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

Optimal. Leaf size=19 \[ 9 x \left (4+x^4 \left (-4+3 e^3+x\right )^2\right ) \]

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Rubi [B]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 3, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6} \begin {gather*} 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 \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.79 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.58, size = 39, normalized size = 2.05 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 0.55, size = 39, normalized size = 2.05 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.02, size = 36, normalized size = 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	\(81 x^{5} {\mathrm e}^{6}+54 x^{6} {\mathrm e}^{3}-216 x^{5} {\mathrm e}^{3}+9 x^{7}-72 x^{6}+144 x^{5}+36 x\)	\(41\)
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	\(81 x^{5} {\mathrm e}^{6}+{\mathrm e}^{3} \left (54 x^{6}-216 x^{5}\right )+9 x^{7}-72 x^{6}+144 x^{5}+36 x\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 0.48, size = 39, normalized size = 2.05 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.05, size = 33, normalized size = 1.74 \begin {gather*} 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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.06, size = 32, normalized size = 1.68 \begin {gather*} 9 x^{7} + x^{6} \left (-72 + 54 e^{3}\right ) + x^{5} \left (- 216 e^{3} + 144 + 81 e^{6}\right ) + 36 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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