3.20.16 \(\int \frac {\sqrt [3]{7} (e^x (4-4 x)-12 x^2)}{3 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {4}{3} \sqrt [3]{7} \left (-3-\frac {e^x}{x^2}+\frac {1}{x}\right ) x \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.04, 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, 2197} \begin {gather*} -4 \sqrt [3]{7} x-\frac {4 \sqrt [3]{7} e^x}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7^(1/3)*(E^x*(4 - 4*x) - 12*x^2))/(3*x^2),x]

[Out]

(-4*7^(1/3)*E^x)/(3*x) - 4*7^(1/3)*x

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 2197

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}{3} \sqrt [3]{7} \int \frac {e^x (4-4 x)-12 x^2}{x^2} \, dx\\ &=\frac {1}{3} \sqrt [3]{7} \int \left (-12-\frac {4 e^x (-1+x)}{x^2}\right ) \, dx\\ &=-4 \sqrt [3]{7} x-\frac {1}{3} \left (4 \sqrt [3]{7}\right ) \int \frac {e^x (-1+x)}{x^2} \, dx\\ &=-\frac {4 \sqrt [3]{7} e^x}{3 x}-4 \sqrt [3]{7} x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} -\frac {4}{3} \sqrt [3]{7} \left (\frac {e^x}{x}+3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7^(1/3)*(E^x*(4 - 4*x) - 12*x^2))/(3*x^2),x]

[Out]

(-4*7^(1/3)*(E^x/x + 3*x))/3

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fricas [A]  time = 1.01, size = 20, normalized size = 0.87 \begin {gather*} -\frac {4 \, {\left (3 \cdot 7^{\frac {1}{3}} x^{2} + 7^{\frac {1}{3}} e^{x}\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(x)-12*x^2)*7^(1/3)/x^2,x, algorithm="fricas")

[Out]

-4/3*(3*7^(1/3)*x^2 + 7^(1/3)*e^x)/x

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giac [A]  time = 1.14, size = 16, normalized size = 0.70 \begin {gather*} -\frac {4 \cdot 7^{\frac {1}{3}} {\left (3 \, x^{2} + e^{x}\right )}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(x)-12*x^2)*7^(1/3)/x^2,x, algorithm="giac")

[Out]

-4/3*7^(1/3)*(3*x^2 + e^x)/x

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maple [A]  time = 0.07, size = 17, normalized size = 0.74

method	result	size
default	\(\frac {7^{\frac {1}{3}} \left (-12 x -\frac {4 \,{\mathrm e}^{x}}{x}\right )}{3}\)	\(17\)
risch	\(-4 \,7^{\frac {1}{3}} x -\frac {4 \,7^{\frac {1}{3}} {\mathrm e}^{x}}{3 x}\)	\(18\)
norman	\(\frac {-4 \,7^{\frac {1}{3}} x^{2}-\frac {4 \,7^{\frac {1}{3}} {\mathrm e}^{x}}{3}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-4*x+4)*exp(x)-12*x^2)*7^(1/3)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/3*7^(1/3)*(-12*x-4*exp(x)/x)

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maxima [C]  time = 0.77, size = 18, normalized size = 0.78 \begin {gather*} -\frac {4}{3} \cdot 7^{\frac {1}{3}} {\left (3 \, x + {\rm Ei}\relax (x) - \Gamma \left (-1, -x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(x)-12*x^2)*7^(1/3)/x^2,x, algorithm="maxima")

[Out]

-4/3*7^(1/3)*(3*x + Ei(x) - gamma(-1, -x))

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mupad [B]  time = 0.07, size = 16, normalized size = 0.70 \begin {gather*} -\frac {4\,7^{1/3}\,\left ({\mathrm {e}}^x+3\,x^2\right )}{3\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(7^(1/3)*(exp(x)*(4*x - 4) + 12*x^2))/(3*x^2),x)

[Out]

-(4*7^(1/3)*(exp(x) + 3*x^2))/(3*x)

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sympy [A]  time = 0.18, size = 22, normalized size = 0.96 \begin {gather*} - 4 \sqrt [3]{7} x - \frac {4 \sqrt [3]{7} e^{x}}{3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-4*x+4)*exp(x)-12*x**2)*7**(1/3)/x**2,x)

[Out]

-4*7**(1/3)*x - 4*7**(1/3)*exp(x)/(3*x)

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