\(\int \frac {3744-7968 x+2880 x^2+\frac {(13 x-6 x^2) (664 x^2-480 x^3)}{3 e^3}+\frac {(13 x-6 x^2)^2 (-26 x^2+24 x^3)}{9 e^6}}{-13 x^3+6 x^4} \, dx\) [8364]

Optimal result

Integrand size = 80, antiderivative size = 34 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\left (4 \left (-5+\frac {3}{x}\right )+\frac {\frac {x}{3}+(4-x) x-x^2}{e^3}\right )^2 \]

[Out]

(exp(ln((-x+4)*x+1/3*x-x^2)-3)+12/x-20)^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1607, 1600, 14} \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {4 x^4}{e^6}-\frac {52 x^3}{3 e^6}+\frac {\left (169+720 e^3\right ) x^2}{9 e^6}+\frac {144}{x^2}-\frac {664 x}{3 e^3}-\frac {480}{x} \]

[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 1600

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 1607

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.62 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {144}{x^2}-\frac {480}{x}-\frac {664 x}{3 e^3}+\frac {169 x^2}{9 e^6}+\frac {80 x^2}{e^3}-\frac {52 x^3}{3 e^6}+\frac {4 x^4}{e^6} \]

[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

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47

method	result	size
risch	\(4 \,{\mathrm e}^{-6} x^{4}-\frac {52 \,{\mathrm e}^{-6} x^{3}}{3}+\frac {169 \,{\mathrm e}^{-6} x^{2}}{9}+80 \,{\mathrm e}^{-3} x^{2}-\frac {664 \,{\mathrm e}^{-3} x}{3}+\frac {-480 x +144}{x^{2}}\)	\(50\)
norman	\(\frac {\left (-\frac {664 x^{3}}{3}-480 x \,{\mathrm e}^{3}-\frac {52 \,{\mathrm e}^{-3} x^{5}}{3}+4 \,{\mathrm e}^{-3} x^{6}+\frac {\left (169+720 \,{\mathrm e}^{3}\right ) {\mathrm e}^{-3} x^{4}}{9}+144 \,{\mathrm e}^{3}\right ) {\mathrm e}^{-3}}{x^{2}}\)	\(57\)
default	\(\frac {144}{x^{2}}-\frac {480}{x}-\frac {2 \left (6480 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{4}-31968 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{3}+84162 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x \right )}{9 x^{2} \left (6 x -13\right )^{2}}+\left (-2 x^{2}+\frac {13}{3} x \right )^{2} {\mathrm e}^{-6}\)	\(92\)
parts	\(\frac {144}{x^{2}}-\frac {480}{x}-\frac {2 \left (6480 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{4}-31968 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{3}+84162 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x \right )}{9 x^{2} \left (6 x -13\right )^{2}}+\left (-2 x^{2}+\frac {13}{3} x \right )^{2} {\mathrm e}^{-6}\)	\(92\)
parallelrisch	\(\frac {3796416-16159104 x -622080 x^{4}+9569664 x^{2}+5616 \left (-2 x^{2}+\frac {13}{3} x \right )^{2} {\mathrm e}^{-6} x^{4}-24336 \left (-2 x^{2}+\frac {13}{3} x \right )^{2} {\mathrm e}^{-6} x^{3}-224640 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{4}+26364 \left (-2 x^{2}+\frac {13}{3} x \right )^{2} {\mathrm e}^{-6} x^{2}+1108224 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x^{3}-2917616 \,{\mathrm e}^{\ln \left (-2 x^{2}+\frac {13}{3} x \right )-3} x}{156 x^{2} \left (6 x -13\right )^{2}}\)	\(140\)

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {{\left (36 \, x^{6} - 156 \, x^{5} + 169 \, x^{4} - 432 \, {\left (10 \, x - 3\right )} e^{6} + 24 \, {\left (30 \, x^{4} - 83 \, x^{3}\right )} e^{3}\right )} e^{\left (-6\right )}}{9 \, x^{2}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {36 x^{4} - 156 x^{3} + x^{2} \cdot \left (169 + 720 e^{3}\right ) - 1992 x e^{3} + \frac {- 4320 x e^{6} + 1296 e^{6}}{x^{2}}}{9 e^{6}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {1}{9} \, {\left (36 \, x^{4} - 156 \, x^{3} + x^{2} {\left (720 \, e^{3} + 169\right )} - 1992 \, x e^{3}\right )} e^{\left (-6\right )} - \frac {48 \, {\left (10 \, x - 3\right )}}{x^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=\frac {1}{9} \, {\left (36 \, x^{4} e^{18} - 156 \, x^{3} e^{18} + 720 \, x^{2} e^{21} + 169 \, x^{2} e^{18} - 1992 \, x e^{21}\right )} e^{\left (-24\right )} - \frac {48 \, {\left (10 \, x - 3\right )}}{x^{2}} \]

[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^18 - 156*x^3*e^18 + 720*x^2*e^21 + 169*x^2*e^18 - 1992*x*e^21)*e^(-24) - 48*(10*x - 3)/x^2

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {3744-7968 x+2880 x^2+\frac {\left (13 x-6 x^2\right ) \left (664 x^2-480 x^3\right )}{3 e^3}+\frac {\left (13 x-6 x^2\right )^2 \left (-26 x^2+24 x^3\right )}{9 e^6}}{-13 x^3+6 x^4} \, dx=x^2\,\left (80\,{\mathrm {e}}^{-3}+\frac {169\,{\mathrm {e}}^{-6}}{9}\right )-\frac {664\,x\,{\mathrm {e}}^{-3}}{3}-\frac {480\,x-144}{x^2}-\frac {52\,x^3\,{\mathrm {e}}^{-6}}{3}+4\,x^4\,{\mathrm {e}}^{-6} \]

[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
)