\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^3 (d+e x)} \, dx\) [463]

Optimal result

Integrand size = 40, antiderivative size = 339 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=-\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\frac {(a e-c d x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 x^2}+\frac {3 \sqrt {c} \sqrt {d} \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) \text {arctanh}\left (\frac {2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {d}} \]

[Out]

-1/2*(-c*d*x+a*e)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^2+3/8*(5*a^2*e^4+10*a*c*d^2*e^2+c^2*d^4)*arctanh(1
/2*(2*c*d*e*x+a*e^2+c*d^2)/c^(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))*c^(1/2)*d^(1/2)/e^
(1/2)-3/8*(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^(1/2)*e^(1/2)/d^(1/2)-3/4*(a*e*(a*e^2+3*c*d^2)-c*d*(3*a*e^2+c*d^2)*x)*
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {863, 826, 857, 635, 212, 738} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\frac {3 \sqrt {c} \sqrt {d} \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (a^2 e^4+10 a c d^2 e^2+5 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt {d}}-\frac {(a e-c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 x^2}-\frac {3 \left (a e \left (a e^2+3 c d^2\right )-c d x \left (3 a e^2+c d^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 x} \]

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x]

[Out]

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

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 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 826

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

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 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^3} \, dx \\ & = -\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}}{2 x^2}-\frac {3}{8} \int \frac {\left (-2 a e \left (3 c d^2+a e^2\right )-2 c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2} \, dx \\ & = -\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\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}}{2 x^2}+\frac {3}{16} \int \frac {2 a e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right )+2 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) x}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \\ & = -\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\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}}{2 x^2}+\frac {1}{8} \left (3 a e \left (5 c^2 d^4+10 a c d^2 e^2+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+\frac {1}{8} \left (3 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx \\ & = -\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\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}}{2 x^2}-\frac {1}{4} \left (3 a e \left (5 c^2 d^4+10 a c d^2 e^2+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 )+\frac {1}{4} \left (3 c d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right ) \\ & = -\frac {3 \left (a e \left (3 c d^2+a e^2\right )-c d \left (c d^2+3 a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 x}-\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}}{2 x^2}+\frac {3 \sqrt {c} \sqrt {d} \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 \sqrt {e}}-\frac {3 \sqrt {a} \sqrt {e} \left (5 c^2 d^4+10 a c d^2 e^2+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 )}{8 \sqrt {d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {d} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x} \left (-9 a c d e x (d-e x)+c^2 d^2 x^2 (5 d+2 e x)-a^2 e^2 (2 d+5 e x)\right )-3 \sqrt {a} e \left (5 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )+3 \sqrt {c} d \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )\right )}{4 \sqrt {d} \sqrt {e} x^2 \sqrt {(a e+c d x) (d+e x)}} \]

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-9*a*c*d*e*x*(d - e*x) + c^
2*d^2*x^2*(5*d + 2*e*x) - a^2*e^2*(2*d + 5*e*x)) - 3*Sqrt[a]*e*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^2*ArcT
anh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])] + 3*Sqrt[c]*d*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a
^2*e^4)*x^2*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])]))/(4*Sqrt[d]*Sqrt[e]*x^2*Sqrt
[(a*e + c*d*x)*(d + e*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3892\) vs. \(2(295)=590\).

Time = 0.81 (sec) , antiderivative size = 3893, normalized size of antiderivative = 11.48

method	result	size
default	\(\text {Expression too large to display}\)	\(3893\)

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a/d/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+3/4*(a*e^2+c*d^2)/a/d/e*(-1/a/d/e/x*(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(7/2)+5/2*(a*e^2+c*d^2)/a/d/e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/2*(a*e^2+c*d^
2)*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2
)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^
2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d
*e)^(1/2)))+a*d*e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*(a*e^2+c*d^2)*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/
c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^
2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))+a*d*e*((a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(1/2)+1/2*(a*e^2+c*d^2)*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2))/(c*d*e)^(1/2)-a*d*e/(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))))+6*c/a*(1/12*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)
+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))))+5/2*c/a*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/
2*(a*e^2+c*d^2)*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/16*(4*a*c*d^2*e^2
-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*
d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2))/(c*d*e)^(1/2)))+a*d*e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*(a*e^2+c*d^2)*(1/4*(2*c*d*e*x+
a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e
^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))+a*d*e*((a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2*(a*e^2+c*d^2)*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-a*d*e/(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)))))+e^2/d^3*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/2*(a*e^2+c*d
^2)*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^
2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e
^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*
d*e)^(1/2)))+a*d*e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/2*(a*e^2+c*d^2)*(1/4*(2*c*d*e*x+a*e^2+c*d^2)
/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d
^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))+a*d*e*((a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)+1/2*(a*e^2+c*d^2)*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2))/(c*d*e)^(1/2)-a*d*e/(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))))-e/d^2*(-1/a/d/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)+5/2*(a*e^2+c*d^2)/a/d/
e*(1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/2*(a*e^2+c*d^2)*(1/8*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a
*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*
x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)))+a*d*e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)+1/2*(a*e^2+c*d^2)*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/
8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2))/(c*d*e)^(1/2))+a*d*e*((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2*(a*e^2+c*d^2)*ln((1/2*e^2*
a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-a*d*e/(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))))+6*c/a*(1/12*(2*c*d*
e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/d/e*(1/8
*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/16*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/c/
d/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)/c/d/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2
)^2)/c/d/e*ln((1/2*e^2*a+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/
2)))))-e^2/d^3*(1/5*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+1/2*(a*e^2-c*d^2)*(1/8*(2*c*d*e*(x+d/e)+e^2*
a-c*d^2)/c/d/e*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/16*(a*e^2-c*d^2)^2/c/d/e*(1/4*(2*c*d*e*(x+d/e)+
e^2*a-c*d^2)/c/d/e*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/c/d/e*ln((1/2*e^2*a-1/2*c
*d^2+c*d*e*(x+d/e))/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 4.42 (sec) , antiderivative size = 1569, normalized size of antiderivative = 4.63 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\text {Too large to display} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d),x, algorithm="fricas")

[Out]

[1/16*(3*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(c*d/e)*x^2*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^
2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c
^2*d^3*e + a*c*d*e^3)*x) + 3*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*e/d)*x^2*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^2*e + (c*d^3 + a*d*e
^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c
*d*e^2)*x^2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^2, -1/16*(6*(c^2*d^4
+ 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(-c*d/e)*x^2*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*
e*x + c*d^2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) - 3*(5*c^2*d^4 + 10*a
*c*d^2*e^2 + a^2*e^4)*sqrt(a*e/d)*x^2*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^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)
*x)/x^2) - 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)*x^2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqr
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^2, 1/16*(6*(5*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(-a*e/d)*x^2
*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*e/d)/(a*c*d*e^2*
x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)*x)) + 3*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(c*d/e)*x^2*log(8*c
^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e
+ (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2*c^2*d^2*e*x^3 - 2*a^2*d*e^2 + (5*c^2*d^
3 + 9*a*c*d*e^2)*x^2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/x^2, -1/8*(3*
(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*sqrt(-c*d/e)*x^2*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d/e)/(c^2*d^2*e*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) - 3*(5*c^2*d^
4 + 10*a*c*d^2*e^2 + a^2*e^4)*sqrt(-a*e/d)*x^2*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*e/d)/(a*c*d*e^2*x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)*x)) - 2*(2*c^2*d^2*e*x^3
- 2*a^2*d*e^2 + (5*c^2*d^3 + 9*a*c*d*e^2)*x^2 - (9*a*c*d^2*e + 5*a^2*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x))/x^2]

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\text {Timed out} \]

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**3/(e*x+d),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{3}} \,d x } \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(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^3), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (295) = 590\).

Time = 0.46 (sec) , antiderivative size = 746, normalized size of antiderivative = 2.20 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\frac {1}{4} \, {\left (2 \, c^{2} d^{2} e x + \frac {5 \, c^{3} d^{4} e + 9 \, a c^{2} d^{2} e^{3}}{c d e}\right )} \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} + \frac {3 \, {\left (5 \, a c^{2} d^{4} e + 10 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{4 \, \sqrt {-a d e}} - \frac {3 \, {\left (c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{8 \, \sqrt {c d e}} - \frac {7 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{2} c^{2} d^{5} e^{2} + 6 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{3} c d^{3} e^{4} + 3 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a^{4} d e^{6} + 16 \, \sqrt {c d e} a^{3} c d^{4} e^{3} + 8 \, \sqrt {c d e} a^{4} d^{2} e^{5} - 9 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a c^{2} d^{4} e - 18 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{2} c d^{2} e^{3} - 5 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{3} a^{3} e^{5} - 24 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{2} c d^{3} e^{2} - 16 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2} a^{3} d e^{4}}{4 \, {\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )}^{2}} \]

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d),x, algorithm="giac")

[Out]

1/4*(2*c^2*d^2*e*x + (5*c^3*d^4*e + 9*a*c^2*d^2*e^3)/(c*d*e))*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e) + 3/
4*(5*a*c^2*d^4*e + 10*a^2*c*d^2*e^3 + a^3*e^5)*arctan(-(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) - 3/8*(c^3*d^5 + 10*a*c^2*d^3*e^2 + 5*a^2*c*d*e^4)*log(abs(-c*d^2 - a*e^2 -
2*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))))/sqrt(c*d*e) - 1/4*(7*(sqrt(c*d*e
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^2*c^2*d^5*e^2 + 6*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2
*x + a*e^2*x + a*d*e))*a^3*c*d^3*e^4 + 3*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*d*e
^6 + 16*sqrt(c*d*e)*a^3*c*d^4*e^3 + 8*sqrt(c*d*e)*a^4*d^2*e^5 - 9*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x +
a*e^2*x + a*d*e))^3*a*c^2*d^4*e - 18*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^2*c*d^2
*e^3 - 5*(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*e^5 - 24*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^2*c*d^3*e^2 - 16*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^3*d*e^4)/(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)^2

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^3\,\left (d+e\,x\right )} \,d x \]

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)),x)

[Out]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)), x)