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

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

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 74, normalized size = 1.32 \begin {gather*} -\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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1-2 x) (2+3 x)^8 (3+5 x)^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.12, size = 64, normalized size = 1.14 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.23, size = 64, normalized size = 1.14 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 65, normalized size = 1.16 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3*x+2)^8*(5*x+3)^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 = 0.52, size = 64, normalized size = 1.14 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.08, size = 64, normalized size = 1.14 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)*(3*x + 2)^8*(5*x + 3)^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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sympy [A]  time = 0.08, size = 71, normalized size = 1.27 \begin {gather*} - \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 {gather*}

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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