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

Optimal. Leaf size=56 \[ -\frac {40 (3 x+2)^{13}}{3159}+\frac {107}{729} (3 x+2)^{12}-\frac {518}{891} (3 x+2)^{11}+\frac {2009 (3 x+2)^{10}}{2430}-\frac {343 (3 x+2)^9}{2187} \]

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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 {40 (3 x+2)^{13}}{3159}+\frac {107}{729} (3 x+2)^{12}-\frac {518}{891} (3 x+2)^{11}+\frac {2009 (3 x+2)^{10}}{2430}-\frac {343 (3 x+2)^9}{2187} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-343*(2 + 3*x)^9)/2187 + (2009*(2 + 3*x)^10)/2430 - (518*(2 + 3*x)^11)/891 + (107*(2 + 3*x)^12)/729 - (40*(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)^3 (2+3 x)^8 (3+5 x) \, dx &=\int \left (-\frac {343}{81} (2+3 x)^8+\frac {2009}{81} (2+3 x)^9-\frac {518}{27} (2+3 x)^{10}+\frac {428}{81} (2+3 x)^{11}-\frac {40}{81} (2+3 x)^{12}\right ) \, dx\\ &=-\frac {343 (2+3 x)^9}{2187}+\frac {2009 (2+3 x)^{10}}{2430}-\frac {518}{891} (2+3 x)^{11}+\frac {107}{729} (2+3 x)^{12}-\frac {40 (2+3 x)^{13}}{3159}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 72, normalized size = 1.29 \begin {gather*} -\frac {262440 x^{13}}{13}-96957 x^{12}-\frac {1966842 x^{11}}{11}-\frac {1290573 x^{10}}{10}+38331 x^9+128412 x^8+67248 x^7-17456 x^6-\frac {159712 x^5}{5}-9216 x^4+3328 x^3+2944 x^2+768 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

768*x + 2944*x^2 + 3328*x^3 - 9216*x^4 - (159712*x^5)/5 - 17456*x^6 + 67248*x^7 + 128412*x^8 + 38331*x^9 - (12
90573*x^10)/10 - (1966842*x^11)/11 - 96957*x^12 - (262440*x^13)/13

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.06, size = 64, normalized size = 1.14 \begin {gather*} -\frac {262440}{13} x^{13} - 96957 x^{12} - \frac {1966842}{11} x^{11} - \frac {1290573}{10} x^{10} + 38331 x^{9} + 128412 x^{8} + 67248 x^{7} - 17456 x^{6} - \frac {159712}{5} x^{5} - 9216 x^{4} + 3328 x^{3} + 2944 x^{2} + 768 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-262440/13*x^13 - 96957*x^12 - 1966842/11*x^11 - 1290573/10*x^10 + 38331*x^9 + 128412*x^8 + 67248*x^7 - 17456*
x^6 - 159712/5*x^5 - 9216*x^4 + 3328*x^3 + 2944*x^2 + 768*x

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giac [A]  time = 0.95, size = 64, normalized size = 1.14 \begin {gather*} -\frac {262440}{13} \, x^{13} - 96957 \, x^{12} - \frac {1966842}{11} \, x^{11} - \frac {1290573}{10} \, x^{10} + 38331 \, x^{9} + 128412 \, x^{8} + 67248 \, x^{7} - 17456 \, x^{6} - \frac {159712}{5} \, x^{5} - 9216 \, x^{4} + 3328 \, x^{3} + 2944 \, x^{2} + 768 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-262440/13*x^13 - 96957*x^12 - 1966842/11*x^11 - 1290573/10*x^10 + 38331*x^9 + 128412*x^8 + 67248*x^7 - 17456*
x^6 - 159712/5*x^5 - 9216*x^4 + 3328*x^3 + 2944*x^2 + 768*x

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maple [A]  time = 0.00, size = 65, normalized size = 1.16 \begin {gather*} -\frac {262440}{13} x^{13}-96957 x^{12}-\frac {1966842}{11} x^{11}-\frac {1290573}{10} x^{10}+38331 x^{9}+128412 x^{8}+67248 x^{7}-17456 x^{6}-\frac {159712}{5} x^{5}-9216 x^{4}+3328 x^{3}+2944 x^{2}+768 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-262440/13*x^13-96957*x^12-1966842/11*x^11-1290573/10*x^10+38331*x^9+128412*x^8+67248*x^7-17456*x^6-159712/5*x
^5-9216*x^4+3328*x^3+2944*x^2+768*x

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maxima [A]  time = 0.73, size = 64, normalized size = 1.14 \begin {gather*} -\frac {262440}{13} \, x^{13} - 96957 \, x^{12} - \frac {1966842}{11} \, x^{11} - \frac {1290573}{10} \, x^{10} + 38331 \, x^{9} + 128412 \, x^{8} + 67248 \, x^{7} - 17456 \, x^{6} - \frac {159712}{5} \, x^{5} - 9216 \, x^{4} + 3328 \, x^{3} + 2944 \, x^{2} + 768 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-262440/13*x^13 - 96957*x^12 - 1966842/11*x^11 - 1290573/10*x^10 + 38331*x^9 + 128412*x^8 + 67248*x^7 - 17456*
x^6 - 159712/5*x^5 - 9216*x^4 + 3328*x^3 + 2944*x^2 + 768*x

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mupad [B]  time = 0.08, size = 64, normalized size = 1.14 \begin {gather*} -\frac {262440\,x^{13}}{13}-96957\,x^{12}-\frac {1966842\,x^{11}}{11}-\frac {1290573\,x^{10}}{10}+38331\,x^9+128412\,x^8+67248\,x^7-17456\,x^6-\frac {159712\,x^5}{5}-9216\,x^4+3328\,x^3+2944\,x^2+768\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

768*x + 2944*x^2 + 3328*x^3 - 9216*x^4 - (159712*x^5)/5 - 17456*x^6 + 67248*x^7 + 128412*x^8 + 38331*x^9 - (12
90573*x^10)/10 - (1966842*x^11)/11 - 96957*x^12 - (262440*x^13)/13

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sympy [A]  time = 0.08, size = 70, normalized size = 1.25 \begin {gather*} - \frac {262440 x^{13}}{13} - 96957 x^{12} - \frac {1966842 x^{11}}{11} - \frac {1290573 x^{10}}{10} + 38331 x^{9} + 128412 x^{8} + 67248 x^{7} - 17456 x^{6} - \frac {159712 x^{5}}{5} - 9216 x^{4} + 3328 x^{3} + 2944 x^{2} + 768 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-262440*x**13/13 - 96957*x**12 - 1966842*x**11/11 - 1290573*x**10/10 + 38331*x**9 + 128412*x**8 + 67248*x**7 -
17456*x**6 - 159712*x**5/5 - 9216*x**4 + 3328*x**3 + 2944*x**2 + 768*x

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