3.1158 \(\int (1-2 x) (2+3 x)^8 (3+5 x)^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{25}{486} (3 x+2)^{12}+\frac{65}{297} (3 x+2)^{11}-\frac{4}{45} (3 x+2)^{10}+\frac{7}{729} (3 x+2)^9 \]

[Out]

(7*(2 + 3*x)^9)/729 - (4*(2 + 3*x)^10)/45 + (65*(2 + 3*x)^11)/297 - (25*(2 + 3*x)^12)/486

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Rubi [A]  time = 0.0308512, 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} \[ -\frac{25}{486} (3 x+2)^{12}+\frac{65}{297} (3 x+2)^{11}-\frac{4}{45} (3 x+2)^{10}+\frac{7}{729} (3 x+2)^9 \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)*(2 + 3*x)^8*(3 + 5*x)^2,x]

[Out]

(7*(2 + 3*x)^9)/729 - (4*(2 + 3*x)^10)/45 + (65*(2 + 3*x)^11)/297 - (25*(2 + 3*x)^12)/486

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)^8 (3+5 x)^2 \, dx &=\int \left (\frac{7}{27} (2+3 x)^8-\frac{8}{3} (2+3 x)^9+\frac{65}{9} (2+3 x)^{10}-\frac{50}{27} (2+3 x)^{11}\right ) \, dx\\ &=\frac{7}{729} (2+3 x)^9-\frac{4}{45} (2+3 x)^{10}+\frac{65}{297} (2+3 x)^{11}-\frac{25}{486} (2+3 x)^{12}\\ \end{align*}

Mathematica [A]  time = 0.0025995, size = 69, normalized size = 1.53 \[ -\frac{54675 x^{12}}{2}-\frac{1979235 x^{11}}{11}-\frac{2614194 x^{10}}{5}-869103 x^9-881442 x^8-507600 x^7-71904 x^6+\frac{679008 x^5}{5}+127168 x^4+\frac{173056 x^3}{3}+15360 x^2+2304 x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^8*(3 + 5*x)^2,x]

[Out]

2304*x + 15360*x^2 + (173056*x^3)/3 + 127168*x^4 + (679008*x^5)/5 - 71904*x^6 - 507600*x^7 - 881442*x^8 - 8691
03*x^9 - (2614194*x^10)/5 - (1979235*x^11)/11 - (54675*x^12)/2

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Maple [A]  time = 0.001, size = 60, normalized size = 1.3 \begin{align*} -{\frac{54675\,{x}^{12}}{2}}-{\frac{1979235\,{x}^{11}}{11}}-{\frac{2614194\,{x}^{10}}{5}}-869103\,{x}^{9}-881442\,{x}^{8}-507600\,{x}^{7}-71904\,{x}^{6}+{\frac{679008\,{x}^{5}}{5}}+127168\,{x}^{4}+{\frac{173056\,{x}^{3}}{3}}+15360\,{x}^{2}+2304\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^8*(3+5*x)^2,x)

[Out]

-54675/2*x^12-1979235/11*x^11-2614194/5*x^10-869103*x^9-881442*x^8-507600*x^7-71904*x^6+679008/5*x^5+127168*x^
4+173056/3*x^3+15360*x^2+2304*x

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Maxima [A]  time = 1.15128, size = 80, normalized size = 1.78 \begin{align*} -\frac{54675}{2} \, x^{12} - \frac{1979235}{11} \, x^{11} - \frac{2614194}{5} \, x^{10} - 869103 \, x^{9} - 881442 \, x^{8} - 507600 \, x^{7} - 71904 \, x^{6} + \frac{679008}{5} \, x^{5} + 127168 \, x^{4} + \frac{173056}{3} \, x^{3} + 15360 \, x^{2} + 2304 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="maxima")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x

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Fricas [A]  time = 1.34552, size = 223, normalized size = 4.96 \begin{align*} -\frac{54675}{2} x^{12} - \frac{1979235}{11} x^{11} - \frac{2614194}{5} x^{10} - 869103 x^{9} - 881442 x^{8} - 507600 x^{7} - 71904 x^{6} + \frac{679008}{5} x^{5} + 127168 x^{4} + \frac{173056}{3} x^{3} + 15360 x^{2} + 2304 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="fricas")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x

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Sympy [A]  time = 0.071487, size = 66, normalized size = 1.47 \begin{align*} - \frac{54675 x^{12}}{2} - \frac{1979235 x^{11}}{11} - \frac{2614194 x^{10}}{5} - 869103 x^{9} - 881442 x^{8} - 507600 x^{7} - 71904 x^{6} + \frac{679008 x^{5}}{5} + 127168 x^{4} + \frac{173056 x^{3}}{3} + 15360 x^{2} + 2304 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**8*(3+5*x)**2,x)

[Out]

-54675*x**12/2 - 1979235*x**11/11 - 2614194*x**10/5 - 869103*x**9 - 881442*x**8 - 507600*x**7 - 71904*x**6 + 6
79008*x**5/5 + 127168*x**4 + 173056*x**3/3 + 15360*x**2 + 2304*x

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Giac [A]  time = 1.75778, size = 80, normalized size = 1.78 \begin{align*} -\frac{54675}{2} \, x^{12} - \frac{1979235}{11} \, x^{11} - \frac{2614194}{5} \, x^{10} - 869103 \, x^{9} - 881442 \, x^{8} - 507600 \, x^{7} - 71904 \, x^{6} + \frac{679008}{5} \, x^{5} + 127168 \, x^{4} + \frac{173056}{3} \, x^{3} + 15360 \, x^{2} + 2304 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^8*(3+5*x)^2,x, algorithm="giac")

[Out]

-54675/2*x^12 - 1979235/11*x^11 - 2614194/5*x^10 - 869103*x^9 - 881442*x^8 - 507600*x^7 - 71904*x^6 + 679008/5
*x^5 + 127168*x^4 + 173056/3*x^3 + 15360*x^2 + 2304*x