3.10.6 \(\int \frac {-392-112 x+1944 e^4 x^8+e^2 (-1008 x^4-432 x^5)}{x^3} \, dx\)

Optimal. Leaf size=20 \[ \frac {4 \left (-7-2 x+9 e^2 x^4\right )^2}{x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.75, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {14} \begin {gather*} 324 e^4 x^6-144 e^2 x^3-504 e^2 x^2+\frac {196}{x^2}+\frac {112}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-392 - 112*x + 1944*E^4*x^8 + E^2*(-1008*x^4 - 432*x^5))/x^3,x]

[Out]

196/x^2 + 112/x - 504*E^2*x^2 - 144*E^2*x^3 + 324*E^4*x^6

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {392}{x^3}-\frac {112}{x^2}-1008 e^2 x-432 e^2 x^2+1944 e^4 x^5\right ) \, dx\\ &=\frac {196}{x^2}+\frac {112}{x}-504 e^2 x^2-144 e^2 x^3+324 e^4 x^6\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.00, size = 41, normalized size = 2.05 \begin {gather*} 8 \left (\frac {49}{2 x^2}+\frac {14}{x}-63 e^2 x^2-18 e^2 x^3+\frac {81 e^4 x^6}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-392 - 112*x + 1944*E^4*x^8 + E^2*(-1008*x^4 - 432*x^5))/x^3,x]

[Out]

8*(49/(2*x^2) + 14/x - 63*E^2*x^2 - 18*E^2*x^3 + (81*E^4*x^6)/2)

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fricas [A]  time = 0.46, size = 32, normalized size = 1.60 \begin {gather*} \frac {4 \, {\left (81 \, x^{8} e^{4} - 18 \, {\left (2 \, x^{5} + 7 \, x^{4}\right )} e^{2} + 28 \, x + 49\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1944*x^8*exp(2)^2+(-432*x^5-1008*x^4)*exp(2)-112*x-392)/x^3,x, algorithm="fricas")

[Out]

4*(81*x^8*e^4 - 18*(2*x^5 + 7*x^4)*e^2 + 28*x + 49)/x^2

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giac [A]  time = 0.75, size = 32, normalized size = 1.60 \begin {gather*} 324 \, x^{6} e^{4} - 144 \, x^{3} e^{2} - 504 \, x^{2} e^{2} + \frac {28 \, {\left (4 \, x + 7\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1944*x^8*exp(2)^2+(-432*x^5-1008*x^4)*exp(2)-112*x-392)/x^3,x, algorithm="giac")

[Out]

324*x^6*e^4 - 144*x^3*e^2 - 504*x^2*e^2 + 28*(4*x + 7)/x^2

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maple [A]  time = 0.04, size = 32, normalized size = 1.60

method	result	size
risch	\(324 x^{6} {\mathrm e}^{4}-144 x^{3} {\mathrm e}^{2}-504 x^{2} {\mathrm e}^{2}+\frac {112 x +196}{x^{2}}\)	\(32\)
default	\(324 x^{6} {\mathrm e}^{4}-144 x^{3} {\mathrm e}^{2}-504 x^{2} {\mathrm e}^{2}+\frac {196}{x^{2}}+\frac {112}{x}\)	\(33\)
norman	\(\frac {196+112 x -504 x^{4} {\mathrm e}^{2}+324 x^{8} {\mathrm e}^{4}-144 \,{\mathrm e}^{2} x^{5}}{x^{2}}\)	\(33\)
gosper	\(\frac {196+112 x -504 x^{4} {\mathrm e}^{2}+324 x^{8} {\mathrm e}^{4}-144 \,{\mathrm e}^{2} x^{5}}{x^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1944*x^8*exp(2)^2+(-432*x^5-1008*x^4)*exp(2)-112*x-392)/x^3,x,method=_RETURNVERBOSE)

[Out]

324*x^6*exp(4)-144*x^3*exp(2)-504*x^2*exp(2)+(112*x+196)/x^2

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maxima [A]  time = 0.57, size = 32, normalized size = 1.60 \begin {gather*} 324 \, x^{6} e^{4} - 144 \, x^{3} e^{2} - 504 \, x^{2} e^{2} + \frac {28 \, {\left (4 \, x + 7\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1944*x^8*exp(2)^2+(-432*x^5-1008*x^4)*exp(2)-112*x-392)/x^3,x, algorithm="maxima")

[Out]

324*x^6*e^4 - 144*x^3*e^2 - 504*x^2*e^2 + 28*(4*x + 7)/x^2

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mupad [B]  time = 0.09, size = 31, normalized size = 1.55 \begin {gather*} \frac {112\,x+196}{x^2}-504\,x^2\,{\mathrm {e}}^2-144\,x^3\,{\mathrm {e}}^2+324\,x^6\,{\mathrm {e}}^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(112*x + exp(2)*(1008*x^4 + 432*x^5) - 1944*x^8*exp(4) + 392)/x^3,x)

[Out]

(112*x + 196)/x^2 - 504*x^2*exp(2) - 144*x^3*exp(2) + 324*x^6*exp(4)

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sympy [A]  time = 0.10, size = 32, normalized size = 1.60 \begin {gather*} 324 x^{6} e^{4} - 144 x^{3} e^{2} - 504 x^{2} e^{2} + \frac {112 x + 196}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1944*x**8*exp(2)**2+(-432*x**5-1008*x**4)*exp(2)-112*x-392)/x**3,x)

[Out]

324*x**6*exp(4) - 144*x**3*exp(2) - 504*x**2*exp(2) + (112*x + 196)/x**2

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