3.1161 \(\int (1-2 x) (2+3 x)^5 (3+5 x)^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{50}{729} (3 x+2)^9+\frac{65}{216} (3 x+2)^8-\frac{8}{63} (3 x+2)^7+\frac{7}{486} (3 x+2)^6 \]

[Out]

(7*(2 + 3*x)^6)/486 - (8*(2 + 3*x)^7)/63 + (65*(2 + 3*x)^8)/216 - (50*(2 + 3*x)^9)/729

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Rubi [A]  time = 0.0217211, 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} \[ -\frac{50}{729} (3 x+2)^9+\frac{65}{216} (3 x+2)^8-\frac{8}{63} (3 x+2)^7+\frac{7}{486} (3 x+2)^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^6)/486 - (8*(2 + 3*x)^7)/63 + (65*(2 + 3*x)^8)/216 - (50*(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+3 x)^5 (3+5 x)^2 \, dx &=\int \left (\frac{7}{27} (2+3 x)^5-\frac{8}{3} (2+3 x)^6+\frac{65}{9} (2+3 x)^7-\frac{50}{27} (2+3 x)^8\right ) \, dx\\ &=\frac{7}{486} (2+3 x)^6-\frac{8}{63} (2+3 x)^7+\frac{65}{216} (2+3 x)^8-\frac{50}{729} (2+3 x)^9\\ \end{align*}

Mathematica [A]  time = 0.0020228, size = 52, normalized size = 1.16 \[ -1350 x^9-\frac{49005 x^8}{8}-\frac{79434 x^7}{7}-\frac{20631 x^6}{2}-3390 x^5+2090 x^4+\frac{8240 x^3}{3}+1272 x^2+288 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

288*x + 1272*x^2 + (8240*x^3)/3 + 2090*x^4 - 3390*x^5 - (20631*x^6)/2 - (79434*x^7)/7 - (49005*x^8)/8 - 1350*x
^9

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Maple [A]  time = 0.002, size = 45, normalized size = 1. \begin{align*} -1350\,{x}^{9}-{\frac{49005\,{x}^{8}}{8}}-{\frac{79434\,{x}^{7}}{7}}-{\frac{20631\,{x}^{6}}{2}}-3390\,{x}^{5}+2090\,{x}^{4}+{\frac{8240\,{x}^{3}}{3}}+1272\,{x}^{2}+288\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1350*x^9-49005/8*x^8-79434/7*x^7-20631/2*x^6-3390*x^5+2090*x^4+8240/3*x^3+1272*x^2+288*x

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Maxima [A]  time = 1.14581, size = 59, normalized size = 1.31 \begin{align*} -1350 \, x^{9} - \frac{49005}{8} \, x^{8} - \frac{79434}{7} \, x^{7} - \frac{20631}{2} \, x^{6} - 3390 \, x^{5} + 2090 \, x^{4} + \frac{8240}{3} \, x^{3} + 1272 \, x^{2} + 288 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x

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Fricas [A]  time = 1.29791, size = 144, normalized size = 3.2 \begin{align*} -1350 x^{9} - \frac{49005}{8} x^{8} - \frac{79434}{7} x^{7} - \frac{20631}{2} x^{6} - 3390 x^{5} + 2090 x^{4} + \frac{8240}{3} x^{3} + 1272 x^{2} + 288 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x

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Sympy [A]  time = 0.065484, size = 49, normalized size = 1.09 \begin{align*} - 1350 x^{9} - \frac{49005 x^{8}}{8} - \frac{79434 x^{7}}{7} - \frac{20631 x^{6}}{2} - 3390 x^{5} + 2090 x^{4} + \frac{8240 x^{3}}{3} + 1272 x^{2} + 288 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1350*x**9 - 49005*x**8/8 - 79434*x**7/7 - 20631*x**6/2 - 3390*x**5 + 2090*x**4 + 8240*x**3/3 + 1272*x**2 + 28
8*x

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Giac [A]  time = 3.79763, size = 59, normalized size = 1.31 \begin{align*} -1350 \, x^{9} - \frac{49005}{8} \, x^{8} - \frac{79434}{7} \, x^{7} - \frac{20631}{2} \, x^{6} - 3390 \, x^{5} + 2090 \, x^{4} + \frac{8240}{3} \, x^{3} + 1272 \, x^{2} + 288 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1350*x^9 - 49005/8*x^8 - 79434/7*x^7 - 20631/2*x^6 - 3390*x^5 + 2090*x^4 + 8240/3*x^3 + 1272*x^2 + 288*x