3.30.74 \(\int \frac {2048+4096 x^2+3072 x \log (2)+e^x (100 x+75 \log (2))}{100 x+75 \log (2)} \, dx\)

Optimal. Leaf size=21 \[ e^x+\frac {512}{25} \left (x^2+\log \left (\frac {4 x}{3}+\log (2)\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6742, 2194, 698} \begin {gather*} \frac {512 x^2}{25}+e^x+\frac {512}{25} \log (4 x+\log (8)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2048 + 4096*x^2 + 3072*x*Log[2] + E^x*(100*x + 75*Log[2]))/(100*x + 75*Log[2]),x]

[Out]

E^x + (512*x^2)/25 + (512*Log[4*x + Log[8]])/25

Rule 698

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

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+\frac {1024 \left (2+4 x^2+x \log (8)\right )}{25 (4 x+\log (8))}\right ) \, dx\\ &=\frac {1024}{25} \int \frac {2+4 x^2+x \log (8)}{4 x+\log (8)} \, dx+\int e^x \, dx\\ &=e^x+\frac {1024}{25} \int \left (x+\frac {2}{4 x+\log (8)}\right ) \, dx\\ &=e^x+\frac {512 x^2}{25}+\frac {512}{25} \log (4 x+\log (8))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 30, normalized size = 1.43 \begin {gather*} \frac {1}{25} \left (25 e^x+512 x^2-32 \log ^2(8)+512 \log (4 x+\log (8))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2048 + 4096*x^2 + 3072*x*Log[2] + E^x*(100*x + 75*Log[2]))/(100*x + 75*Log[2]),x]

[Out]

(25*E^x + 512*x^2 - 32*Log[8]^2 + 512*Log[4*x + Log[8]])/25

________________________________________________________________________________________

fricas [A]  time = 1.11, size = 19, normalized size = 0.90 \begin {gather*} \frac {512}{25} \, x^{2} + e^{x} + \frac {512}{25} \, \log \left (4 \, x + 3 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*log(2)+100*x)*exp(x)+3072*x*log(2)+4096*x^2+2048)/(75*log(2)+100*x),x, algorithm="fricas")

[Out]

512/25*x^2 + e^x + 512/25*log(4*x + 3*log(2))

________________________________________________________________________________________

giac [A]  time = 0.28, size = 19, normalized size = 0.90 \begin {gather*} \frac {512}{25} \, x^{2} + e^{x} + \frac {512}{25} \, \log \left (4 \, x + 3 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*log(2)+100*x)*exp(x)+3072*x*log(2)+4096*x^2+2048)/(75*log(2)+100*x),x, algorithm="giac")

[Out]

512/25*x^2 + e^x + 512/25*log(4*x + 3*log(2))

________________________________________________________________________________________

maple [A]  time = 0.44, size = 20, normalized size = 0.95

method	result	size
default	\(\frac {512 \ln \left (3 \ln \relax (2)+4 x \right )}{25}+\frac {512 x^{2}}{25}+{\mathrm e}^{x}\)	\(20\)
norman	\(\frac {512 x^{2}}{25}+{\mathrm e}^{x}+\frac {512 \ln \left (75 \ln \relax (2)+100 x \right )}{25}\)	\(20\)
risch	\(\frac {512 \ln \left (3 \ln \relax (2)+4 x \right )}{25}+\frac {512 x^{2}}{25}+{\mathrm e}^{x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((75*ln(2)+100*x)*exp(x)+3072*x*ln(2)+4096*x^2+2048)/(75*ln(2)+100*x),x,method=_RETURNVERBOSE)

[Out]

512/25*ln(3*ln(2)+4*x)+512/25*x^2+exp(x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {576}{25} \, \log \relax (2)^{2} \log \left (4 \, x + 3 \, \log \relax (2)\right ) - \frac {3}{8} \cdot 2^{\frac {1}{4}} E_{1}\left (-x - \frac {3}{4} \, \log \relax (2)\right ) \log \relax (2) + \frac {512}{25} \, x^{2} - \frac {192}{25} \, {\left (3 \, \log \relax (2) \log \left (4 \, x + 3 \, \log \relax (2)\right ) - 4 \, x\right )} \log \relax (2) - \frac {768}{25} \, x \log \relax (2) - 12 \, \int \frac {e^{x}}{16 \, x^{2} + 24 \, x \log \relax (2) + 9 \, \log \relax (2)^{2}}\,{d x} \log \relax (2) + \frac {4 \, x e^{x}}{4 \, x + 3 \, \log \relax (2)} + \frac {512}{25} \, \log \left (4 \, x + 3 \, \log \relax (2)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*log(2)+100*x)*exp(x)+3072*x*log(2)+4096*x^2+2048)/(75*log(2)+100*x),x, algorithm="maxima")

[Out]

576/25*log(2)^2*log(4*x + 3*log(2)) - 3/8*2^(1/4)*exp_integral_e(1, -x - 3/4*log(2))*log(2) + 512/25*x^2 - 192
/25*(3*log(2)*log(4*x + 3*log(2)) - 4*x)*log(2) - 768/25*x*log(2) - 12*integrate(e^x/(16*x^2 + 24*x*log(2) + 9
*log(2)^2), x)*log(2) + 4*x*e^x/(4*x + 3*log(2)) + 512/25*log(4*x + 3*log(2))

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 17, normalized size = 0.81 \begin {gather*} \frac {512\,\ln \left (4\,x+\ln \relax (8)\right )}{25}+{\mathrm {e}}^x+\frac {512\,x^2}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(100*x + 75*log(2)) + 3072*x*log(2) + 4096*x^2 + 2048)/(100*x + 75*log(2)),x)

[Out]

(512*log(4*x + log(8)))/25 + exp(x) + (512*x^2)/25

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 22, normalized size = 1.05 \begin {gather*} \frac {512 x^{2}}{25} + e^{x} + \frac {512 \log {\left (4 x + 3 \log {\relax (2 )} \right )}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*ln(2)+100*x)*exp(x)+3072*x*ln(2)+4096*x**2+2048)/(75*ln(2)+100*x),x)

[Out]

512*x**2/25 + exp(x) + 512*log(4*x + 3*log(2))/25

________________________________________________________________________________________