3.82.100 \(\int \frac {1}{3} 2^{\frac {2}{3} (-6 x^2+18 x^3+x^4-3 x^5)} (-12 x+54 x^2+4 x^3-15 x^4) \log (4) \, dx\)

Optimal. Leaf size=18 \[ 4^{\left (-\frac {1}{3}+x\right ) x^2 \left (6-x^2\right )} \]

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Rubi [A]  time = 0.16, antiderivative size = 36, normalized size of antiderivative = 2.00, number of steps used = 2, number of rules used = 2, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {12, 6706} \begin {gather*} \frac {2^{-\frac {2}{3} \left (3 x^5-x^4-18 x^3+6 x^2\right )-1} \log (4)}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2^((2*(-6*x^2 + 18*x^3 + x^4 - 3*x^5))/3)*(-12*x + 54*x^2 + 4*x^3 - 15*x^4)*Log[4])/3,x]

[Out]

(2^(-1 - (2*(6*x^2 - 18*x^3 - x^4 + 3*x^5))/3)*Log[4])/Log[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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \log (4) \int 2^{\frac {2}{3} \left (-6 x^2+18 x^3+x^4-3 x^5\right )} \left (-12 x+54 x^2+4 x^3-15 x^4\right ) \, dx\\ &=\frac {2^{-1-\frac {2}{3} \left (6 x^2-18 x^3-x^4+3 x^5\right )} \log (4)}{\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 33, normalized size = 1.83 \begin {gather*} \frac {2^{-1-4 x^2+12 x^3+\frac {2 x^4}{3}-2 x^5} \log (4)}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2^((2*(-6*x^2 + 18*x^3 + x^4 - 3*x^5))/3)*(-12*x + 54*x^2 + 4*x^3 - 15*x^4)*Log[4])/3,x]

[Out]

(2^(-1 - 4*x^2 + 12*x^3 + (2*x^4)/3 - 2*x^5)*Log[4])/Log[2]

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fricas [A]  time = 0.88, size = 23, normalized size = 1.28 \begin {gather*} 2^{-2 \, x^{5} + \frac {2}{3} \, x^{4} + 12 \, x^{3} - 4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*(-15*x^4+4*x^3+54*x^2-12*x)*log(2)*exp(2/3*(-3*x^5+x^4+18*x^3-6*x^2)*log(2)),x, algorithm="frica
s")

[Out]

2^(-2*x^5 + 2/3*x^4 + 12*x^3 - 4*x^2)

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giac [A]  time = 0.16, size = 23, normalized size = 1.28 \begin {gather*} 2^{-2 \, x^{5} + \frac {2}{3} \, x^{4} + 12 \, x^{3} - 4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*(-15*x^4+4*x^3+54*x^2-12*x)*log(2)*exp(2/3*(-3*x^5+x^4+18*x^3-6*x^2)*log(2)),x, algorithm="giac"
)

[Out]

2^(-2*x^5 + 2/3*x^4 + 12*x^3 - 4*x^2)

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maple [A]  time = 0.05, size = 18, normalized size = 1.00

method	result	size
risch	\(2^{-\frac {2 x^{2} \left (3 x -1\right ) \left (x^{2}-6\right )}{3}}\)	\(18\)
gosper	\({\mathrm e}^{-\frac {2 x^{2} \left (3 x^{3}-x^{2}-18 x +6\right ) \ln \relax (2)}{3}}\)	\(24\)
derivativedivides	\({\mathrm e}^{\frac {2 \left (-3 x^{5}+x^{4}+18 x^{3}-6 x^{2}\right ) \ln \relax (2)}{3}}\)	\(25\)
default	\({\mathrm e}^{\frac {2 \left (-3 x^{5}+x^{4}+18 x^{3}-6 x^{2}\right ) \ln \relax (2)}{3}}\)	\(25\)
norman	\({\mathrm e}^{\frac {2 \left (-3 x^{5}+x^{4}+18 x^{3}-6 x^{2}\right ) \ln \relax (2)}{3}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/3*(-15*x^4+4*x^3+54*x^2-12*x)*ln(2)*exp(2/3*(-3*x^5+x^4+18*x^3-6*x^2)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

2^(-2/3*x^2*(3*x-1)*(x^2-6))

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maxima [A]  time = 0.37, size = 23, normalized size = 1.28 \begin {gather*} 2^{-2 \, x^{5} + \frac {2}{3} \, x^{4} + 12 \, x^{3} - 4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*(-15*x^4+4*x^3+54*x^2-12*x)*log(2)*exp(2/3*(-3*x^5+x^4+18*x^3-6*x^2)*log(2)),x, algorithm="maxim
a")

[Out]

2^(-2*x^5 + 2/3*x^4 + 12*x^3 - 4*x^2)

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mupad [B]  time = 0.16, size = 33, normalized size = 1.83 \begin {gather*} \frac {2^{\frac {2\,x^4}{3}}\,2^{12\,x^3}}{2^{4\,x^2}\,2^{2\,x^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(-(2*log(2)*(6*x^2 - 18*x^3 - x^4 + 3*x^5))/3)*log(2)*(12*x - 54*x^2 - 4*x^3 + 15*x^4))/3,x)

[Out]

(2^((2*x^4)/3)*2^(12*x^3))/(2^(4*x^2)*2^(2*x^5))

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sympy [A]  time = 0.17, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (- 2 x^{5} + \frac {2 x^{4}}{3} + 12 x^{3} - 4 x^{2}\right ) \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/3*(-15*x**4+4*x**3+54*x**2-12*x)*ln(2)*exp(2/3*(-3*x**5+x**4+18*x**3-6*x**2)*ln(2)),x)

[Out]

exp((-2*x**5 + 2*x**4/3 + 12*x**3 - 4*x**2)*log(2))

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