\(\int (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 (-1080 x^4+324 x^5)) \, dx\) [50]

Optimal result

Integrand size = 40, antiderivative size = 19 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 x \left (4+x^4 \left (-4+3 e^3+x\right )^2\right ) \]

[Out]

9*x*((x+3*exp(3)-4)^2*x^4+4)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(19)=38\).

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6} \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 x^7+54 e^3 x^6-72 x^6+9 \left (16+9 e^6\right ) x^5-216 e^3 x^5+36 x \]

[In]

Int[36 + 720*x^4 + 405*E^6*x^4 - 432*x^5 + 63*x^6 + E^3*(-1080*x^4 + 324*x^5),x]

[Out]

36*x - 216*E^3*x^5 + 9*(16 + 9*E^6)*x^5 - 72*x^6 + 54*E^3*x^6 + 9*x^7

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (36+\left (720+405 e^6\right ) x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx \\ & = 36 x+9 \left (16+9 e^6\right ) x^5-72 x^6+9 x^7+e^3 \int \left (-1080 x^4+324 x^5\right ) \, dx \\ & = 36 x-216 e^3 x^5+9 \left (16+9 e^6\right ) x^5-72 x^6+54 e^3 x^6+9 x^7 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 \left (4 x+\left (-4+3 e^3\right )^2 x^5+2 \left (-4+3 e^3\right ) x^6+x^7\right ) \]

[In]

Integrate[36 + 720*x^4 + 405*E^6*x^4 - 432*x^5 + 63*x^6 + E^3*(-1080*x^4 + 324*x^5),x]

[Out]

9*(4*x + (-4 + 3*E^3)^2*x^5 + 2*(-4 + 3*E^3)*x^6 + x^7)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89

method	result	size
norman	\(\left (54 \,{\mathrm e}^{3}-72\right ) x^{6}+\left (81 \,{\mathrm e}^{6}-216 \,{\mathrm e}^{3}+144\right ) x^{5}+36 x +9 x^{7}\)	\(36\)
risch	\(81 x^{5} {\mathrm e}^{6}+54 x^{6} {\mathrm e}^{3}+9 x^{7}-216 x^{5} {\mathrm e}^{3}-72 x^{6}+144 x^{5}+36 x\)	\(41\)
gosper	\(9 x \left (9 x^{4} {\mathrm e}^{6}+6 x^{5} {\mathrm e}^{3}+x^{6}-24 x^{4} {\mathrm e}^{3}-8 x^{5}+16 x^{4}+4\right )\)	\(42\)
default	\(81 x^{5} {\mathrm e}^{6}+54 x^{6} {\mathrm e}^{3}+9 x^{7}-216 x^{5} {\mathrm e}^{3}-72 x^{6}+144 x^{5}+36 x\)	\(43\)
parallelrisch	\(81 x^{5} {\mathrm e}^{6}+54 x^{6} {\mathrm e}^{3}+9 x^{7}-216 x^{5} {\mathrm e}^{3}-72 x^{6}+144 x^{5}+36 x\)	\(43\)
parts	\(81 x^{5} {\mathrm e}^{6}+54 x^{6} {\mathrm e}^{3}+9 x^{7}-216 x^{5} {\mathrm e}^{3}-72 x^{6}+144 x^{5}+36 x\)	\(43\)

[In]

int(405*x^4*exp(3)^2+(324*x^5-1080*x^4)*exp(3)+63*x^6-432*x^5+720*x^4+36,x,method=_RETURNVERBOSE)

[Out]

(54*exp(3)-72)*x^6+(81*exp(3)^2-216*exp(3)+144)*x^5+36*x+9*x^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 \, x^{7} - 72 \, x^{6} + 81 \, x^{5} e^{6} + 144 \, x^{5} + 54 \, {\left (x^{6} - 4 \, x^{5}\right )} e^{3} + 36 \, x \]

[In]

integrate(405*x^4*exp(3)^2+(324*x^5-1080*x^4)*exp(3)+63*x^6-432*x^5+720*x^4+36,x, algorithm="fricas")

[Out]

9*x^7 - 72*x^6 + 81*x^5*e^6 + 144*x^5 + 54*(x^6 - 4*x^5)*e^3 + 36*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 x^{7} + x^{6} \left (-72 + 54 e^{3}\right ) + x^{5} \left (- 216 e^{3} + 144 + 81 e^{6}\right ) + 36 x \]

[In]

integrate(405*x**4*exp(3)**2+(324*x**5-1080*x**4)*exp(3)+63*x**6-432*x**5+720*x**4+36,x)

[Out]

9*x**7 + x**6*(-72 + 54*exp(3)) + x**5*(-216*exp(3) + 144 + 81*exp(6)) + 36*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 \, x^{7} - 72 \, x^{6} + 81 \, x^{5} e^{6} + 144 \, x^{5} + 54 \, {\left (x^{6} - 4 \, x^{5}\right )} e^{3} + 36 \, x \]

[In]

integrate(405*x^4*exp(3)^2+(324*x^5-1080*x^4)*exp(3)+63*x^6-432*x^5+720*x^4+36,x, algorithm="maxima")

[Out]

9*x^7 - 72*x^6 + 81*x^5*e^6 + 144*x^5 + 54*(x^6 - 4*x^5)*e^3 + 36*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9 \, x^{7} - 72 \, x^{6} + 81 \, x^{5} e^{6} + 144 \, x^{5} + 54 \, {\left (x^{6} - 4 \, x^{5}\right )} e^{3} + 36 \, x \]

[In]

integrate(405*x^4*exp(3)^2+(324*x^5-1080*x^4)*exp(3)+63*x^6-432*x^5+720*x^4+36,x, algorithm="giac")

[Out]

9*x^7 - 72*x^6 + 81*x^5*e^6 + 144*x^5 + 54*(x^6 - 4*x^5)*e^3 + 36*x

Mupad [B] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \left (36+720 x^4+405 e^6 x^4-432 x^5+63 x^6+e^3 \left (-1080 x^4+324 x^5\right )\right ) \, dx=9\,x^7+\left (54\,{\mathrm {e}}^3-72\right )\,x^6+\left (81\,{\mathrm {e}}^6-216\,{\mathrm {e}}^3+144\right )\,x^5+36\,x \]

[In]

int(405*x^4*exp(6) - exp(3)*(1080*x^4 - 324*x^5) + 720*x^4 - 432*x^5 + 63*x^6 + 36,x)

[Out]

36*x + x^5*(81*exp(6) - 216*exp(3) + 144) + x^6*(54*exp(3) - 72) + 9*x^7