3.1239 \(\int (1-2 x)^2 (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=45 \[ \frac{20}{729} (3 x+2)^9-\frac{2}{9} (3 x+2)^8+\frac{13}{27} (3 x+2)^7-\frac{49}{486} (3 x+2)^6 \]

[Out]

(-49*(2 + 3*x)^6)/486 + (13*(2 + 3*x)^7)/27 - (2*(2 + 3*x)^8)/9 + (20*(2 + 3*x)^9)/729

________________________________________________________________________________________

Rubi [A]  time = 0.0228015, 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} \[ \frac{20}{729} (3 x+2)^9-\frac{2}{9} (3 x+2)^8+\frac{13}{27} (3 x+2)^7-\frac{49}{486} (3 x+2)^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-49*(2 + 3*x)^6)/486 + (13*(2 + 3*x)^7)/27 - (2*(2 + 3*x)^8)/9 + (20*(2 + 3*x)^9)/729

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)^5 (3+5 x) \, dx &=\int \left (-\frac{49}{27} (2+3 x)^5+\frac{91}{9} (2+3 x)^6-\frac{16}{3} (2+3 x)^7+\frac{20}{27} (2+3 x)^8\right ) \, dx\\ &=-\frac{49}{486} (2+3 x)^6+\frac{13}{27} (2+3 x)^7-\frac{2}{9} (2+3 x)^8+\frac{20}{729} (2+3 x)^9\\ \end{align*}

Mathematica [A]  time = 0.0022539, size = 48, normalized size = 1.07 \[ 540 x^9+1782 x^8+1917 x^7+\frac{273 x^6}{2}-1218 x^5-770 x^4+\frac{224 x^3}{3}+248 x^2+96 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

96*x + 248*x^2 + (224*x^3)/3 - 770*x^4 - 1218*x^5 + (273*x^6)/2 + 1917*x^7 + 1782*x^8 + 540*x^9

________________________________________________________________________________________

Maple [A]  time = 0., size = 45, normalized size = 1. \begin{align*} 540\,{x}^{9}+1782\,{x}^{8}+1917\,{x}^{7}+{\frac{273\,{x}^{6}}{2}}-1218\,{x}^{5}-770\,{x}^{4}+{\frac{224\,{x}^{3}}{3}}+248\,{x}^{2}+96\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9+1782*x^8+1917*x^7+273/2*x^6-1218*x^5-770*x^4+224/3*x^3+248*x^2+96*x

________________________________________________________________________________________

Maxima [A]  time = 1.09928, size = 59, normalized size = 1.31 \begin{align*} 540 \, x^{9} + 1782 \, x^{8} + 1917 \, x^{7} + \frac{273}{2} \, x^{6} - 1218 \, x^{5} - 770 \, x^{4} + \frac{224}{3} \, x^{3} + 248 \, x^{2} + 96 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x

________________________________________________________________________________________

Fricas [A]  time = 1.35906, size = 126, normalized size = 2.8 \begin{align*} 540 x^{9} + 1782 x^{8} + 1917 x^{7} + \frac{273}{2} x^{6} - 1218 x^{5} - 770 x^{4} + \frac{224}{3} x^{3} + 248 x^{2} + 96 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x

________________________________________________________________________________________

Sympy [A]  time = 0.065173, size = 46, normalized size = 1.02 \begin{align*} 540 x^{9} + 1782 x^{8} + 1917 x^{7} + \frac{273 x^{6}}{2} - 1218 x^{5} - 770 x^{4} + \frac{224 x^{3}}{3} + 248 x^{2} + 96 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x**9 + 1782*x**8 + 1917*x**7 + 273*x**6/2 - 1218*x**5 - 770*x**4 + 224*x**3/3 + 248*x**2 + 96*x

________________________________________________________________________________________

Giac [A]  time = 2.94897, size = 59, normalized size = 1.31 \begin{align*} 540 \, x^{9} + 1782 \, x^{8} + 1917 \, x^{7} + \frac{273}{2} \, x^{6} - 1218 \, x^{5} - 770 \, x^{4} + \frac{224}{3} \, x^{3} + 248 \, x^{2} + 96 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x