3.1338 \(\int (1-2 x)^3 (2+3 x)^8 (3+5 x) \, dx\)

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

[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

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Rubi [A]  time = 0.0304658, 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} \[ -\frac{40 (3 x+2)^{13}}{3159}+\frac{107}{729} (3 x+2)^{12}-\frac{518}{891} (3 x+2)^{11}+\frac{2009 (3 x+2)^{10}}{2430}-\frac{343 (3 x+2)^9}{2187} \]

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*}

Mathematica [A]  time = 0.0026748, size = 72, normalized size = 1.29 \[ -\frac{262440 x^{13}}{13}-96957 x^{12}-\frac{1966842 x^{11}}{11}-\frac{1290573 x^{10}}{10}+38331 x^9+128412 x^8+67248 x^7-17456 x^6-\frac{159712 x^5}{5}-9216 x^4+3328 x^3+2944 x^2+768 x \]

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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Maple [A]  time = 0.002, size = 65, normalized size = 1.2 \begin{align*} -{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(2+3*x)^8*(3+5*x),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 = 1.07067, size = 86, normalized size = 1.54 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.0598, size = 231, normalized size = 4.12 \begin{align*} -\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{align*}

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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Sympy [A]  time = 0.073841, size = 70, normalized size = 1.25 \begin{align*} - \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{align*}

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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Giac [A]  time = 3.43355, size = 86, normalized size = 1.54 \begin{align*} -\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{align*}

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