^2+c*d^2)^3*(33*a^3*e^6+45*a^2*c*d^2*
e^4+35*a*c^2*d^4*e^2+15*c^3*d^6)*(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^5/d^6/e^5
/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.57, antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 9, 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 (-33 a^2 e^4+10 a c d^2 e^2+15 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (-231 a^3 e^6+15 a^2 c d^2 e^4+95 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {3 \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\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 )}{32768 a^{11/2} d^{13/2} e^{11/2}}-\frac {3 \left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (33 a^3 e^6+45 a^2 c d^2 e^4+35 a c^2 d^4 e^2+15 c^3 d^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 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^9*(d + e*x)),x]

[Out]

(-3*(c*d^2 - a*e^2)^3*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*(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])/(16384*a^5*d^6*e^5*x^2) + ((c*d^2 - a*e^2)*(15*c^3*d^6 + 35
*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*(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))/(2048*a^4*d^5*e^4*x^4) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(8*d*x^8) - (((5*c)/(a*e
) - (11*e)/d^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(112*x^7) + ((15*c^2*d^4 + 10*a*c*d^2*e^2 - 33*
a^2*e^4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(448*a^2*d^3*e^2*x^6) - ((105*c^3*d^6 + 95*a*c^2*d^4*e
^2 + 15*a^2*c*d^2*e^4 - 231*a^3*e^6)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4480*a^3*d^4*e^3*x^5) + (
3*(c*d^2 - a*e^2)^5*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*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])])/(32768*a^(11/2)*d^(13/2)
*e^(11/2))

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^9 (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^9} \, dx\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-11 a e^2\right )+3 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^8} \, dx}{8 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}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\int \frac {\left (-\frac {3}{4} a e \left (15 c^2 d^4+10 a c d^2 e^2-33 a^2 e^4\right )-a c d e^2 \left (5 c d^2-11 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^7} \, dx}{56 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}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\int \frac {\left (-\frac {3}{8} a e \left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right )-\frac {3}{4} a c d e^2 \left (15 c^2 d^4+10 a c d^2 e^2-33 a^2 e^4\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}{336 a^3 d^3 e^3}\\ &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}-\frac {\left (\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}{256 a^3 d^4 e^3}\\ &=\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {\left (3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}{4096 a^4 d^5 e^4}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}-\frac {\left (3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}{32768 a^5 d^6 e^5}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {\left (3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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 )}{16384 a^5 d^6 e^5}\\ &=-\frac {3 \left (c d^2-a e^2\right )^3 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{16384 a^5 d^6 e^5 x^2}+\frac {\left (c d^2-a e^2\right ) \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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}}{2048 a^4 d^5 e^4 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{8 d x^8}-\frac {\left (\frac {5 c}{a e}-\frac {11 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}}{112 x^7}+\frac {\left (15 c^2 d^4+10 a c d^2 e^2-33 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}}{448 a^2 d^3 e^2 x^6}-\frac {\left (105 c^3 d^6+95 a c^2 d^4 e^2+15 a^2 c d^2 e^4-231 a^3 e^6\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4480 a^3 d^4 e^3 x^5}+\frac {3 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\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 )}{32768 a^{11/2} d^{13/2} e^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.67, size = 572, normalized size = 0.91 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (1575 c^7 d^{14} x^7-525 a c^6 d^{12} e x^6 (2 d+7 e x)+35 a^2 c^5 d^{10} e^2 x^5 \left (24 d^2+68 d e x+29 e^2 x^2\right )-5 a^3 c^4 d^8 e^3 x^4 \left (144 d^3+376 d^2 e x+110 d e^2 x^2-185 e^3 x^3\right )+5 a^4 c^3 d^6 e^4 x^3 \left (128 d^4+320 d^3 e x+80 d^2 e^2 x^2-120 d e^3 x^3+265 e^4 x^4\right )+a^5 c^2 d^4 e^5 x^2 \left (103680 d^5+137600 d^4 e x+4640 d^3 e^2 x^2-5488 d^2 e^3 x^3+7034 d e^4 x^4-11193 e^5 x^5\right )+a^6 c d^2 e^6 x \left (168960 d^6+212480 d^5 e x+4480 d^4 e^2 x^2-5056 d^3 e^3 x^3+5928 d^2 e^4 x^4-7476 d e^5 x^5+11445 e^6 x^6\right )+a^7 e^7 \left (71680 d^7+87040 d^6 e x+1280 d^5 e^2 x^2-1408 d^4 e^3 x^3+1584 d^3 e^4 x^4-1848 d^2 e^5 x^5+2310 d e^6 x^6-3465 e^7 x^7\right )\right )}{x^8}+\frac {105 \left (c d^2-a e^2\right )^5 \left (15 c^3 d^6+35 a c^2 d^4 e^2+45 a^2 c d^2 e^4+33 a^3 e^6\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{573440 a^{11/2} d^{13/2} e^{11/2}} \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^9*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(1575*c^7*d^14*x^7 - 525*a*c^6*d^12*e*x^6*(2*d + 7*
e*x) + 35*a^2*c^5*d^10*e^2*x^5*(24*d^2 + 68*d*e*x + 29*e^2*x^2) - 5*a^3*c^4*d^8*e^3*x^4*(144*d^3 + 376*d^2*e*x
+ 110*d*e^2*x^2 - 185*e^3*x^3) + 5*a^4*c^3*d^6*e^4*x^3*(128*d^4 + 320*d^3*e*x + 80*d^2*e^2*x^2 - 120*d*e^3*x^
3 + 265*e^4*x^4) + a^5*c^2*d^4*e^5*x^2*(103680*d^5 + 137600*d^4*e*x + 4640*d^3*e^2*x^2 - 5488*d^2*e^3*x^3 + 70
34*d*e^4*x^4 - 11193*e^5*x^5) + a^6*c*d^2*e^6*x*(168960*d^6 + 212480*d^5*e*x + 4480*d^4*e^2*x^2 - 5056*d^3*e^3
*x^3 + 5928*d^2*e^4*x^4 - 7476*d*e^5*x^5 + 11445*e^6*x^6) + a^7*e^7*(71680*d^7 + 87040*d^6*e*x + 1280*d^5*e^2*
x^2 - 1408*d^4*e^3*x^3 + 1584*d^3*e^4*x^4 - 1848*d^2*e^5*x^5 + 2310*d*e^6*x^6 - 3465*e^7*x^7)))/x^8) + (105*(c
*d^2 - a*e^2)^5*(15*c^3*d^6 + 35*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + 33*a^3*e^6)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*
d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(573440*a^(11/2)*d^(13/2)*e^(11/2)
)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70735\) vs. \(2(586)=1172\).
time = 0.08, size = 70736, normalized size = 112.64

method	result	size
default	\(\text {Expression too large to display}\)	\(70736\)

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^9/(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^9/(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^9), x)

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Fricas [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^9/(e*x+d),x, algorithm="fricas")

[Out]

Timed out

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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**9/(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. 5747 vs. \(2 (570) = 1140\).
time = 2.25, size = 5747, normalized size = 9.15 \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^9/(e*x+d),x, algorithm="giac")

[Out]

-3/16384*(15*c^8*d^16 - 40*a*c^7*d^14*e^2 + 20*a^2*c^6*d^12*e^4 + 8*a^3*c^5*d^10*e^6 + 10*a^4*c^4*d^8*e^8 + 40
*a^5*c^3*d^6*e^10 - 140*a^6*c^2*d^4*e^12 + 120*a^7*c*d^2*e^14 - 33*a^8*e^16)*arctan(-(sqrt(c*d)*x*e^(1/2) - sq
rt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))/sqrt(-a*d*e))*e^(-5)/(sqrt(-a*d*e)*a^5*d^6) + 1/573440*(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^8*d^23*e^7 - 12075*(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^8*d^22*e^6 + 40215*(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^8*d^21*e^5 + 88045*(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^8*d^20*e^4 - 75795*(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^8*d^19*e^3 + 40215*(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^8*d^18
*e^2 - 12075*(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^8*d^17*e + 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))^15*c^8*d^16 + 1146880*(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^5*c^7*d^19*e^(11/2) - 4200*(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^7*d^21*e^9 + 32200*(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^7*d^20*e^8 + 3333400*(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^7*d^19*e^7 + 5117320*(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^7*d^18*e^6 + 202120*(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^7*d^17*e
^5 - 107240*(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^3*c^7*d^16*e^4 + 32200*(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))^13*a^2*c^7*d^15*e^3 - 4200*(sqrt(c*d)*x*e^(1
/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^15*a*c^7*d^14*e^2 + 5734400*(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^6*d^18*e^(17/2) + 20643840*(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^6*c^6*d^17*e^(15/2) + 13762560*(sqrt(c*d)*x*e^(1/2) - sq
rt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^8*sqrt(c*d)*a^5*c^6*d^16*e^(13/2) + 2100*(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))*a^9*c^6*d^19*e^11 + 5718300*(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^6*d^18*e^10 + 41341300*(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^6*d^17*e^9 + 62321980*(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^6*d^16*e^8 + 21034300*(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
^6*d^15*e^7 + 53620*(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^6*d^14*e^6 -
16100*(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^6*d^13*e^5 + 2100*(sqrt(c*d
)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^15*a^2*c^6*d^12*e^4 + 3440640*(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^9*c^5*d^17*e^(23/2) + 48168960*(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^5*d^16*e^(21/2) + 136478720*(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^5*d^15*e^(19/2) + 105512960*(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^5*d^14*e^(17/2) + 19496960*(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^5*d^13*e^(15/2) + 1147720*(sqr
t(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^5*d^17*e^13 + 34399960*(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^5*d^16*e^12 + 175494088*(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))^5*a^8*c^5*d^15*e^11 + 248341016*(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^5*d^14*e^10 + 104522264*(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^6*c^5*d^13*e^9 + 10113992*(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^5*c^5*d^12*e^8 - 6440*(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^
4*c^5*d^11*e^7 + 840*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^15*a^3*c^5*d^10*e^6 +
163840*sqrt(c*d)*a^11*c^4*d^16*e^(29/2) + 14745600*(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^10*c^4*d^15*e^(27/2) + 141066240*(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^9*c^4*d^14*e^(25/2) + 344064000*(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^8*c^4*d^13*e^(23/2) + 260341760*(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^7*c^4*d^12*e^(21/2) + 57344000*(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^6*c^4*...

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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^9\,\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^9*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^9*(d + e*x)), x)

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