3.1175 \(\int (1-2 x) (2+3 x)^8 (3+5 x)^3 \, dx\)

Optimal. Leaf size=56 \[ -\frac{250 (3 x+2)^{13}}{3159}+\frac{1025 (3 x+2)^{12}}{2916}-\frac{185}{891} (3 x+2)^{11}+\frac{107 (3 x+2)^{10}}{2430}-\frac{7 (3 x+2)^9}{2187} \]

[Out]

(-7*(2 + 3*x)^9)/2187 + (107*(2 + 3*x)^10)/2430 - (185*(2 + 3*x)^11)/891 + (1025*(2 + 3*x)^12)/2916 - (250*(2
+ 3*x)^13)/3159

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Rubi [A]  time = 0.0320184, antiderivative size = 56, 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} \[ -\frac{250 (3 x+2)^{13}}{3159}+\frac{1025 (3 x+2)^{12}}{2916}-\frac{185}{891} (3 x+2)^{11}+\frac{107 (3 x+2)^{10}}{2430}-\frac{7 (3 x+2)^9}{2187} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 + 3*x)^9)/2187 + (107*(2 + 3*x)^10)/2430 - (185*(2 + 3*x)^11)/891 + (1025*(2 + 3*x)^12)/2916 - (250*(2
+ 3*x)^13)/3159

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)^3 \, dx &=\int \left (-\frac{7}{81} (2+3 x)^8+\frac{107}{81} (2+3 x)^9-\frac{185}{27} (2+3 x)^{10}+\frac{1025}{81} (2+3 x)^{11}-\frac{250}{81} (2+3 x)^{12}\right ) \, dx\\ &=-\frac{7 (2+3 x)^9}{2187}+\frac{107 (2+3 x)^{10}}{2430}-\frac{185}{891} (2+3 x)^{11}+\frac{1025 (2+3 x)^{12}}{2916}-\frac{250 (2+3 x)^{13}}{3159}\\ \end{align*}

Mathematica [A]  time = 0.0024004, size = 74, normalized size = 1.32 \[ -\frac{1640250 x^{13}}{13}-\frac{3626775 x^{12}}{4}-\frac{32079645 x^{11}}{11}-\frac{54794799 x^{10}}{10}-6524829 x^9-4865076 x^8-1830960 x^7+350128 x^6+\frac{4580384 x^5}{5}+597824 x^4+224256 x^3+51840 x^2+6912 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

6912*x + 51840*x^2 + 224256*x^3 + 597824*x^4 + (4580384*x^5)/5 + 350128*x^6 - 1830960*x^7 - 4865076*x^8 - 6524
829*x^9 - (54794799*x^10)/10 - (32079645*x^11)/11 - (3626775*x^12)/4 - (1640250*x^13)/13

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Maple [A]  time = 0., size = 65, normalized size = 1.2 \begin{align*} -{\frac{1640250\,{x}^{13}}{13}}-{\frac{3626775\,{x}^{12}}{4}}-{\frac{32079645\,{x}^{11}}{11}}-{\frac{54794799\,{x}^{10}}{10}}-6524829\,{x}^{9}-4865076\,{x}^{8}-1830960\,{x}^{7}+350128\,{x}^{6}+{\frac{4580384\,{x}^{5}}{5}}+597824\,{x}^{4}+224256\,{x}^{3}+51840\,{x}^{2}+6912\,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)^3,x)

[Out]

-1640250/13*x^13-3626775/4*x^12-32079645/11*x^11-54794799/10*x^10-6524829*x^9-4865076*x^8-1830960*x^7+350128*x
^6+4580384/5*x^5+597824*x^4+224256*x^3+51840*x^2+6912*x

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Maxima [A]  time = 1.11687, size = 86, normalized size = 1.54 \begin{align*} -\frac{1640250}{13} \, x^{13} - \frac{3626775}{4} \, x^{12} - \frac{32079645}{11} \, x^{11} - \frac{54794799}{10} \, x^{10} - 6524829 \, x^{9} - 4865076 \, x^{8} - 1830960 \, x^{7} + 350128 \, x^{6} + \frac{4580384}{5} \, x^{5} + 597824 \, x^{4} + 224256 \, x^{3} + 51840 \, x^{2} + 6912 \, 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)^3,x, algorithm="maxima")

[Out]

-1640250/13*x^13 - 3626775/4*x^12 - 32079645/11*x^11 - 54794799/10*x^10 - 6524829*x^9 - 4865076*x^8 - 1830960*
x^7 + 350128*x^6 + 4580384/5*x^5 + 597824*x^4 + 224256*x^3 + 51840*x^2 + 6912*x

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Fricas [A]  time = 1.61024, size = 258, normalized size = 4.61 \begin{align*} -\frac{1640250}{13} x^{13} - \frac{3626775}{4} x^{12} - \frac{32079645}{11} x^{11} - \frac{54794799}{10} x^{10} - 6524829 x^{9} - 4865076 x^{8} - 1830960 x^{7} + 350128 x^{6} + \frac{4580384}{5} x^{5} + 597824 x^{4} + 224256 x^{3} + 51840 x^{2} + 6912 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)^3,x, algorithm="fricas")

[Out]

-1640250/13*x^13 - 3626775/4*x^12 - 32079645/11*x^11 - 54794799/10*x^10 - 6524829*x^9 - 4865076*x^8 - 1830960*
x^7 + 350128*x^6 + 4580384/5*x^5 + 597824*x^4 + 224256*x^3 + 51840*x^2 + 6912*x

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Sympy [A]  time = 0.075412, size = 71, normalized size = 1.27 \begin{align*} - \frac{1640250 x^{13}}{13} - \frac{3626775 x^{12}}{4} - \frac{32079645 x^{11}}{11} - \frac{54794799 x^{10}}{10} - 6524829 x^{9} - 4865076 x^{8} - 1830960 x^{7} + 350128 x^{6} + \frac{4580384 x^{5}}{5} + 597824 x^{4} + 224256 x^{3} + 51840 x^{2} + 6912 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)**3,x)

[Out]

-1640250*x**13/13 - 3626775*x**12/4 - 32079645*x**11/11 - 54794799*x**10/10 - 6524829*x**9 - 4865076*x**8 - 18
30960*x**7 + 350128*x**6 + 4580384*x**5/5 + 597824*x**4 + 224256*x**3 + 51840*x**2 + 6912*x

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Giac [A]  time = 3.18005, size = 86, normalized size = 1.54 \begin{align*} -\frac{1640250}{13} \, x^{13} - \frac{3626775}{4} \, x^{12} - \frac{32079645}{11} \, x^{11} - \frac{54794799}{10} \, x^{10} - 6524829 \, x^{9} - 4865076 \, x^{8} - 1830960 \, x^{7} + 350128 \, x^{6} + \frac{4580384}{5} \, x^{5} + 597824 \, x^{4} + 224256 \, x^{3} + 51840 \, x^{2} + 6912 \, 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)^3,x, algorithm="giac")

[Out]

-1640250/13*x^13 - 3626775/4*x^12 - 32079645/11*x^11 - 54794799/10*x^10 - 6524829*x^9 - 4865076*x^8 - 1830960*
x^7 + 350128*x^6 + 4580384/5*x^5 + 597824*x^4 + 224256*x^3 + 51840*x^2 + 6912*x