3.1159 \(\int (1-2 x) (2+3 x)^7 (3+5 x)^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{50}{891} (3 x+2)^{11}+\frac{13}{54} (3 x+2)^{10}-\frac{8}{81} (3 x+2)^9+\frac{7}{648} (3 x+2)^8 \]

[Out]

(7*(2 + 3*x)^8)/648 - (8*(2 + 3*x)^9)/81 + (13*(2 + 3*x)^10)/54 - (50*(2 + 3*x)^11)/891

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Rubi [A]  time = 0.0284157, 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} \[ -\frac{50}{891} (3 x+2)^{11}+\frac{13}{54} (3 x+2)^{10}-\frac{8}{81} (3 x+2)^9+\frac{7}{648} (3 x+2)^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^8)/648 - (8*(2 + 3*x)^9)/81 + (13*(2 + 3*x)^10)/54 - (50*(2 + 3*x)^11)/891

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)^2 \, dx &=\int \left (\frac{7}{27} (2+3 x)^7-\frac{8}{3} (2+3 x)^8+\frac{65}{9} (2+3 x)^9-\frac{50}{27} (2+3 x)^{10}\right ) \, dx\\ &=\frac{7}{648} (2+3 x)^8-\frac{8}{81} (2+3 x)^9+\frac{13}{54} (2+3 x)^{10}-\frac{50}{891} (2+3 x)^{11}\\ \end{align*}

Mathematica [A]  time = 0.0022983, size = 62, normalized size = 1.38 \[ -\frac{109350 x^{11}}{11}-\frac{117369 x^{10}}{2}-150174 x^9-\frac{1706265 x^8}{8}-173286 x^7-62622 x^6+21336 x^5+38804 x^4+\frac{66080 x^3}{3}+6816 x^2+1152 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

1152*x + 6816*x^2 + (66080*x^3)/3 + 38804*x^4 + 21336*x^5 - 62622*x^6 - 173286*x^7 - (1706265*x^8)/8 - 150174*
x^9 - (117369*x^10)/2 - (109350*x^11)/11

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Maple [A]  time = 0.001, size = 55, normalized size = 1.2 \begin{align*} -{\frac{109350\,{x}^{11}}{11}}-{\frac{117369\,{x}^{10}}{2}}-150174\,{x}^{9}-{\frac{1706265\,{x}^{8}}{8}}-173286\,{x}^{7}-62622\,{x}^{6}+21336\,{x}^{5}+38804\,{x}^{4}+{\frac{66080\,{x}^{3}}{3}}+6816\,{x}^{2}+1152\,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)^2,x)

[Out]

-109350/11*x^11-117369/2*x^10-150174*x^9-1706265/8*x^8-173286*x^7-62622*x^6+21336*x^5+38804*x^4+66080/3*x^3+68
16*x^2+1152*x

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Maxima [A]  time = 1.25851, size = 73, normalized size = 1.62 \begin{align*} -\frac{109350}{11} \, x^{11} - \frac{117369}{2} \, x^{10} - 150174 \, x^{9} - \frac{1706265}{8} \, x^{8} - 173286 \, x^{7} - 62622 \, x^{6} + 21336 \, x^{5} + 38804 \, x^{4} + \frac{66080}{3} \, x^{3} + 6816 \, x^{2} + 1152 \, 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)^2,x, algorithm="maxima")

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x

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Fricas [A]  time = 1.292, size = 196, normalized size = 4.36 \begin{align*} -\frac{109350}{11} x^{11} - \frac{117369}{2} x^{10} - 150174 x^{9} - \frac{1706265}{8} x^{8} - 173286 x^{7} - 62622 x^{6} + 21336 x^{5} + 38804 x^{4} + \frac{66080}{3} x^{3} + 6816 x^{2} + 1152 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)^2,x, algorithm="fricas")

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x

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Sympy [A]  time = 0.06802, size = 60, normalized size = 1.33 \begin{align*} - \frac{109350 x^{11}}{11} - \frac{117369 x^{10}}{2} - 150174 x^{9} - \frac{1706265 x^{8}}{8} - 173286 x^{7} - 62622 x^{6} + 21336 x^{5} + 38804 x^{4} + \frac{66080 x^{3}}{3} + 6816 x^{2} + 1152 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)**2,x)

[Out]

-109350*x**11/11 - 117369*x**10/2 - 150174*x**9 - 1706265*x**8/8 - 173286*x**7 - 62622*x**6 + 21336*x**5 + 388
04*x**4 + 66080*x**3/3 + 6816*x**2 + 1152*x

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Giac [A]  time = 2.45369, size = 73, normalized size = 1.62 \begin{align*} -\frac{109350}{11} \, x^{11} - \frac{117369}{2} \, x^{10} - 150174 \, x^{9} - \frac{1706265}{8} \, x^{8} - 173286 \, x^{7} - 62622 \, x^{6} + 21336 \, x^{5} + 38804 \, x^{4} + \frac{66080}{3} \, x^{3} + 6816 \, x^{2} + 1152 \, 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)^2,x, algorithm="giac")

[Out]

-109350/11*x^11 - 117369/2*x^10 - 150174*x^9 - 1706265/8*x^8 - 173286*x^7 - 62622*x^6 + 21336*x^5 + 38804*x^4
+ 66080/3*x^3 + 6816*x^2 + 1152*x