3.84.64 \(\int \frac {3744-7968 x+2880 x^2+\frac {(13 x-6 x^2) (664 x^2-480 x^3)}{3 e^3}+\frac {(13 x-6 x^2)^2 (-26 x^2+24 x^3)}{9 e^6}}{-13 x^3+6 x^4} \, dx\)

Optimal. Leaf size=34 \[ \left (4 \left (-5+\frac {3}{x}\right )+\frac {\frac {x}{3}+(4-x) x-x^2}{e^3}\right )^2 \]

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Rubi [A]  time = 0.09, antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1593, 1586, 14} \begin {gather*} \frac {4 x^4}{e^6}-\frac {52 x^3}{3 e^6}+\frac {\left (169+720 e^3\right ) x^2}{9 e^6}+\frac {144}{x^2}-\frac {664 x}{3 e^3}-\frac {480}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3744 - 7968*x + 2880*x^2 + ((13*x - 6*x^2)*(664*x^2 - 480*x^3))/(3*E^3) + ((13*x - 6*x^2)^2*(-26*x^2 + 24
*x^3))/(9*E^6))/(-13*x^3 + 6*x^4),x]

[Out]

144/x^2 - 480/x - (664*x)/(3*E^3) + ((169 + 720*E^3)*x^2)/(9*E^6) - (52*x^3)/(3*E^6) + (4*x^4)/E^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]]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{x^3 (-13+6 x)} \, dx\\ &=\int \frac {-288+480 x-\frac {664 x^3}{3 e^3}+\left (\frac {338}{9 e^6}+\frac {160}{e^3}\right ) x^4-\frac {52 x^5}{e^6}+\frac {16 x^6}{e^6}}{x^3} \, dx\\ &=\int \left (-\frac {664}{3 e^3}-\frac {288}{x^3}+\frac {480}{x^2}+\frac {2 \left (169+720 e^3\right ) x}{9 e^6}-\frac {52 x^2}{e^6}+\frac {16 x^3}{e^6}\right ) \, dx\\ &=\frac {144}{x^2}-\frac {480}{x}-\frac {664 x}{3 e^3}+\frac {\left (169+720 e^3\right ) x^2}{9 e^6}-\frac {52 x^3}{3 e^6}+\frac {4 x^4}{e^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 55, normalized size = 1.62 \begin {gather*} \frac {144}{x^2}-\frac {480}{x}-\frac {664 x}{3 e^3}+\frac {169 x^2}{9 e^6}+\frac {80 x^2}{e^3}-\frac {52 x^3}{3 e^6}+\frac {4 x^4}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3744 - 7968*x + 2880*x^2 + ((13*x - 6*x^2)*(664*x^2 - 480*x^3))/(3*E^3) + ((13*x - 6*x^2)^2*(-26*x^
2 + 24*x^3))/(9*E^6))/(-13*x^3 + 6*x^4),x]

[Out]

144/x^2 - 480/x - (664*x)/(3*E^3) + (169*x^2)/(9*E^6) + (80*x^2)/E^3 - (52*x^3)/(3*E^6) + (4*x^4)/E^6

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fricas [A]  time = 0.66, size = 47, normalized size = 1.38 \begin {gather*} \frac {{\left (36 \, x^{6} - 156 \, x^{5} + 169 \, x^{4} - 432 \, {\left (10 \, x - 3\right )} e^{6} + 24 \, {\left (30 \, x^{4} - 83 \, x^{3}\right )} e^{3}\right )} e^{\left (-6\right )}}{9 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3-26*x^2)*exp(log(-2*x^2+13/3*x)-3)^2+(-480*x^3+664*x^2)*exp(log(-2*x^2+13/3*x)-3)+2880*x^2-7
968*x+3744)/(6*x^4-13*x^3),x, algorithm="fricas")

[Out]

1/9*(36*x^6 - 156*x^5 + 169*x^4 - 432*(10*x - 3)*e^6 + 24*(30*x^4 - 83*x^3)*e^3)*e^(-6)/x^2

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giac [A]  time = 0.24, size = 49, normalized size = 1.44 \begin {gather*} \frac {1}{9} \, {\left (36 \, x^{4} e^{30} - 156 \, x^{3} e^{30} + 720 \, x^{2} e^{33} + 169 \, x^{2} e^{30} - 1992 \, x e^{33}\right )} e^{\left (-36\right )} - \frac {48 \, {\left (10 \, x - 3\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3-26*x^2)*exp(log(-2*x^2+13/3*x)-3)^2+(-480*x^3+664*x^2)*exp(log(-2*x^2+13/3*x)-3)+2880*x^2-7
968*x+3744)/(6*x^4-13*x^3),x, algorithm="giac")

[Out]

1/9*(36*x^4*e^30 - 156*x^3*e^30 + 720*x^2*e^33 + 169*x^2*e^30 - 1992*x*e^33)*e^(-36) - 48*(10*x - 3)/x^2

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maple [A]  time = 0.43, size = 44, normalized size = 1.29

method	result	size
risch	\(4 \,{\mathrm e}^{-6} x^{4}-\frac {52 \,{\mathrm e}^{-6} x^{3}}{3}+\frac {169 \,{\mathrm e}^{-6} x^{2}}{9}+80 \,{\mathrm e}^{-3} x^{2}-\frac {664 x \,{\mathrm e}^{-3}}{3}+\frac {-480 x +144}{x^{2}}\)	\(44\)
default	\(-\frac {480}{x}+\frac {144}{x^{2}}-\frac {8 \,{\mathrm e}^{-3} \left (-30 x^{2}+83 x \right )}{3}+\frac {2 \,{\mathrm e}^{-6} \left (18 x^{4}-78 x^{3}+\frac {169}{2} x^{2}\right )}{9}\)	\(45\)
norman	\(\frac {\left (-\frac {664 x^{3}}{3}-480 x \,{\mathrm e}^{3}-\frac {52 \,{\mathrm e}^{-3} x^{5}}{3}+4 \,{\mathrm e}^{-3} x^{6}+\frac {\left (169+720 \,{\mathrm e}^{3}\right ) {\mathrm e}^{-3} x^{4}}{9}+144 \,{\mathrm e}^{3}\right ) {\mathrm e}^{-3}}{x^{2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^3-26*x^2)*exp(ln(-2*x^2+13/3*x)-3)^2+(-480*x^3+664*x^2)*exp(ln(-2*x^2+13/3*x)-3)+2880*x^2-7968*x+37
44)/(6*x^4-13*x^3),x,method=_RETURNVERBOSE)

[Out]

4*exp(-6)*x^4-52/3*exp(-6)*x^3+169/9*exp(-6)*x^2+80*exp(-3)*x^2-664/3*x*exp(-3)+(-480*x+144)/x^2

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maxima [A]  time = 0.36, size = 41, normalized size = 1.21 \begin {gather*} \frac {1}{9} \, {\left (36 \, x^{4} - 156 \, x^{3} + x^{2} {\left (720 \, e^{3} + 169\right )} - 1992 \, x e^{3}\right )} e^{\left (-6\right )} - \frac {48 \, {\left (10 \, x - 3\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^3-26*x^2)*exp(log(-2*x^2+13/3*x)-3)^2+(-480*x^3+664*x^2)*exp(log(-2*x^2+13/3*x)-3)+2880*x^2-7
968*x+3744)/(6*x^4-13*x^3),x, algorithm="maxima")

[Out]

1/9*(36*x^4 - 156*x^3 + x^2*(720*e^3 + 169) - 1992*x*e^3)*e^(-6) - 48*(10*x - 3)/x^2

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mupad [B]  time = 0.11, size = 43, normalized size = 1.26 \begin {gather*} x^2\,\left (80\,{\mathrm {e}}^{-3}+\frac {169\,{\mathrm {e}}^{-6}}{9}\right )-\frac {664\,x\,{\mathrm {e}}^{-3}}{3}-\frac {480\,x-144}{x^2}-\frac {52\,x^3\,{\mathrm {e}}^{-6}}{3}+4\,x^4\,{\mathrm {e}}^{-6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(log((13*x)/3 - 2*x^2) - 3)*(664*x^2 - 480*x^3) - exp(2*log((13*x)/3 - 2*x^2) - 6)*(26*x^2 - 24*x^3)
- 7968*x + 2880*x^2 + 3744)/(13*x^3 - 6*x^4),x)

[Out]

x^2*(80*exp(-3) + (169*exp(-6))/9) - (664*x*exp(-3))/3 - (480*x - 144)/x^2 - (52*x^3*exp(-6))/3 + 4*x^4*exp(-6
)

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sympy [B]  time = 0.19, size = 46, normalized size = 1.35 \begin {gather*} \frac {36 x^{4} - 156 x^{3} + x^{2} \left (169 + 720 e^{3}\right ) - 1992 x e^{3} + \frac {- 4320 x e^{6} + 1296 e^{6}}{x^{2}}}{9 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**3-26*x**2)*exp(ln(-2*x**2+13/3*x)-3)**2+(-480*x**3+664*x**2)*exp(ln(-2*x**2+13/3*x)-3)+2880*
x**2-7968*x+3744)/(6*x**4-13*x**3),x)

[Out]

(36*x**4 - 156*x**3 + x**2*(169 + 720*exp(3)) - 1992*x*exp(3) + (-4320*x*exp(6) + 1296*exp(6))/x**2)*exp(-6)/9

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