3.73.88 $\int (150+300x+225x^2+e^{3x^2}(75+900x+450x^2))dx$

Optimal. Leaf size=17 $$75(2+x)(2+e^{3x^2}+x^2)$$

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Rubi [A]  time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.94, number of steps used = 7, number of rules used = 4, integrand size = 28, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.143, Rules used = {2226, 2204, 2209, 2212} $$\begin{array}{c}75x^3+150x^2+75e^{3x^2}x+150e^{3x^2}+150x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[150 + 300*x + 225*x^2 + E^(3*x^2)*(75 + 900*x + 450*x^2),x]

[Out]

150*E^(3*x^2) + 150*x + 75*E^(3*x^2)*x + 150*x^2 + 75*x^3

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

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]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =150x+150x^2+75x^3+\int e^{3x^2}(75+900x+450x^2)dx\\ & =150x+150x^2+75x^3+\int (75e^{3x^2}+900e^{3x^2}x+450e^{3x^2}{x}^2)dx\\ & =150x+150x^2+75x^3+75\int e^{3x^2}dx+450\int e^{3x^2}{x}^{2}dx+900\int e^{3x^2}xdx\\ & =150e^{3x^2}+150x+75e^{3x^2}x+150x^2+75x^3+\frac{25}{2}\sqrt{3\pi }\text{erfi}\left(\sqrt{3}x\right)-75\int e^{3x^2}dx\\ & =150e^{3x^2}+150x+75e^{3x^2}x+150x^2+75x^3\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.88 $$\begin{array}{c}75(2e^{3x^2}+2x+e^{3x^2}x+2x^2+x^3)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[150 + 300*x + 225*x^2 + E^(3*x^2)*(75 + 900*x + 450*x^2),x]

[Out]

75*(2*E^(3*x^2) + 2*x + E^(3*x^2)*x + 2*x^2 + x^3)

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fricas [A]  time = 0.76, size = 25, normalized size = 1.47 $$\begin{array}{c}75x^3+150x^2+75(x+2)e^{\left(3x^2\right)}+150x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((450*x^2+900*x+75)*exp(3*x^2)+225*x^2+300*x+150,x, algorithm="fricas")

[Out]

75*x^3 + 150*x^2 + 75*(x + 2)*e^(3*x^2) + 150*x

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giac [A]  time = 0.16, size = 25, normalized size = 1.47 $$\begin{array}{c}75x^3+150x^2+75(x+2)e^{\left(3x^2\right)}+150x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((450*x^2+900*x+75)*exp(3*x^2)+225*x^2+300*x+150,x, algorithm="giac")

[Out]

75*x^3 + 150*x^2 + 75*(x + 2)*e^(3*x^2) + 150*x

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maple [A]  time = 0.02, size = 27, normalized size = 1.59

method	result	size
risch	$(150+75x){\mathrm{e}}^{3x^2}+75x^3+150x^2+150x$	$27$
default	$150x+150x^2+75x^3+75x{\mathrm{e}}^{3x^2}+150{\mathrm{e}}^{3x^2}$	$32$
norman	$150x+150x^2+75x^3+75x{\mathrm{e}}^{3x^2}+150{\mathrm{e}}^{3x^2}$	$32$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((450*x^2+900*x+75)*exp(3*x^2)+225*x^2+300*x+150,x,method=_RETURNVERBOSE)

[Out]

(150+75*x)*exp(3*x^2)+75*x^3+150*x^2+150*x

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maxima [A]  time = 0.35, size = 25, normalized size = 1.47 $$\begin{array}{c}75x^3+150x^2+75(x+2)e^{\left(3x^2\right)}+150x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((450*x^2+900*x+75)*exp(3*x^2)+225*x^2+300*x+150,x, algorithm="maxima")

[Out]

75*x^3 + 150*x^2 + 75*(x + 2)*e^(3*x^2) + 150*x

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mupad [B]  time = 4.34, size = 31, normalized size = 1.82 $$\begin{array}{c}150x+150{\mathrm{e}}^{3x^2}+75x{\mathrm{e}}^{3x^2}+150x^2+75x^3\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(300*x + exp(3*x^2)*(900*x + 450*x^2 + 75) + 225*x^2 + 150,x)

[Out]

150*x + 150*exp(3*x^2) + 75*x*exp(3*x^2) + 150*x^2 + 75*x^3

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sympy [A]  time = 0.10, size = 24, normalized size = 1.41 $$\begin{array}{c}75x^3+150x^2+150x+(75x+150)e^{3x^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((450*x**2+900*x+75)*exp(3*x**2)+225*x**2+300*x+150,x)

[Out]

75*x**3 + 150*x**2 + 150*x + (75*x + 150)*exp(3*x**2)

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