3.101.58 \(\int \frac {9 x^2+26 e^x x^2-6 x^3-6 x^4+e^{2 x} (-13+26 x)}{x^2} \, dx\) [10058]

Optimal. Leaf size=29 \[ 4-x+x \left (-3+\frac {13 \left (e^x+x\right )^2}{x^2}-x (3+2 x)\right ) \]

[Out]

4+x*(13*(exp(x)+x)^2/x^2-x*(3+2*x)-3)-x

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Rubi [A]
time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14, 2225, 2228} \begin {gather*} -2 x^3-3 x^2+9 x+26 e^x+\frac {13 e^{2 x}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9*x^2 + 26*E^x*x^2 - 6*x^3 - 6*x^4 + E^(2*x)*(-13 + 26*x))/x^2,x]

[Out]

26*E^x + (13*E^(2*x))/x + 9*x - 3*x^2 - 2*x^3

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 2225

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 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} &=\int \left (26 e^x+\frac {13 e^{2 x} (-1+2 x)}{x^2}-3 \left (-3+2 x+2 x^2\right )\right ) \, dx\\ &=-\left (3 \int \left (-3+2 x+2 x^2\right ) \, dx\right )+13 \int \frac {e^{2 x} (-1+2 x)}{x^2} \, dx+26 \int e^x \, dx\\ &=26 e^x+\frac {13 e^{2 x}}{x}+9 x-3 x^2-2 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 29, normalized size = 1.00 \begin {gather*} 26 e^x+\frac {13 e^{2 x}}{x}+9 x-3 x^2-2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x^2 + 26*E^x*x^2 - 6*x^3 - 6*x^4 + E^(2*x)*(-13 + 26*x))/x^2,x]

[Out]

26*E^x + (13*E^(2*x))/x + 9*x - 3*x^2 - 2*x^3

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Maple [A]
time = 0.06, size = 28, normalized size = 0.97

method	result	size
default	\(-2 x^{3}-3 x^{2}+9 x +\frac {13 \,{\mathrm e}^{2 x}}{x}+26 \,{\mathrm e}^{x}\)	\(28\)
risch	\(-2 x^{3}-3 x^{2}+9 x +\frac {13 \,{\mathrm e}^{2 x}}{x}+26 \,{\mathrm e}^{x}\)	\(28\)
norman	\(\frac {9 x^{2}-3 x^{3}-2 x^{4}+13 \,{\mathrm e}^{2 x}+26 \,{\mathrm e}^{x} x}{x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((26*x-13)*exp(x)^2+26*exp(x)*x^2-6*x^4-6*x^3+9*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-2*x^3-3*x^2+9*x+13*exp(x)^2/x+26*exp(x)

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.29, size = 31, normalized size = 1.07 \begin {gather*} -2 \, x^{3} - 3 \, x^{2} + 9 \, x + 26 \, {\rm Ei}\left (2 \, x\right ) + 26 \, e^{x} - 26 \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x-13)*exp(x)^2+26*exp(x)*x^2-6*x^4-6*x^3+9*x^2)/x^2,x, algorithm="maxima")

[Out]

-2*x^3 - 3*x^2 + 9*x + 26*Ei(2*x) + 26*e^x - 26*gamma(-1, -2*x)

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Fricas [A]
time = 0.41, size = 32, normalized size = 1.10 \begin {gather*} -\frac {2 \, x^{4} + 3 \, x^{3} - 9 \, x^{2} - 26 \, x e^{x} - 13 \, e^{\left (2 \, x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x-13)*exp(x)^2+26*exp(x)*x^2-6*x^4-6*x^3+9*x^2)/x^2,x, algorithm="fricas")

[Out]

-(2*x^4 + 3*x^3 - 9*x^2 - 26*x*e^x - 13*e^(2*x))/x

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Sympy [A]
time = 0.05, size = 27, normalized size = 0.93 \begin {gather*} - 2 x^{3} - 3 x^{2} + 9 x + \frac {26 x e^{x} + 13 e^{2 x}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x-13)*exp(x)**2+26*exp(x)*x**2-6*x**4-6*x**3+9*x**2)/x**2,x)

[Out]

-2*x**3 - 3*x**2 + 9*x + (26*x*exp(x) + 13*exp(2*x))/x

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Giac [A]
time = 0.41, size = 32, normalized size = 1.10 \begin {gather*} -\frac {2 \, x^{4} + 3 \, x^{3} - 9 \, x^{2} - 26 \, x e^{x} - 13 \, e^{\left (2 \, x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((26*x-13)*exp(x)^2+26*exp(x)*x^2-6*x^4-6*x^3+9*x^2)/x^2,x, algorithm="giac")

[Out]

-(2*x^4 + 3*x^3 - 9*x^2 - 26*x*e^x - 13*e^(2*x))/x

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Mupad [B]
time = 7.77, size = 27, normalized size = 0.93 \begin {gather*} 9\,x+26\,{\mathrm {e}}^x+\frac {13\,{\mathrm {e}}^{2\,x}}{x}-3\,x^2-2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((26*x^2*exp(x) + exp(2*x)*(26*x - 13) + 9*x^2 - 6*x^3 - 6*x^4)/x^2,x)

[Out]

9*x + 26*exp(x) + (13*exp(2*x))/x - 3*x^2 - 2*x^3

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