3.21.90 \(\int \frac {21+e^x (7 x+7 x^2+(4+4 x) \log (2))}{7 x+4 \log (2)} \, dx\)

Optimal. Leaf size=22 \[ 3+e^x x+3 \log (2 (x-4 (2 x+\log (2)))) \]

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Rubi [A]  time = 0.10, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6688, 2176, 2194} \begin {gather*} e^x (x+1)-e^x+3 \log (7 x+\log (16)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(21 + E^x*(7*x + 7*x^2 + (4 + 4*x)*Log[2]))/(7*x + 4*Log[2]),x]

[Out]

-E^x + E^x*(1 + x) + 3*Log[7*x + Log[16]]

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]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x (1+x)+\frac {21}{7 x+\log (16)}\right ) \, dx\\ &=3 \log (7 x+\log (16))+\int e^x (1+x) \, dx\\ &=e^x (1+x)+3 \log (7 x+\log (16))-\int e^x \, dx\\ &=-e^x+e^x (1+x)+3 \log (7 x+\log (16))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.68 \begin {gather*} e^x x+3 \log (7 x+\log (16)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(21 + E^x*(7*x + 7*x^2 + (4 + 4*x)*Log[2]))/(7*x + 4*Log[2]),x]

[Out]

E^x*x + 3*Log[7*x + Log[16]]

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fricas [A]  time = 1.00, size = 16, normalized size = 0.73 \begin {gather*} x e^{x} + 3 \, \log \left (7 \, x + 4 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*log(2)+7*x^2+7*x)*exp(x)+21)/(4*log(2)+7*x),x, algorithm="fricas")

[Out]

x*e^x + 3*log(7*x + 4*log(2))

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giac [A]  time = 0.14, size = 16, normalized size = 0.73 \begin {gather*} x e^{x} + 3 \, \log \left (7 \, x + 4 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*log(2)+7*x^2+7*x)*exp(x)+21)/(4*log(2)+7*x),x, algorithm="giac")

[Out]

x*e^x + 3*log(7*x + 4*log(2))

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maple [A]  time = 0.36, size = 17, normalized size = 0.77

method	result	size
default	\(3 \ln \left (4 \ln \relax (2)+7 x \right )+{\mathrm e}^{x} x\)	\(17\)
norman	\(3 \ln \left (4 \ln \relax (2)+7 x \right )+{\mathrm e}^{x} x\)	\(17\)
risch	\(3 \ln \left (4 \ln \relax (2)+7 x \right )+{\mathrm e}^{x} x\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x+4)*ln(2)+7*x^2+7*x)*exp(x)+21)/(4*ln(2)+7*x),x,method=_RETURNVERBOSE)

[Out]

3*ln(4*ln(2)+7*x)+exp(x)*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2}{7} \cdot 2^{\frac {3}{7}} E_{1}\left (-x - \frac {4}{7} \, \log \relax (2)\right ) \log \relax (2) + x e^{x} - 4 \, \int \frac {e^{x}}{7 \, x + 4 \, \log \relax (2)}\,{d x} \log \relax (2) + 3 \, \log \left (7 \, x + 4 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*log(2)+7*x^2+7*x)*exp(x)+21)/(4*log(2)+7*x),x, algorithm="maxima")

[Out]

-2/7*2^(3/7)*exp_integral_e(1, -x - 4/7*log(2))*log(2) + x*e^x - 4*integrate(e^x/(7*x + 4*log(2)), x)*log(2) +
3*log(7*x + 4*log(2))

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mupad [B]  time = 0.08, size = 14, normalized size = 0.64 \begin {gather*} 3\,\ln \left (7\,x+\ln \left (16\right )\right )+x\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(7*x + log(2)*(4*x + 4) + 7*x^2) + 21)/(7*x + 4*log(2)),x)

[Out]

3*log(7*x + log(16)) + x*exp(x)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.68 \begin {gather*} x e^{x} + 3 \log {\left (7 x + 4 \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+4)*ln(2)+7*x**2+7*x)*exp(x)+21)/(4*ln(2)+7*x),x)

[Out]

x*exp(x) + 3*log(7*x + 4*log(2))

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