3.1144 \(\int (1-2 x) (2+3 x)^7 (3+5 x) \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{27} (3 x+2)^{10}+\frac{37}{243} (3 x+2)^9-\frac{7}{216} (3 x+2)^8 \]

[Out]

(-7*(2 + 3*x)^8)/216 + (37*(2 + 3*x)^9)/243 - (2 + 3*x)^10/27

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Rubi [A]  time = 0.021603, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{1}{27} (3 x+2)^{10}+\frac{37}{243} (3 x+2)^9-\frac{7}{216} (3 x+2)^8 \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)*(2 + 3*x)^7*(3 + 5*x),x]

[Out]

(-7*(2 + 3*x)^8)/216 + (37*(2 + 3*x)^9)/243 - (2 + 3*x)^10/27

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+3 x)^7 (3+5 x) \, dx &=\int \left (-\frac{7}{9} (2+3 x)^7+\frac{37}{9} (2+3 x)^8-\frac{10}{9} (2+3 x)^9\right ) \, dx\\ &=-\frac{7}{216} (2+3 x)^8+\frac{37}{243} (2+3 x)^9-\frac{1}{27} (2+3 x)^{10}\\ \end{align*}

Mathematica [A]  time = 0.0021647, size = 53, normalized size = 1.56 \[ -2187 x^{10}-11583 x^9-\frac{207765 x^8}{8}-30942 x^7-18774 x^6-1512 x^5+6468 x^4+\frac{15520 x^3}{3}+1952 x^2+384 x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^7*(3 + 5*x),x]

[Out]

384*x + 1952*x^2 + (15520*x^3)/3 + 6468*x^4 - 1512*x^5 - 18774*x^6 - 30942*x^7 - (207765*x^8)/8 - 11583*x^9 -
2187*x^10

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Maple [A]  time = 0.002, size = 50, normalized size = 1.5 \begin{align*} -2187\,{x}^{10}-11583\,{x}^{9}-{\frac{207765\,{x}^{8}}{8}}-30942\,{x}^{7}-18774\,{x}^{6}-1512\,{x}^{5}+6468\,{x}^{4}+{\frac{15520\,{x}^{3}}{3}}+1952\,{x}^{2}+384\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^7*(3+5*x),x)

[Out]

-2187*x^10-11583*x^9-207765/8*x^8-30942*x^7-18774*x^6-1512*x^5+6468*x^4+15520/3*x^3+1952*x^2+384*x

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Maxima [A]  time = 1.3477, size = 66, normalized size = 1.94 \begin{align*} -2187 \, x^{10} - 11583 \, x^{9} - \frac{207765}{8} \, x^{8} - 30942 \, x^{7} - 18774 \, x^{6} - 1512 \, x^{5} + 6468 \, x^{4} + \frac{15520}{3} \, x^{3} + 1952 \, x^{2} + 384 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x),x, algorithm="maxima")

[Out]

-2187*x^10 - 11583*x^9 - 207765/8*x^8 - 30942*x^7 - 18774*x^6 - 1512*x^5 + 6468*x^4 + 15520/3*x^3 + 1952*x^2 +
384*x

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Fricas [A]  time = 1.416, size = 159, normalized size = 4.68 \begin{align*} -2187 x^{10} - 11583 x^{9} - \frac{207765}{8} x^{8} - 30942 x^{7} - 18774 x^{6} - 1512 x^{5} + 6468 x^{4} + \frac{15520}{3} x^{3} + 1952 x^{2} + 384 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x),x, algorithm="fricas")

[Out]

-2187*x^10 - 11583*x^9 - 207765/8*x^8 - 30942*x^7 - 18774*x^6 - 1512*x^5 + 6468*x^4 + 15520/3*x^3 + 1952*x^2 +
384*x

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Sympy [A]  time = 0.065858, size = 51, normalized size = 1.5 \begin{align*} - 2187 x^{10} - 11583 x^{9} - \frac{207765 x^{8}}{8} - 30942 x^{7} - 18774 x^{6} - 1512 x^{5} + 6468 x^{4} + \frac{15520 x^{3}}{3} + 1952 x^{2} + 384 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**7*(3+5*x),x)

[Out]

-2187*x**10 - 11583*x**9 - 207765*x**8/8 - 30942*x**7 - 18774*x**6 - 1512*x**5 + 6468*x**4 + 15520*x**3/3 + 19
52*x**2 + 384*x

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Giac [A]  time = 1.86379, size = 66, normalized size = 1.94 \begin{align*} -2187 \, x^{10} - 11583 \, x^{9} - \frac{207765}{8} \, x^{8} - 30942 \, x^{7} - 18774 \, x^{6} - 1512 \, x^{5} + 6468 \, x^{4} + \frac{15520}{3} \, x^{3} + 1952 \, x^{2} + 384 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^7*(3+5*x),x, algorithm="giac")

[Out]

-2187*x^10 - 11583*x^9 - 207765/8*x^8 - 30942*x^7 - 18774*x^6 - 1512*x^5 + 6468*x^4 + 15520/3*x^3 + 1952*x^2 +
384*x