3.54.97 \(\int \frac {-300+200 x+400 x^3+e^5 (12 x^2-4 x^3+8 x^5)+e^5 (-6 x^2+4 x^3+8 x^5) \log (\frac {-3+2 x+4 x^3}{4 x^2})}{e^5 (-3 x^2+2 x^3+4 x^5)} \, dx\)

Optimal. Leaf size=27 \[ -\frac {100}{e^5 x}+2 x \log \left (x+\frac {-3+2 x}{4 x^2}\right ) \]

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Rubi [A]  time = 4.44, antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 23, number of rules used = 10, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 1594, 6742, 2079, 800, 634, 618, 204, 628, 2523} \begin {gather*} 2 x \log \left (-\frac {-4 x^3-2 x+3}{4 x^2}\right )-\frac {100}{e^5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-300 + 200*x + 400*x^3 + E^5*(12*x^2 - 4*x^3 + 8*x^5) + E^5*(-6*x^2 + 4*x^3 + 8*x^5)*Log[(-3 + 2*x + 4*x^
3)/(4*x^2)])/(E^5*(-3*x^2 + 2*x^3 + 4*x^5)),x]

[Out]

-100/(E^5*x) + 2*x*Log[-1/4*(3 - 2*x - 4*x^3)/x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

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

Rule 618

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), 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] && IntegerQ[m]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/
3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2523

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[(x*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-300+200 x+400 x^3+e^5 \left (12 x^2-4 x^3+8 x^5\right )+e^5 \left (-6 x^2+4 x^3+8 x^5\right ) \log \left (\frac {-3+2 x+4 x^3}{4 x^2}\right )}{-3 x^2+2 x^3+4 x^5} \, dx}{e^5}\\ &=\frac {\int \frac {-300+200 x+400 x^3+e^5 \left (12 x^2-4 x^3+8 x^5\right )+e^5 \left (-6 x^2+4 x^3+8 x^5\right ) \log \left (\frac {-3+2 x+4 x^3}{4 x^2}\right )}{x^2 \left (-3+2 x+4 x^3\right )} \, dx}{e^5}\\ &=\frac {\int \left (\frac {4 \left (75-50 x-3 e^5 x^2-100 \left (1-\frac {e^5}{100}\right ) x^3-2 e^5 x^5\right )}{x^2 \left (3-2 x-4 x^3\right )}+2 e^5 \log \left (\frac {-3+2 x+4 x^3}{4 x^2}\right )\right ) \, dx}{e^5}\\ &=2 \int \log \left (\frac {-3+2 x+4 x^3}{4 x^2}\right ) \, dx+\frac {4 \int \frac {75-50 x-3 e^5 x^2-100 \left (1-\frac {e^5}{100}\right ) x^3-2 e^5 x^5}{x^2 \left (3-2 x-4 x^3\right )} \, dx}{e^5}\\ &=2 x \log \left (-\frac {3-2 x-4 x^3}{4 x^2}\right )-2 \int \frac {-6+2 x-4 x^3}{3-2 x-4 x^3} \, dx+\frac {4 \int \left (\frac {e^5}{2}+\frac {25}{x^2}-\frac {e^5 (-9+4 x)}{2 \left (-3+2 x+4 x^3\right )}\right ) \, dx}{e^5}\\ &=-\frac {100}{e^5 x}+2 x+2 x \log \left (-\frac {3-2 x-4 x^3}{4 x^2}\right )-2 \int \frac {-9+4 x}{-3+2 x+4 x^3} \, dx-2 \int \left (1-\frac {9-4 x}{3-2 x-4 x^3}\right ) \, dx\\ &=-\frac {100}{e^5 x}+2 x \log \left (-\frac {3-2 x-4 x^3}{4 x^2}\right )+2 \int \frac {9-4 x}{3-2 x-4 x^3} \, dx-32 \int \frac {-9+4 x}{\left (\frac {2}{3} \left (\frac {2\ 3^{2/3}}{\sqrt [3]{27+\sqrt {753}}}-\sqrt [3]{3 \left (27+\sqrt {753}\right )}\right )+4 x\right ) \left (\frac {4}{9} \left (6+\frac {12 \sqrt [3]{3}}{\left (27+\sqrt {753}\right )^{2/3}}+\left (3 \left (27+\sqrt {753}\right )\right )^{2/3}\right )-\frac {8 \left (2 \sqrt [3]{\frac {3}{27+\sqrt {753}}}-\sqrt [3]{27+\sqrt {753}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx\\ &=-\frac {100}{e^5 x}+2 x \log \left (-\frac {3-2 x-4 x^3}{4 x^2}\right )+32 \int \frac {9-4 x}{\left (-\frac {2}{3} \left (\frac {2\ 3^{2/3}}{\sqrt [3]{27+\sqrt {753}}}-\sqrt [3]{3 \left (27+\sqrt {753}\right )}\right )-4 x\right ) \left (\frac {4}{9} \left (6+\frac {12 \sqrt [3]{3}}{\left (27+\sqrt {753}\right )^{2/3}}+\left (3 \left (27+\sqrt {753}\right )\right )^{2/3}\right )-\frac {8 \left (2 \sqrt [3]{\frac {3}{27+\sqrt {753}}}-\sqrt [3]{27+\sqrt {753}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx-32 \int \left (\frac {\left (27+\sqrt {753}\right )^{2/3} \left (-4 3^{2/3}-27 \sqrt [3]{27+\sqrt {753}}+2 \sqrt [3]{3} \left (27+\sqrt {753}\right )^{2/3}\right )}{8 \left (4 \sqrt [3]{3}+\sqrt [6]{3} \left (9 \sqrt {3}+\sqrt {251}\right ) \sqrt [3]{27+\sqrt {753}}-2 \left (27+\sqrt {753}\right )^{2/3}\right ) \left (2\ 3^{2/3}-\sqrt [3]{3} \left (27+\sqrt {753}\right )^{2/3}+6 \sqrt [3]{27+\sqrt {753}} x\right )}+\frac {\left (27+\sqrt {753}\right )^{2/3} \left (-\left (\left (9\ 3^{2/3}-\sqrt [6]{3} \sqrt {251}\right ) \sqrt [3]{27+\sqrt {753}}\right )+2 \left (27+\sqrt {753}\right )^{2/3}+\sqrt [3]{3} \left (247+9 \sqrt {753}\right )+\sqrt [3]{27+\sqrt {753}} \left (4\ 3^{2/3}+27 \sqrt [3]{27+\sqrt {753}}-2 \sqrt [3]{3} \left (27+\sqrt {753}\right )^{2/3}\right ) x\right )}{4 \left (4 \sqrt [3]{3}+\sqrt [6]{3} \left (9 \sqrt {3}+\sqrt {251}\right ) \sqrt [3]{27+\sqrt {753}}-2 \left (27+\sqrt {753}\right )^{2/3}\right ) \left (4 \sqrt [3]{3}+\sqrt [6]{3} \left (9 \sqrt {3}+\sqrt {251}\right ) \sqrt [3]{27+\sqrt {753}}+2 \left (27+\sqrt {753}\right )^{2/3}+2\ 3^{2/3} \left (9\ 3^{2/3}+\sqrt [6]{3} \sqrt {251}-2 \sqrt [3]{27+\sqrt {753}}\right ) x+12 \left (27+\sqrt {753}\right )^{2/3} x^2\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 30, normalized size = 1.11 \begin {gather*} -\frac {100}{e^5 x}+2 x \log \left (\frac {-3+2 x+4 x^3}{4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-300 + 200*x + 400*x^3 + E^5*(12*x^2 - 4*x^3 + 8*x^5) + E^5*(-6*x^2 + 4*x^3 + 8*x^5)*Log[(-3 + 2*x
+ 4*x^3)/(4*x^2)])/(E^5*(-3*x^2 + 2*x^3 + 4*x^5)),x]

[Out]

-100/(E^5*x) + 2*x*Log[(-3 + 2*x + 4*x^3)/(4*x^2)]

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fricas [A]  time = 0.62, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (x^{2} e^{5} \log \left (\frac {4 \, x^{3} + 2 \, x - 3}{4 \, x^{2}}\right ) - 50\right )} e^{\left (-5\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+4*x^3-6*x^2)*exp(5)*log(1/4*(4*x^3+2*x-3)/x^2)+(8*x^5-4*x^3+12*x^2)*exp(5)+400*x^3+200*x-300
)/(4*x^5+2*x^3-3*x^2)/exp(5),x, algorithm="fricas")

[Out]

2*(x^2*e^5*log(1/4*(4*x^3 + 2*x - 3)/x^2) - 50)*e^(-5)/x

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giac [A]  time = 0.16, size = 31, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (x^{2} e^{5} \log \left (\frac {4 \, x^{3} + 2 \, x - 3}{4 \, x^{2}}\right ) - 50\right )} e^{\left (-5\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+4*x^3-6*x^2)*exp(5)*log(1/4*(4*x^3+2*x-3)/x^2)+(8*x^5-4*x^3+12*x^2)*exp(5)+400*x^3+200*x-300
)/(4*x^5+2*x^3-3*x^2)/exp(5),x, algorithm="giac")

[Out]

2*(x^2*e^5*log(1/4*(4*x^3 + 2*x - 3)/x^2) - 50)*e^(-5)/x

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maple [A]  time = 0.15, size = 28, normalized size = 1.04

method	result	size
risch	\(2 x \ln \left (\frac {4 x^{3}+2 x -3}{4 x^{2}}\right )-\frac {100 \,{\mathrm e}^{-5}}{x}\)	\(28\)
norman	\(\frac {-100 \,{\mathrm e}^{-5}+2 x^{2} \ln \left (\frac {4 x^{3}+2 x -3}{4 x^{2}}\right )}{x}\)	\(33\)
default	\({\mathrm e}^{-5} \left (-4 x \,{\mathrm e}^{5} \ln \relax (2)+2 \,{\mathrm e}^{5} x \ln \left (\frac {4 x^{3}+2 x -3}{x^{2}}\right )-{\mathrm e}^{5} \left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{3}+2 \textit {\_Z} -3\right )}{\sum }\frac {\left (-4 \textit {\_R} +9\right ) \ln \left (x -\textit {\_R} \right )}{6 \textit {\_R}^{2}+1}\right )-\frac {100}{x}-\left (\munderset {\textit {\_R} =\RootOf \left (4 \textit {\_Z}^{3}+2 \textit {\_Z} -3\right )}{\sum }\frac {\left (4 \textit {\_R} -9\right ) \ln \left (x -\textit {\_R} \right )}{6 \textit {\_R}^{2}+1}\right ) {\mathrm e}^{5}\right )\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^5+4*x^3-6*x^2)*exp(5)*ln(1/4*(4*x^3+2*x-3)/x^2)+(8*x^5-4*x^3+12*x^2)*exp(5)+400*x^3+200*x-300)/(4*x^
5+2*x^3-3*x^2)/exp(5),x,method=_RETURNVERBOSE)

[Out]

2*x*ln(1/4*(4*x^3+2*x-3)/x^2)-100/x*exp(-5)

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maxima [A]  time = 0.47, size = 45, normalized size = 1.67 \begin {gather*} -\frac {2 \, {\left (2 \, x^{2} e^{5} \log \relax (2) - x^{2} e^{5} \log \left (4 \, x^{3} + 2 \, x - 3\right ) + 2 \, x^{2} e^{5} \log \relax (x) + 50\right )} e^{\left (-5\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^5+4*x^3-6*x^2)*exp(5)*log(1/4*(4*x^3+2*x-3)/x^2)+(8*x^5-4*x^3+12*x^2)*exp(5)+400*x^3+200*x-300
)/(4*x^5+2*x^3-3*x^2)/exp(5),x, algorithm="maxima")

[Out]

-2*(2*x^2*e^5*log(2) - x^2*e^5*log(4*x^3 + 2*x - 3) + 2*x^2*e^5*log(x) + 50)*e^(-5)/x

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mupad [B]  time = 3.79, size = 24, normalized size = 0.89 \begin {gather*} 2\,x\,\ln \left (\frac {x^3+\frac {x}{2}-\frac {3}{4}}{x^2}\right )-\frac {100\,{\mathrm {e}}^{-5}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-5)*(200*x + exp(5)*(12*x^2 - 4*x^3 + 8*x^5) + 400*x^3 + exp(5)*log((x/2 + x^3 - 3/4)/x^2)*(4*x^3 - 6
*x^2 + 8*x^5) - 300))/(2*x^3 - 3*x^2 + 4*x^5),x)

[Out]

2*x*log((x/2 + x^3 - 3/4)/x^2) - (100*exp(-5))/x

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sympy [A]  time = 0.20, size = 24, normalized size = 0.89 \begin {gather*} 2 x \log {\left (\frac {x^{3} + \frac {x}{2} - \frac {3}{4}}{x^{2}} \right )} - \frac {100}{x e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**5+4*x**3-6*x**2)*exp(5)*ln(1/4*(4*x**3+2*x-3)/x**2)+(8*x**5-4*x**3+12*x**2)*exp(5)+400*x**3+2
00*x-300)/(4*x**5+2*x**3-3*x**2)/exp(5),x)

[Out]

2*x*log((x**3 + x/2 - 3/4)/x**2) - 100*exp(-5)/x

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