3.5.1 \(\int \frac {1}{625} (5625+33750 x+6 e^{2 x^3} x^2) \, dx\)

Optimal. Leaf size=20 \[ \frac {e^{2 x^3}}{625}+9 x+27 x^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {12, 2209} \begin {gather*} \frac {e^{2 x^3}}{625}+27 x^2+9 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5625 + 33750*x + 6*E^(2*x^3)*x^2)/625,x]

[Out]

E^(2*x^3)/625 + 9*x + 27*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 2209

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{625} \int \left (5625+33750 x+6 e^{2 x^3} x^2\right ) \, dx\\ &=9 x+27 x^2+\frac {6}{625} \int e^{2 x^3} x^2 \, dx\\ &=\frac {e^{2 x^3}}{625}+9 x+27 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {e^{2 x^3}}{625}+9 x+27 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5625 + 33750*x + 6*E^(2*x^3)*x^2)/625,x]

[Out]

E^(2*x^3)/625 + 9*x + 27*x^2

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fricas [A]  time = 0.54, size = 17, normalized size = 0.85 \begin {gather*} 27 \, x^{2} + 9 \, x + \frac {1}{625} \, e^{\left (2 \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/625*x^2*exp(x^3)^2+54*x+9,x, algorithm="fricas")

[Out]

27*x^2 + 9*x + 1/625*e^(2*x^3)

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giac [A]  time = 0.35, size = 17, normalized size = 0.85 \begin {gather*} 27 \, x^{2} + 9 \, x + \frac {1}{625} \, e^{\left (2 \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/625*x^2*exp(x^3)^2+54*x+9,x, algorithm="giac")

[Out]

27*x^2 + 9*x + 1/625*e^(2*x^3)

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maple [A]  time = 0.02, size = 18, normalized size = 0.90

method	result	size
default	\(\frac {{\mathrm e}^{2 x^{3}}}{625}+27 x^{2}+9 x\)	\(18\)
norman	\(\frac {{\mathrm e}^{2 x^{3}}}{625}+27 x^{2}+9 x\)	\(18\)
risch	\(\frac {{\mathrm e}^{2 x^{3}}}{625}+27 x^{2}+9 x\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6/625*x^2*exp(x^3)^2+54*x+9,x,method=_RETURNVERBOSE)

[Out]

1/625*exp(x^3)^2+27*x^2+9*x

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maxima [A]  time = 0.56, size = 17, normalized size = 0.85 \begin {gather*} 27 \, x^{2} + 9 \, x + \frac {1}{625} \, e^{\left (2 \, x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/625*x^2*exp(x^3)^2+54*x+9,x, algorithm="maxima")

[Out]

27*x^2 + 9*x + 1/625*e^(2*x^3)

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mupad [B]  time = 0.43, size = 17, normalized size = 0.85 \begin {gather*} 9\,x+\frac {{\mathrm {e}}^{2\,x^3}}{625}+27\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(54*x + (6*x^2*exp(2*x^3))/625 + 9,x)

[Out]

9*x + exp(2*x^3)/625 + 27*x^2

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sympy [A]  time = 0.10, size = 15, normalized size = 0.75 \begin {gather*} 27 x^{2} + 9 x + \frac {e^{2 x^{3}}}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6/625*x**2*exp(x**3)**2+54*x+9,x)

[Out]

27*x**2 + 9*x + exp(2*x**3)/625

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