\(\int \frac {-7200 x+e^{2 x} (125000+30000 x+2400 x^2+64 x^3)}{15625+3750 x+300 x^2+8 x^3} \, dx\) [6933]

Optimal result

Integrand size = 43, antiderivative size = 26 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=4 e^{2 x}-\frac {16 x^2}{\left (\frac {25-x}{3}+x\right )^2} \]

[Out]

-x^2/(1/6*x+25/12)^2+4*exp(x)^2

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {6820, 12, 6874, 2225, 37} \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=4 e^{2 x}-\frac {144 x^2}{(2 x+25)^2} \]

[In]

Int[(-7200*x + E^(2*x)*(125000 + 30000*x + 2400*x^2 + 64*x^3))/(15625 + 3750*x + 300*x^2 + 8*x^3),x]

[Out]

4*E^(2*x) - (144*x^2)/(25 + 2*x)^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \left (-900 x+e^{2 x} (25+2 x)^3\right )}{(25+2 x)^3} \, dx \\ & = 8 \int \frac {-900 x+e^{2 x} (25+2 x)^3}{(25+2 x)^3} \, dx \\ & = 8 \int \left (e^{2 x}-\frac {900 x}{(25+2 x)^3}\right ) \, dx \\ & = 8 \int e^{2 x} \, dx-7200 \int \frac {x}{(25+2 x)^3} \, dx \\ & = 4 e^{2 x}-\frac {144 x^2}{(25+2 x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=8 \left (\frac {e^{2 x}}{2}-\frac {18 x^2}{(25+2 x)^2}\right ) \]

[In]

Integrate[(-7200*x + E^(2*x)*(125000 + 30000*x + 2400*x^2 + 64*x^3))/(15625 + 3750*x + 300*x^2 + 8*x^3),x]

[Out]

8*(E^(2*x)/2 - (18*x^2)/(25 + 2*x)^2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
risch	\(\frac {900 x +5625}{x^{2}+25 x +\frac {625}{4}}+4 \,{\mathrm e}^{2 x}\)	\(24\)
default	\(\frac {1800}{2 x +25}-\frac {22500}{\left (2 x +25\right )^{2}}+4 \,{\mathrm e}^{2 x}\)	\(26\)
parts	\(\frac {1800}{2 x +25}-\frac {22500}{\left (2 x +25\right )^{2}}+4 \,{\mathrm e}^{2 x}\)	\(26\)
norman	\(\frac {3600 x +2500 \,{\mathrm e}^{2 x}+400 x \,{\mathrm e}^{2 x}+16 \,{\mathrm e}^{2 x} x^{2}+22500}{\left (2 x +25\right )^{2}}\)	\(36\)
parallelrisch	\(\frac {64 \,{\mathrm e}^{2 x} x^{2}+90000+1600 x \,{\mathrm e}^{2 x}+10000 \,{\mathrm e}^{2 x}+14400 x}{16 x^{2}+400 x +2500}\)	\(42\)

[In]

int(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x,method=_RETURNVERBOSE)

[Out]

(900*x+5625)/(x^2+25*x+625/4)+4*exp(2*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=\frac {4 \, {\left ({\left (4 \, x^{2} + 100 \, x + 625\right )} e^{\left (2 \, x\right )} + 900 \, x + 5625\right )}}{4 \, x^{2} + 100 \, x + 625} \]

[In]

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="fricas
")

[Out]

4*((4*x^2 + 100*x + 625)*e^(2*x) + 900*x + 5625)/(4*x^2 + 100*x + 625)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=- \frac {7200 \left (- 4 x - 25\right )}{32 x^{2} + 800 x + 5000} + 4 e^{2 x} \]

[In]

integrate(((64*x**3+2400*x**2+30000*x+125000)*exp(x)**2-7200*x)/(8*x**3+300*x**2+3750*x+15625),x)

[Out]

-7200*(-4*x - 25)/(32*x**2 + 800*x + 5000) + 4*exp(2*x)

Maxima [F]

\[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=\int { \frac {8 \, {\left ({\left (8 \, x^{3} + 300 \, x^{2} + 3750 \, x + 15625\right )} e^{\left (2 \, x\right )} - 900 \, x\right )}}{8 \, x^{3} + 300 \, x^{2} + 3750 \, x + 15625} \,d x } \]

[In]

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="maxima
")

[Out]

8*(4*x^3 + 150*x^2 + 1875*x)*e^(2*x)/(8*x^3 + 300*x^2 + 3750*x + 15625) + 900*(4*x + 25)/(4*x^2 + 100*x + 625)
- 62500*e^(-25)*exp_integral_e(3, -2*x - 25)/(2*x + 25)^2 - 375000*integrate(e^(2*x)/(16*x^4 + 800*x^3 + 1500
0*x^2 + 125000*x + 390625), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=\frac {4 \, {\left (4 \, x^{2} e^{\left (2 \, x\right )} + 100 \, x e^{\left (2 \, x\right )} + 900 \, x + 625 \, e^{\left (2 \, x\right )} + 5625\right )}}{4 \, x^{2} + 100 \, x + 625} \]

[In]

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="giac")

[Out]

4*(4*x^2*e^(2*x) + 100*x*e^(2*x) + 900*x + 625*e^(2*x) + 5625)/(4*x^2 + 100*x + 625)

Mupad [B] (verification not implemented)

Time = 12.57 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {-7200 x+e^{2 x} \left (125000+30000 x+2400 x^2+64 x^3\right )}{15625+3750 x+300 x^2+8 x^3} \, dx=4\,{\mathrm {e}}^{2\,x}+\frac {3600\,x+22500}{{\left (2\,x+25\right )}^2} \]

[In]

int(-(7200*x - exp(2*x)*(30000*x + 2400*x^2 + 64*x^3 + 125000))/(3750*x + 300*x^2 + 8*x^3 + 15625),x)

[Out]

4*exp(2*x) + (3600*x + 22500)/(2*x + 25)^2