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

Optimal. Leaf size=404 \[ -\frac {\left (x \left (-3 a^3 e^6+11 a^2 c d^2 e^4+83 a c^2 d^4 e^2+5 c^3 d^6\right )+2 a d e \left (5 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 a d^2 e x^2}+\frac {\left (-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac {\left (x \left (3 a e^2+11 c d^2\right )+6 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 d x^4} \]

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Rubi [A]  time = 0.46, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {849, 810, 843, 621, 206, 724} \begin {gather*} -\frac {\left (x \left (11 a^2 c d^2 e^4-3 a^3 e^6+83 a c^2 d^4 e^2+5 c^3 d^6\right )+2 a d e \left (5 c d^2-a e^2\right ) \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}}{64 a d^2 e x^2}+\frac {\left (-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \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 )}{128 a^{3/2} d^{5/2} e^{3/2}}+c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\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 )-\frac {\left (x \left (3 a e^2+11 c d^2\right )+6 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}}{24 d x^4} \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^5*(d + e*x)),x]

[Out]

-((2*a*d*e*(5*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2) + (5*c^3*d^6 + 83*a*c^2*d^4*e^2 + 11*a^2*c*d^2*e^4 - 3*a^3*e^6)
*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(64*a*d^2*e*x^2) - ((6*a*d*e + (11*c*d^2 + 3*a*e^2)*x)*(a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(24*d*x^4) + c^(5/2)*d^(5/2)*e^(3/2)*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])] + ((5*c^4*d^8 - 60*a*c^3*d^6*e^2
- 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*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])])/(128*a^(3/2)*d^(5/2)*e^(3/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 810

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

Rule 843

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 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^5 (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^5} \, dx\\ &=-\frac {\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}-\frac {\int \frac {\left (-\frac {1}{2} a e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )-8 a c^2 d^3 e^2 x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^3} \, dx}{8 a d e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\frac {\int \frac {-\frac {1}{4} a e \left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right )+32 a^2 c^3 d^5 e^4 x}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{32 a^2 d^2 e^2}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\left (c^3 d^3 e^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx-\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \int \frac {1}{x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 a d^2 e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+\left (2 c^3 d^3 e^2\right ) \operatorname {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 {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\right ) \operatorname {Subst}\left (\int \frac {1}{4 a d e-x^2} \, dx,x,\frac {2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{64 a d^2 e}\\ &=-\frac {\left (2 a d e \left (5 c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (5 c^3 d^6+83 a c^2 d^4 e^2+11 a^2 c d^2 e^4-3 a^3 e^6\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 a d^2 e x^2}-\frac {\left (6 a d e+\left (11 c d^2+3 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}}{24 d x^4}+c^{5/2} d^{5/2} e^{3/2} \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 )+\frac {\left (5 c^4 d^8-60 a c^3 d^6 e^2-90 a^2 c^2 d^4 e^4+20 a^3 c d^2 e^6-3 a^4 e^8\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 )}{128 a^{3/2} d^{5/2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 3.48, size = 404, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a e+c d x} \left (-\frac {\sqrt {d} \sqrt {e} (d+e x) \sqrt {a e+c d x} \left (3 a^3 e^3 \left (16 d^3+24 d^2 e x+2 d e^2 x^2-3 e^3 x^3\right )+a^2 c d^2 e^2 x \left (136 d^2+244 d e x+57 e^2 x^2\right )+a c^2 d^4 e x^2 (118 d+337 e x)+15 c^3 d^6 x^3\right )}{a x^4}+\frac {3 \sqrt {d+e x} \left (-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{a^{3/2}}+384 c^{3/2} d^4 e^3 \sqrt {c d} \sqrt {c d^2-a e^2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )\right )}{192 d^{5/2} e^{3/2} \sqrt {(d+e x) (a e+c d x)}} \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^5*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*(-((Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)*(15*c^3*d^6*x^3 + a*c^2*d^4*e*x^2*(118*d +
337*e*x) + a^2*c*d^2*e^2*x*(136*d^2 + 244*d*e*x + 57*e^2*x^2) + 3*a^3*e^3*(16*d^3 + 24*d^2*e*x + 2*d*e^2*x^2 -
3*e^3*x^3)))/(a*x^4)) + 384*c^(3/2)*d^4*Sqrt[c*d]*e^3*Sqrt[c*d^2 - a*e^2]*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2
)]*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])] + (3*(5*c^4*d^8 - 60*a
*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*
d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/a^(3/2)))/(192*d^(5/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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IntegrateAlgebraic [A]  time = 2.57, size = 544, normalized size = 1.35 \begin {gather*} -\frac {1}{2} c^2 d^2 e \sqrt {c d e} \log \left (a^2 e^4+8 c d e x \sqrt {c d e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-2 a c d^2 e^2-4 a c d e^3 x+c^2 d^4-4 c^2 d^3 e x-8 c^2 d^2 e^2 x^2\right )+\frac {\sqrt {a d e+a e^2 x+c d^2 x+c d e x^2} \left (-48 a^3 d^3 e^3-72 a^3 d^2 e^4 x-6 a^3 d e^5 x^2+9 a^3 e^6 x^3-136 a^2 c d^4 e^2 x-244 a^2 c d^3 e^3 x^2-57 a^2 c d^2 e^4 x^3-118 a c^2 d^5 e x^2-337 a c^2 d^4 e^2 x^3-15 c^3 d^6 x^3\right )}{192 a d^2 e x^4}+\frac {\left (-3 a^4 e^8+20 a^3 c d^2 e^6-90 a^2 c^2 d^4 e^4-60 a c^3 d^6 e^2+5 c^4 d^8\right ) \tanh ^{-1}\left (\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}-x \sqrt {c d e}}{\sqrt {a} \sqrt {d} \sqrt {e}}\right )}{64 a^{3/2} d^{5/2} e^{3/2}}-c^{5/2} d^{5/2} e^{3/2} \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} x \sqrt {c d e}}{a e^2+c d^2}-\frac {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}}{a e^2+c d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*d*e + c*d^2*x + a*e^2*x + c*d*e*x^2]*(-48*a^3*d^3*e^3 - 136*a^2*c*d^4*e^2*x - 72*a^3*d^2*e^4*x - 118*a
*c^2*d^5*e*x^2 - 244*a^2*c*d^3*e^3*x^2 - 6*a^3*d*e^5*x^2 - 15*c^3*d^6*x^3 - 337*a*c^2*d^4*e^2*x^3 - 57*a^2*c*d
^2*e^4*x^3 + 9*a^3*e^6*x^3))/(192*a*d^2*e*x^4) + ((5*c^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*
c*d^2*e^6 - 3*a^4*e^8)*ArcTanh[(-(Sqrt[c*d*e]*x) + Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[a]*Sqrt[
d]*Sqrt[e])])/(64*a^(3/2)*d^(5/2)*e^(3/2)) - c^(5/2)*d^(5/2)*e^(3/2)*ArcTanh[(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[c
*d*e]*x)/(c*d^2 + a*e^2) - (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 + a*
e^2)] - (c^2*d^2*e*Sqrt[c*d*e]*Log[c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4 - 4*c^2*d^3*e*x - 4*a*c*d*e^3*x - 8*c^2*d
^2*e^2*x^2 + 8*c*d*e*Sqrt[c*d*e]*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]])/2

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fricas [A]  time = 24.35, size = 1917, normalized size = 4.75

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(384*sqrt(c*d*e)*a^2*c^2*d^5*e^3*x^4*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*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) + 8*(c^2*d^3*e + a*c*d*e^3)*x)
- 3*(5*c^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*sqrt(a*d*e)*x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e +
337*a^2*c^2*d^5*e^3 + 57*a^3*c*d^3*e^5 - 9*a^4*d*e^7)*x^3 + 2*(59*a^2*c^2*d^6*e^2 + 122*a^3*c*d^4*e^4 + 3*a^4
*d^2*e^6)*x^2 + 8*(17*a^3*c*d^5*e^3 + 9*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d^3*
e^2*x^4), -1/768*(768*sqrt(-c*d*e)*a^2*c^2*d^5*e^3*x^4*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^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 3*(5*c
^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*sqrt(a*d*e)*x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 337*a^2*
c^2*d^5*e^3 + 57*a^3*c*d^3*e^5 - 9*a^4*d*e^7)*x^3 + 2*(59*a^2*c^2*d^6*e^2 + 122*a^3*c*d^4*e^4 + 3*a^4*d^2*e^6)
*x^2 + 8*(17*a^3*c*d^5*e^3 + 9*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d^3*e^2*x^4),
1/384*(192*sqrt(c*d*e)*a^2*c^2*d^5*e^3*x^4*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*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) + 8*(c^2*d^3*e + a*c*d*e^3)*x)
- 3*(5*c^4*d^8 - 60*a*c^3*d^6*e^2 - 90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*sqrt(-a*d*e)*x^4*arcta
n(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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 337*a^2*c^2*d^5*e^3 + 57*a
^3*c*d^3*e^5 - 9*a^4*d*e^7)*x^3 + 2*(59*a^2*c^2*d^6*e^2 + 122*a^3*c*d^4*e^4 + 3*a^4*d^2*e^6)*x^2 + 8*(17*a^3*c
*d^5*e^3 + 9*a^4*d^3*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d^3*e^2*x^4), -1/384*(384*sqrt(
-c*d*e)*a^2*c^2*d^5*e^3*x^4*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^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 3*(5*c^4*d^8 - 60*a*c^3*d^6*e^2 -
90*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 3*a^4*e^8)*sqrt(-a*d*e)*x^4*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*(48*a^4*d^4*e^4 + (15*a*c^3*d^7*e + 337*a^2*c^2*d^5*e^3 + 57*a^3*c*d^3*e^5 - 9*a^4*d*e^7)*x^3 + 2*
(59*a^2*c^2*d^6*e^2 + 122*a^3*c*d^4*e^4 + 3*a^4*d^2*e^6)*x^2 + 8*(17*a^3*c*d^5*e^3 + 9*a^4*d^3*e^5)*x)*sqrt(c*
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*d^3*e^2*x^4)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^5/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 1.26Error: Bad Argument Type

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maple [B]  time = 0.03, size = 3646, normalized size = 9.02 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/d^2*e^3*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-3/128*d*e^2*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x
+d/e))^(1/2)-3/64*e^3*c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+25/32*d^3/a*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(1/2)*c^3+3/8/d^3/a/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)+1/96/d/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(5/2)*c^2+35/96*e/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^2-1/8*e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(1/2)*x*c^2+127/128*d*e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^2+3/64/d^3*e^6*(c*d*e*x^2+a*d*e+(a*e^
2+c*d^2)*x)^(1/2)*a^2+1/64/d^4*e^5*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-19/96/d^2*e^3*(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(3/2)*c-1/5/d^5*e^4*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(5/2)-61/320/d^5*e^4*(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(5/2)+15/256/d^5*e^10*a^4/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^
2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/64/d^6*e^9*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))
^(1/2)*x+9/64/d^2*e^5*a*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-3/256/d^7*e^12*a^5/c^2*ln((1/2*a*e^2
-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+15/128/d*
e^6*a^2*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/
(c*d*e)^(1/2)-15/256*d*e^4*a*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-
c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-45/64*d^2*e^3*a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^2-31/64/d^2*e/a^2*c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x
-35/192/d*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c^2-3/32/d^2*e^5*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
1/2)*x*a*c-15/256/d^5*e^10/c*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(1/2))/(c*d*e)^(1/2)*a^4-15/128/d*e^6*c*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2
+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^2+15/256*d*e^4*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d
*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a*c^2-25/64/d^4*e^3*c/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x-5/6
4*d^4/a^2/e*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*c^4+5/128*d^6/a/e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2
)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c^4-5/192*d^3/a^3/e^2*(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(3/2)*x*c^4-1/64*d^2/a^4/e^3*c^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*x+1/64*d/a^4/e^4/x*(c*d*e*x
^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^3+1/24/d/a^2/e^2/x^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c-3/64/d^6*e^9*
a^3/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+3/256/d^7*e^12*a^5/c^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e
)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+45/64*e*d^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(1/2)*x*c^3-13/48/e/d^2/a^2/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c-43/192/e^2/d/a^3/x*(c*d*e*x^2+a*d*e
+(a*e^2+c*d^2)*x)^(7/2)*c^2-5/192*d^4/a^3/e^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^4-1/64*d^3/a^4/e^4*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c^4-5/64*d^5/a^2/e^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*c^4+1/96/a^3
/e^3/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c^2+5/32*e^5*a^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*
(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)*c+9/64/d^4*e^7*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/
2)*x*a^2+25/64/d^5*e^2/a/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)-7/64/d^3*e^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(3/2)*x*c+15/128/d^3*e^8*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^
(1/2))/(c*d*e)^(1/2)*a^3-3/128/d^2*e^7*a^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^
2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/x)-15/32/d^3*e^2/a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2)*c-17/64/d*e^4*c*(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a+253/256*d^3*e^2*c^3*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d
*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+3/64/d^5*e^8/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+
1/8/d^5*e^6*a*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x+1/16/d^6*e^7*a^2/c*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
3/2)-3/128/d^7*e^10*a^4/c^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+1/8/d^3*e^4*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^
2)*(x+d/e))^(3/2)*x-1/16/d^6*e^7*a^2/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)-9/64/d^4*e^7*a^2*((x+d/e)
^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+3/128/d^7*e^10*a^4/c^2*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3
/64/d*e^4*a*c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)-15/128/d^3*e^8*a^3*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)
*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/256*d^3*e^2*c^3*ln((1/2*a
*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+31/64
/d^3/a^2/x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)*c-13/32/d^4*e/a/x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)
-15/32*e*d^4/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
)/x)*c^3+35/96/e*d^2/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*c^3+19/96/e^2*d/a^3*(c*d*e*x^2+a*d*e+(a*e^2+c
*d^2)*x)^(5/2)*c^3+85/192*d/a^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*c^3+43/192/e/a^3*c^3*(c*d*e*x^2+a*d*
e+(a*e^2+c*d^2)*x)^(5/2)*x-1/4/d^2/a/e/x^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(7/2)-1/8/d^5*e^6*a*((x+d/e)^2*c*
d*e+(a*e^2-c*d^2)*(x+d/e))^(3/2)*x-3/64/d^5*e^8*a^3/c*((x+d/e)^2*c*d*e+(a*e^2-c*d^2)*(x+d/e))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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