3.1238 \(\int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{81} (3 x+2)^{10}-\frac{16}{81} (3 x+2)^9+\frac{91}{216} (3 x+2)^8-\frac{7}{81} (3 x+2)^7 \]

[Out]

(-7*(2 + 3*x)^7)/81 + (91*(2 + 3*x)^8)/216 - (16*(2 + 3*x)^9)/81 + (2*(2 + 3*x)^10)/81

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Rubi [A]  time = 0.0219301, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{2}{81} (3 x+2)^{10}-\frac{16}{81} (3 x+2)^9+\frac{91}{216} (3 x+2)^8-\frac{7}{81} (3 x+2)^7 \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

(-7*(2 + 3*x)^7)/81 + (91*(2 + 3*x)^8)/216 - (16*(2 + 3*x)^9)/81 + (2*(2 + 3*x)^10)/81

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int (1-2 x)^2 (2+3 x)^6 (3+5 x) \, dx &=\int \left (-\frac{49}{27} (2+3 x)^6+\frac{91}{9} (2+3 x)^7-\frac{16}{3} (2+3 x)^8+\frac{20}{27} (2+3 x)^9\right ) \, dx\\ &=-\frac{7}{81} (2+3 x)^7+\frac{91}{216} (2+3 x)^8-\frac{16}{81} (2+3 x)^9+\frac{2}{81} (2+3 x)^{10}\\ \end{align*}

Mathematica [A]  time = 0.0021429, size = 53, normalized size = 1.18 \[ 1458 x^{10}+5832 x^9+\frac{68769 x^8}{8}+4185 x^7-2772 x^6-4284 x^5-1372 x^4+\frac{1936 x^3}{3}+640 x^2+192 x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

192*x + 640*x^2 + (1936*x^3)/3 - 1372*x^4 - 4284*x^5 - 2772*x^6 + 4185*x^7 + (68769*x^8)/8 + 5832*x^9 + 1458*x
^10

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Maple [A]  time = 0.001, size = 50, normalized size = 1.1 \begin{align*} 1458\,{x}^{10}+5832\,{x}^{9}+{\frac{68769\,{x}^{8}}{8}}+4185\,{x}^{7}-2772\,{x}^{6}-4284\,{x}^{5}-1372\,{x}^{4}+{\frac{1936\,{x}^{3}}{3}}+640\,{x}^{2}+192\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(2+3*x)^6*(3+5*x),x)

[Out]

1458*x^10+5832*x^9+68769/8*x^8+4185*x^7-2772*x^6-4284*x^5-1372*x^4+1936/3*x^3+640*x^2+192*x

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Maxima [A]  time = 1.08892, size = 66, normalized size = 1.47 \begin{align*} 1458 \, x^{10} + 5832 \, x^{9} + \frac{68769}{8} \, x^{8} + 4185 \, x^{7} - 2772 \, x^{6} - 4284 \, x^{5} - 1372 \, x^{4} + \frac{1936}{3} \, x^{3} + 640 \, x^{2} + 192 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^6*(3+5*x),x, algorithm="maxima")

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x

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Fricas [A]  time = 1.27745, size = 150, normalized size = 3.33 \begin{align*} 1458 x^{10} + 5832 x^{9} + \frac{68769}{8} x^{8} + 4185 x^{7} - 2772 x^{6} - 4284 x^{5} - 1372 x^{4} + \frac{1936}{3} x^{3} + 640 x^{2} + 192 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^6*(3+5*x),x, algorithm="fricas")

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x

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Sympy [A]  time = 0.067641, size = 51, normalized size = 1.13 \begin{align*} 1458 x^{10} + 5832 x^{9} + \frac{68769 x^{8}}{8} + 4185 x^{7} - 2772 x^{6} - 4284 x^{5} - 1372 x^{4} + \frac{1936 x^{3}}{3} + 640 x^{2} + 192 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(2+3*x)**6*(3+5*x),x)

[Out]

1458*x**10 + 5832*x**9 + 68769*x**8/8 + 4185*x**7 - 2772*x**6 - 4284*x**5 - 1372*x**4 + 1936*x**3/3 + 640*x**2
+ 192*x

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Giac [A]  time = 2.18838, size = 66, normalized size = 1.47 \begin{align*} 1458 \, x^{10} + 5832 \, x^{9} + \frac{68769}{8} \, x^{8} + 4185 \, x^{7} - 2772 \, x^{6} - 4284 \, x^{5} - 1372 \, x^{4} + \frac{1936}{3} \, x^{3} + 640 \, x^{2} + 192 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(2+3*x)^6*(3+5*x),x, algorithm="giac")

[Out]

1458*x^10 + 5832*x^9 + 68769/8*x^8 + 4185*x^7 - 2772*x^6 - 4284*x^5 - 1372*x^4 + 1936/3*x^3 + 640*x^2 + 192*x