3.57.12 \(\int \frac {30+28 e^2+6 e^4+e^5 (10+6 e^2)}{x^4} \, dx\)

Optimal. Leaf size=24 \[ 2 \left (3+e^2+e^5\right ) \left (1-\frac {\frac {5}{3}+e^2}{x^3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 30} \begin {gather*} -\frac {2 \left (5+3 e^2\right ) \left (3+e^2+e^5\right )}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(30 + 28*E^2 + 6*E^4 + E^5*(10 + 6*E^2))/x^4,x]

[Out]

(-2*(5 + 3*E^2)*(3 + E^2 + E^5))/(3*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 29, normalized size = 1.21 \begin {gather*} -\frac {30+28 e^2+6 e^4+10 e^5+6 e^7}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(30 + 28*E^2 + 6*E^4 + E^5*(10 + 6*E^2))/x^4,x]

[Out]

-1/3*(30 + 28*E^2 + 6*E^4 + 10*E^5 + 6*E^7)/x^3

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fricas [A]  time = 0.69, size = 23, normalized size = 0.96 \begin {gather*} -\frac {2 \, {\left (3 \, e^{7} + 5 \, e^{5} + 3 \, e^{4} + 14 \, e^{2} + 15\right )}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(2)+10)*exp(5)+6*exp(2)^2+28*exp(2)+30)/x^4,x, algorithm="fricas")

[Out]

-2/3*(3*e^7 + 5*e^5 + 3*e^4 + 14*e^2 + 15)/x^3

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giac [A]  time = 0.12, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, e^{2} + 5\right )} e^{5} + 3 \, e^{4} + 14 \, e^{2} + 15\right )}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(2)+10)*exp(5)+6*exp(2)^2+28*exp(2)+30)/x^4,x, algorithm="giac")

[Out]

-2/3*((3*e^2 + 5)*e^5 + 3*e^4 + 14*e^2 + 15)/x^3

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maple [A]  time = 0.04, size = 27, normalized size = 1.12

method	result	size
default	\(-\frac {\left (6 \,{\mathrm e}^{2}+10\right ) {\mathrm e}^{5}+6 \,{\mathrm e}^{4}+28 \,{\mathrm e}^{2}+30}{3 x^{3}}\)	\(27\)
norman	\(\frac {-2 \,{\mathrm e}^{2} {\mathrm e}^{5}-2 \,{\mathrm e}^{4}-\frac {10 \,{\mathrm e}^{5}}{3}-\frac {28 \,{\mathrm e}^{2}}{3}-10}{x^{3}}\)	\(27\)
gosper	\(-\frac {2 \left (3 \,{\mathrm e}^{2} {\mathrm e}^{5}+3 \,{\mathrm e}^{4}+5 \,{\mathrm e}^{5}+14 \,{\mathrm e}^{2}+15\right )}{3 x^{3}}\)	\(28\)
risch	\(-\frac {2 \,{\mathrm e}^{5} {\mathrm e}^{2}}{x^{3}}-\frac {10 \,{\mathrm e}^{5}}{3 x^{3}}-\frac {2 \,{\mathrm e}^{4}}{x^{3}}-\frac {28 \,{\mathrm e}^{2}}{3 x^{3}}-\frac {10}{x^{3}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*exp(2)+10)*exp(5)+6*exp(2)^2+28*exp(2)+30)/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3/x^3*((6*exp(2)+10)*exp(5)+6*exp(2)^2+28*exp(2)+30)

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maxima [A]  time = 0.37, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, e^{2} + 5\right )} e^{5} + 3 \, e^{4} + 14 \, e^{2} + 15\right )}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(2)+10)*exp(5)+6*exp(2)^2+28*exp(2)+30)/x^4,x, algorithm="maxima")

[Out]

-2/3*((3*e^2 + 5)*e^5 + 3*e^4 + 14*e^2 + 15)/x^3

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mupad [B]  time = 3.48, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\frac {28\,{\mathrm {e}}^2}{3}+2\,{\mathrm {e}}^4+\frac {10\,{\mathrm {e}}^5}{3}+2\,{\mathrm {e}}^7+10}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((28*exp(2) + 6*exp(4) + exp(5)*(6*exp(2) + 10) + 30)/x^4,x)

[Out]

-((28*exp(2))/3 + 2*exp(4) + (10*exp(5))/3 + 2*exp(7) + 10)/x^3

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sympy [A]  time = 0.06, size = 27, normalized size = 1.12 \begin {gather*} - \frac {30 + 28 e^{2} + 6 e^{4} + 10 e^{5} + 6 e^{7}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(2)+10)*exp(5)+6*exp(2)**2+28*exp(2)+30)/x**4,x)

[Out]

-(30 + 28*exp(2) + 6*exp(4) + 10*exp(5) + 6*exp(7))/(3*x**3)

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