3.29.39 \(\int \frac {-1+2 x-2 x^2-3 x^3+e^{2 x} (25 x+50 x^2)}{x} \, dx\)

Optimal. Leaf size=26 \[ -30+2 x-x^3-x \left (-25 e^{2 x}+x\right )-\log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.58, number of steps used = 6, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {14, 2176, 2194} \begin {gather*} -x^3-x^2+2 x-\frac {25 e^{2 x}}{2}+\frac {25}{2} e^{2 x} (2 x+1)-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x - 2*x^2 - 3*x^3 + E^(2*x)*(25*x + 50*x^2))/x,x]

[Out]

(-25*E^(2*x))/2 + 2*x - x^2 - x^3 + (25*E^(2*x)*(1 + 2*x))/2 - Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (25 e^{2 x} (1+2 x)+\frac {-1+2 x-2 x^2-3 x^3}{x}\right ) \, dx\\ &=25 \int e^{2 x} (1+2 x) \, dx+\int \frac {-1+2 x-2 x^2-3 x^3}{x} \, dx\\ &=\frac {25}{2} e^{2 x} (1+2 x)-25 \int e^{2 x} \, dx+\int \left (2-\frac {1}{x}-2 x-3 x^2\right ) \, dx\\ &=-\frac {25 e^{2 x}}{2}+2 x-x^2-x^3+\frac {25}{2} e^{2 x} (1+2 x)-\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 1.00 \begin {gather*} 2 x+25 e^{2 x} x-x^2-x^3-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x - 2*x^2 - 3*x^3 + E^(2*x)*(25*x + 50*x^2))/x,x]

[Out]

2*x + 25*E^(2*x)*x - x^2 - x^3 - Log[x]

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fricas [A]  time = 0.90, size = 25, normalized size = 0.96 \begin {gather*} -x^{3} - x^{2} + 25 \, x e^{\left (2 \, x\right )} + 2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^2+25*x)*exp(x)^2-3*x^3-2*x^2+2*x-1)/x,x, algorithm="fricas")

[Out]

-x^3 - x^2 + 25*x*e^(2*x) + 2*x - log(x)

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giac [A]  time = 0.25, size = 25, normalized size = 0.96 \begin {gather*} -x^{3} - x^{2} + 25 \, x e^{\left (2 \, x\right )} + 2 \, x - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^2+25*x)*exp(x)^2-3*x^3-2*x^2+2*x-1)/x,x, algorithm="giac")

[Out]

-x^3 - x^2 + 25*x*e^(2*x) + 2*x - log(x)

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maple [A]  time = 0.03, size = 26, normalized size = 1.00

method	result	size
default	\(-x^{2}+2 x -\ln \relax (x )-x^{3}+25 x \,{\mathrm e}^{2 x}\)	\(26\)
norman	\(-x^{2}+2 x -\ln \relax (x )-x^{3}+25 x \,{\mathrm e}^{2 x}\)	\(26\)
risch	\(-x^{2}+2 x -\ln \relax (x )-x^{3}+25 x \,{\mathrm e}^{2 x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((50*x^2+25*x)*exp(x)^2-3*x^3-2*x^2+2*x-1)/x,x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x-ln(x)-x^3+25*x*exp(x)^2

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maxima [A]  time = 0.82, size = 35, normalized size = 1.35 \begin {gather*} -x^{3} - x^{2} + \frac {25}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 2 \, x + \frac {25}{2} \, e^{\left (2 \, x\right )} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x^2+25*x)*exp(x)^2-3*x^3-2*x^2+2*x-1)/x,x, algorithm="maxima")

[Out]

-x^3 - x^2 + 25/2*(2*x - 1)*e^(2*x) + 2*x + 25/2*e^(2*x) - log(x)

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mupad [B]  time = 0.05, size = 25, normalized size = 0.96 \begin {gather*} 2\,x-\ln \relax (x)+25\,x\,{\mathrm {e}}^{2\,x}-x^2-x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - exp(2*x)*(25*x + 50*x^2) - 2*x + 3*x^3 + 1)/x,x)

[Out]

2*x - log(x) + 25*x*exp(2*x) - x^2 - x^3

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sympy [A]  time = 0.11, size = 20, normalized size = 0.77 \begin {gather*} - x^{3} - x^{2} + 25 x e^{2 x} + 2 x - \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((50*x**2+25*x)*exp(x)**2-3*x**3-2*x**2+2*x-1)/x,x)

[Out]

-x**3 - x**2 + 25*x*exp(2*x) + 2*x - log(x)

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