3.18.23 \(\int (-2+4 e^{1+4 x}+2 x+\log (x^3)) \, dx\)

Optimal. Leaf size=17 \[ e^{1+4 x}+x \left (-5+x+\log \left (x^3\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2194, 2295} \begin {gather*} x \log \left (x^3\right )+x^2-5 x+e^{4 x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 + 4*E^(1 + 4*x) + 2*x + Log[x^3],x]

[Out]

E^(1 + 4*x) - 5*x + x^2 + x*Log[x^3]

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 x+x^2+4 \int e^{1+4 x} \, dx+\int \log \left (x^3\right ) \, dx\\ &=e^{1+4 x}-5 x+x^2+x \log \left (x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.18 \begin {gather*} e^{1+4 x}-5 x+x^2+x \log \left (x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 + 4*E^(1 + 4*x) + 2*x + Log[x^3],x]

[Out]

E^(1 + 4*x) - 5*x + x^2 + x*Log[x^3]

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fricas [A]  time = 0.66, size = 19, normalized size = 1.12 \begin {gather*} x^{2} + x \log \left (x^{3}\right ) - 5 \, x + e^{\left (4 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^3)+4*exp(4*x+1)+2*x-2,x, algorithm="fricas")

[Out]

x^2 + x*log(x^3) - 5*x + e^(4*x + 1)

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giac [A]  time = 0.20, size = 19, normalized size = 1.12 \begin {gather*} x^{2} + x \log \left (x^{3}\right ) - 5 \, x + e^{\left (4 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^3)+4*exp(4*x+1)+2*x-2,x, algorithm="giac")

[Out]

x^2 + x*log(x^3) - 5*x + e^(4*x + 1)

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maple [A]  time = 0.02, size = 20, normalized size = 1.18

method	result	size
default	\(x^{2}-5 x +{\mathrm e}^{4 x +1}+x \ln \left (x^{3}\right )\)	\(20\)
norman	\(x^{2}-5 x +{\mathrm e}^{4 x +1}+x \ln \left (x^{3}\right )\)	\(20\)
risch	\(x^{2}-5 x +{\mathrm e}^{4 x +1}+x \ln \left (x^{3}\right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x^3)+4*exp(4*x+1)+2*x-2,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 19, normalized size = 1.12 \begin {gather*} x^{2} + x \log \left (x^{3}\right ) - 5 \, x + e^{\left (4 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(x^3)+4*exp(4*x+1)+2*x-2,x, algorithm="maxima")

[Out]

x^2 + x*log(x^3) - 5*x + e^(4*x + 1)

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mupad [B]  time = 1.10, size = 19, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^{4\,x+1}-5\,x+x\,\ln \left (x^3\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x + log(x^3) + 4*exp(4*x + 1) - 2,x)

[Out]

exp(4*x + 1) - 5*x + x*log(x^3) + x^2

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sympy [A]  time = 0.25, size = 19, normalized size = 1.12 \begin {gather*} x^{2} + x \log {\left (x^{3} \right )} - 5 x + e^{4 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x**3)+4*exp(4*x+1)+2*x-2,x)

[Out]

x**2 + x*log(x**3) - 5*x + exp(4*x + 1)

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