\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^6 (d+e x)} \, dx\) [466]

Optimal result

Integrand size = 40, antiderivative size = 289 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\frac {3 \left (c d^2-a e^2\right )^3 \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{16 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}} \]

[Out]

-1/16*(c/a/e-e/d^2)*(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)/x^4-1/5*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(5/2)/d/x^5-3/256*(-a*e^2+c*d^2)^5*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^(5/2)/d^(7/2)/e^(5/2)+3/128*(-a*e^2+c*d^2)^3*(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^2/d^3/e^2/x^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {863, 820, 734, 738, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 a^{5/2} d^{7/2} e^{5/2}}+\frac {3 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 a^2 d^3 e^2 x^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 d x^5}-\frac {\left (\frac {c}{a e}-\frac {e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{16 x^4} \]

[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 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 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*} \text {integral}& = \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 ) \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 )}{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] (verified)

Time = 0.71 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} (d+e x)^3 \left (15 a^4 e^4-\frac {15 d^4 (a e+c d x)^4}{(d+e x)^4}+\frac {70 a d^3 e (a e+c d x)^3}{(d+e x)^3}+\frac {128 a^2 d^2 e^2 (a e+c d x)^2}{(d+e x)^2}-\frac {70 a^3 d e^3 (a e+c d x)}{d+e x}\right )}{\left (-c d^2+a e^2\right )^5 x^5 (a e+c d x)}+\frac {15 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 a^{5/2} d^{7/2} e^{5/2}} \]

[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]

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)^3*(15*a^4*e^4 - (1
5*d^4*(a*e + c*d*x)^4)/(d + e*x)^4 + (70*a*d^3*e*(a*e + c*d*x)^3)/(d + e*x)^3 + (128*a^2*d^2*e^2*(a*e + c*d*x)
^2)/(d + e*x)^2 - (70*a^3*d*e^3*(a*e + c*d*x))/(d + e*x)))/((-(c*d^2) + a*e^2)^5*x^5*(a*e + c*d*x))) + (15*Arc
Tanh[(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))))/(64
0*a^(5/2)*d^(7/2)*e^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(19538\) vs. \(2(259)=518\).

Time = 1.57 (sec) , antiderivative size = 19539, normalized size of antiderivative = 67.61

method	result	size
default	\(\text {Expression too large to display}\)	\(19539\)

[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,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 13.96 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.02 \[ \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=\left [\frac {15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {a d e} x^{5} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (128 \, a^{5} d^{5} e^{5} - {\left (15 \, a c^{4} d^{9} e - 70 \, a^{2} c^{3} d^{7} e^{3} - 128 \, a^{3} c^{2} d^{5} e^{5} + 70 \, a^{4} c d^{3} e^{7} - 15 \, a^{5} d e^{9}\right )} x^{4} + 2 \, {\left (5 \, a^{2} c^{3} d^{8} e^{2} + 233 \, a^{3} c^{2} d^{6} e^{4} + 23 \, a^{4} c d^{4} e^{6} - 5 \, a^{5} d^{2} e^{8}\right )} x^{3} + 8 \, {\left (31 \, a^{3} c^{2} d^{7} e^{3} + 64 \, a^{4} c d^{5} e^{5} + a^{5} d^{3} e^{7}\right )} x^{2} + 16 \, {\left (21 \, a^{4} c d^{6} e^{4} + 11 \, a^{5} d^{4} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2560 \, a^{3} d^{4} e^{3} x^{5}}, \frac {15 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {-a d e} x^{5} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (128 \, a^{5} d^{5} e^{5} - {\left (15 \, a c^{4} d^{9} e - 70 \, a^{2} c^{3} d^{7} e^{3} - 128 \, a^{3} c^{2} d^{5} e^{5} + 70 \, a^{4} c d^{3} e^{7} - 15 \, a^{5} d e^{9}\right )} x^{4} + 2 \, {\left (5 \, a^{2} c^{3} d^{8} e^{2} + 233 \, a^{3} c^{2} d^{6} e^{4} + 23 \, a^{4} c d^{4} e^{6} - 5 \, a^{5} d^{2} e^{8}\right )} x^{3} + 8 \, {\left (31 \, a^{3} c^{2} d^{7} e^{3} + 64 \, a^{4} c d^{5} e^{5} + a^{5} d^{3} e^{7}\right )} x^{2} + 16 \, {\left (21 \, a^{4} c d^{6} e^{4} + 11 \, a^{5} d^{4} e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1280 \, a^{3} d^{4} e^{3} x^{5}}\right ] \]

[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]

[1/2560*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10
)*sqrt(a*d*e)*x^5*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*(128*a^5*d^
5*e^5 - (15*a*c^4*d^9*e - 70*a^2*c^3*d^7*e^3 - 128*a^3*c^2*d^5*e^5 + 70*a^4*c*d^3*e^7 - 15*a^5*d*e^9)*x^4 + 2*
(5*a^2*c^3*d^8*e^2 + 233*a^3*c^2*d^6*e^4 + 23*a^4*c*d^4*e^6 - 5*a^5*d^2*e^8)*x^3 + 8*(31*a^3*c^2*d^7*e^3 + 64*
a^4*c*d^5*e^5 + a^5*d^3*e^7)*x^2 + 16*(21*a^4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 +
a*e^2)*x))/(a^3*d^4*e^3*x^5), 1/1280*(15*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^
6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(-a*d*e)*x^5*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*(128*a^5
*d^5*e^5 - (15*a*c^4*d^9*e - 70*a^2*c^3*d^7*e^3 - 128*a^3*c^2*d^5*e^5 + 70*a^4*c*d^3*e^7 - 15*a^5*d*e^9)*x^4 +
2*(5*a^2*c^3*d^8*e^2 + 233*a^3*c^2*d^6*e^4 + 23*a^4*c*d^4*e^6 - 5*a^5*d^2*e^8)*x^3 + 8*(31*a^3*c^2*d^7*e^3 +
64*a^4*c*d^5*e^5 + a^5*d^3*e^7)*x^2 + 16*(21*a^4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x))/(a^3*d^4*e^3*x^5)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{6}} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2449 vs. \(2 (259) = 518\).

Time = 0.40 (sec) , antiderivative size = 2449, normalized size of antiderivative = 8.47 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\text {Too large to display} \]

[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]

3/128*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*arct
an(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a^2*d^3*e^2) - 1
/640*(15*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c^5*d^14*e^4 - 75*(sqrt(c*d*e)*x -
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^5*c^4*d^12*e^6 + 150*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))*a^6*c^3*d^10*e^8 + 1130*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*
c^2*d^8*e^10 + 75*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c*d^6*e^12 - 15*(sqrt(c*d*
e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^9*d^4*e^14 + 256*sqrt(c*d*e)*a^7*c^2*d^9*e^9 - 70*(sqrt(
c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^3*c^5*d^13*e^3 + 350*(sqrt(c*d*e)*x - sqrt(c*d*e*x
^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4*c^4*d^11*e^5 + 5700*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x
+ a*d*e))^3*a^5*c^3*d^9*e^7 + 7100*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^6*c^2*d^
7*e^9 + 2210*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c*d^5*e^11 + 70*(sqrt(c*d*e)*
x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*d^3*e^13 + 2560*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e
*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^6*c^2*d^8*e^8 + 2560*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^
2*x + a*e^2*x + a*d*e))^2*a^7*c*d^6*e^10 + 128*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5
*a^2*c^5*d^12*e^2 + 3200*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^3*c^4*d^10*e^4 + 12
800*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^4*c^3*d^8*e^6 + 12800*(sqrt(c*d*e)*x - s
qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^5*c^2*d^6*e^8 + 3200*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^5*a^6*c*d^4*e^10 + 128*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^
7*d^2*e^12 + 6400*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^4*c^3*d^9*e^5
+ 14080*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^5*c^2*d^7*e^7 + 6400*sqr
t(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^6*c*d^5*e^9 + 1280*sqrt(c*d*e)*(sqr
t(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^4*a^7*d^3*e^11 + 70*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2
+ c*d^2*x + a*e^2*x + a*d*e))^7*a*c^5*d^11*e + 2210*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*
e))^7*a^2*c^4*d^9*e^3 + 7100*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^3*c^3*d^7*e^5 +
5700*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^4*c^2*d^5*e^7 + 350*(sqrt(c*d*e)*x - s
qrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^7*a^5*c*d^3*e^9 - 70*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a
*e^2*x + a*d*e))^7*a^6*d*e^11 + 1280*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))
^6*a^2*c^4*d^10*e^2 + 6400*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^3*c^3
*d^8*e^4 + 11520*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^4*c^2*d^6*e^6 +
6400*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^6*a^5*c*d^4*e^8 - 15*(sqrt(c*d
*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*c^5*d^10 + 75*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x
+ a*e^2*x + a*d*e))^9*a*c^4*d^8*e^2 + 1130*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^
2*c^3*d^6*e^4 + 150*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^3*c^2*d^4*e^6 - 75*(sqrt
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^9*a^4*c*d^2*e^8 + 15*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2
+ c*d^2*x + a*e^2*x + a*d*e))^9*a^5*e^10 + 2560*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*
x + a*d*e))^8*a^2*c^3*d^7*e^3 + 3840*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))
^8*a^3*c^2*d^5*e^5)/((a*d*e - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^5*a^2*d^3*e^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^6 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^6\,\left (d+e\,x\right )} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^6*(d + e*x)), x)