3.87.39 \(\int \frac {243 x^4+3^x (-4 e+4 e x \log (3))}{9 x^2} \, dx\) [8639]

Optimal. Leaf size=21 \[ 9 x^3+\frac {e \left (4\ 3^x+x\right )}{9 x} \]

[Out]

1/9*exp(1)*(4*exp(x*ln(3))+x)/x+9*x^3

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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 2228} \begin {gather*} 9 x^3+\frac {4 e 3^{x-2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(243*x^4 + 3^x*(-4*E + 4*E*x*Log[3]))/(9*x^2),x]

[Out]

(4*3^(-2 + x)*E)/x + 9*x^3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2228

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[g*u^(m + 1)*(F^(c*v)/(b*c*
e*Log[F])), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {243 x^4+3^x (-4 e+4 e x \log (3))}{x^2} \, dx\\ &=\frac {1}{9} \int \left (243 x^2+\frac {4\ 3^x e (-1+x \log (3))}{x^2}\right ) \, dx\\ &=9 x^3+\frac {1}{9} (4 e) \int \frac {3^x (-1+x \log (3))}{x^2} \, dx\\ &=\frac {4\ 3^{-2+x} e}{x}+9 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {4\ 3^{-2+x} e}{x}+9 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(243*x^4 + 3^x*(-4*E + 4*E*x*Log[3]))/(9*x^2),x]

[Out]

(4*3^(-2 + x)*E)/x + 9*x^3

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.96, size = 54, normalized size = 2.57

method	result	size
risch	\(9 x^{3}+\frac {4 \,{\mathrm e} 3^{x}}{9 x}\)	\(17\)
norman	\(\frac {9 x^{4}+\frac {4 \,{\mathrm e} \,{\mathrm e}^{x \ln \left (3\right )}}{9}}{x}\)	\(20\)
derivativedivides	\(\frac {\ln \left (3\right ) \left (\frac {81 x^{3}}{\ln \left (3\right )}-4 \,{\mathrm e} \left (-\frac {{\mathrm e}^{x \ln \left (3\right )}}{x \ln \left (3\right )}-\expIntegral \left (1, -x \ln \left (3\right )\right )\right )-4 \,{\mathrm e} \expIntegral \left (1, -x \ln \left (3\right )\right )\right )}{9}\)	\(54\)
default	\(\frac {\ln \left (3\right ) \left (\frac {81 x^{3}}{\ln \left (3\right )}-4 \,{\mathrm e} \left (-\frac {{\mathrm e}^{x \ln \left (3\right )}}{x \ln \left (3\right )}-\expIntegral \left (1, -x \ln \left (3\right )\right )\right )-4 \,{\mathrm e} \expIntegral \left (1, -x \ln \left (3\right )\right )\right )}{9}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((4*x*exp(1)*ln(3)-4*exp(1))*exp(x*ln(3))+243*x^4)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/9*ln(3)*(81/ln(3)*x^3-4*exp(1)*(-1/x/ln(3)*exp(x*ln(3))-Ei(1,-x*ln(3)))-4*exp(1)*Ei(1,-x*ln(3)))

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.48, size = 30, normalized size = 1.43 \begin {gather*} 9 \, x^{3} + \frac {4}{9} \, {\rm Ei}\left (x \log \left (3\right )\right ) e \log \left (3\right ) - \frac {4}{9} \, e \Gamma \left (-1, -x \log \left (3\right )\right ) \log \left (3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((4*x*exp(1)*log(3)-4*exp(1))*exp(x*log(3))+243*x^4)/x^2,x, algorithm="maxima")

[Out]

9*x^3 + 4/9*Ei(x*log(3))*e*log(3) - 4/9*e*gamma(-1, -x*log(3))*log(3)

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Fricas [A]
time = 0.39, size = 18, normalized size = 0.86 \begin {gather*} \frac {81 \, x^{4} + 4 \cdot 3^{x} e}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((4*x*exp(1)*log(3)-4*exp(1))*exp(x*log(3))+243*x^4)/x^2,x, algorithm="fricas")

[Out]

1/9*(81*x^4 + 4*3^x*e)/x

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Sympy [A]
time = 0.09, size = 19, normalized size = 0.90 \begin {gather*} 9 x^{3} + \frac {4 e e^{x \log {\left (3 \right )}}}{9 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((4*x*exp(1)*ln(3)-4*exp(1))*exp(x*ln(3))+243*x**4)/x**2,x)

[Out]

9*x**3 + 4*E*exp(x*log(3))/(9*x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((4*x*exp(1)*log(3)-4*exp(1))*exp(x*log(3))+243*x^4)/x^2,x, algorithm="giac")

[Out]

integrate(1/9*(243*x^4 + 4*(x*e*log(3) - e)*3^x)/x^2, x)

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Mupad [B]
time = 5.45, size = 16, normalized size = 0.76 \begin {gather*} 9\,x^3+\frac {4\,3^x\,\mathrm {e}}{9\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(x*log(3))*(4*exp(1) - 4*x*exp(1)*log(3)))/9 - 27*x^4)/x^2,x)

[Out]

9*x^3 + (4*3^x*exp(1))/(9*x)

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