3.1176 \(\int (1-2 x) (2+3 x)^7 (3+5 x)^3 \, dx\)

Optimal. Leaf size=56 \[ -\frac{125 (3 x+2)^{12}}{1458}+\frac{1025 (3 x+2)^{11}}{2673}-\frac{37}{162} (3 x+2)^{10}+\frac{107 (3 x+2)^9}{2187}-\frac{7 (3 x+2)^8}{1944} \]

[Out]

(-7*(2 + 3*x)^8)/1944 + (107*(2 + 3*x)^9)/2187 - (37*(2 + 3*x)^10)/162 + (1025*(2 + 3*x)^11)/2673 - (125*(2 +
3*x)^12)/1458

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Rubi [A]  time = 0.028159, 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} \[ -\frac{125 (3 x+2)^{12}}{1458}+\frac{1025 (3 x+2)^{11}}{2673}-\frac{37}{162} (3 x+2)^{10}+\frac{107 (3 x+2)^9}{2187}-\frac{7 (3 x+2)^8}{1944} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 + 3*x)^8)/1944 + (107*(2 + 3*x)^9)/2187 - (37*(2 + 3*x)^10)/162 + (1025*(2 + 3*x)^11)/2673 - (125*(2 +
3*x)^12)/1458

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)^3 \, dx &=\int \left (-\frac{7}{81} (2+3 x)^7+\frac{107}{81} (2+3 x)^8-\frac{185}{27} (2+3 x)^9+\frac{1025}{81} (2+3 x)^{10}-\frac{250}{81} (2+3 x)^{11}\right ) \, dx\\ &=-\frac{7 (2+3 x)^8}{1944}+\frac{107 (2+3 x)^9}{2187}-\frac{37}{162} (2+3 x)^{10}+\frac{1025 (2+3 x)^{11}}{2673}-\frac{125 (2+3 x)^{12}}{1458}\\ \end{align*}

Mathematica [A]  time = 0.0023952, size = 67, normalized size = 1.2 \[ -\frac{91125 x^{12}}{2}-\frac{3262275 x^{11}}{11}-\frac{1703673 x^{10}}{2}-1398447 x^9-\frac{11183805 x^8}{8}-788238 x^7-98966 x^6+219224 x^5+199012 x^4+88800 x^3+23328 x^2+3456 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

3456*x + 23328*x^2 + 88800*x^3 + 199012*x^4 + 219224*x^5 - 98966*x^6 - 788238*x^7 - (11183805*x^8)/8 - 1398447
*x^9 - (1703673*x^10)/2 - (3262275*x^11)/11 - (91125*x^12)/2

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Maple [A]  time = 0.001, size = 60, normalized size = 1.1 \begin{align*} -{\frac{91125\,{x}^{12}}{2}}-{\frac{3262275\,{x}^{11}}{11}}-{\frac{1703673\,{x}^{10}}{2}}-1398447\,{x}^{9}-{\frac{11183805\,{x}^{8}}{8}}-788238\,{x}^{7}-98966\,{x}^{6}+219224\,{x}^{5}+199012\,{x}^{4}+88800\,{x}^{3}+23328\,{x}^{2}+3456\,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)^3,x)

[Out]

-91125/2*x^12-3262275/11*x^11-1703673/2*x^10-1398447*x^9-11183805/8*x^8-788238*x^7-98966*x^6+219224*x^5+199012
*x^4+88800*x^3+23328*x^2+3456*x

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Maxima [A]  time = 1.11678, size = 80, normalized size = 1.43 \begin{align*} -\frac{91125}{2} \, x^{12} - \frac{3262275}{11} \, x^{11} - \frac{1703673}{2} \, x^{10} - 1398447 \, x^{9} - \frac{11183805}{8} \, x^{8} - 788238 \, x^{7} - 98966 \, x^{6} + 219224 \, x^{5} + 199012 \, x^{4} + 88800 \, x^{3} + 23328 \, x^{2} + 3456 \, 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)^3,x, algorithm="maxima")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x

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Fricas [A]  time = 1.52814, size = 223, normalized size = 3.98 \begin{align*} -\frac{91125}{2} x^{12} - \frac{3262275}{11} x^{11} - \frac{1703673}{2} x^{10} - 1398447 x^{9} - \frac{11183805}{8} x^{8} - 788238 x^{7} - 98966 x^{6} + 219224 x^{5} + 199012 x^{4} + 88800 x^{3} + 23328 x^{2} + 3456 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)^3,x, algorithm="fricas")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x

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Sympy [A]  time = 0.073014, size = 65, normalized size = 1.16 \begin{align*} - \frac{91125 x^{12}}{2} - \frac{3262275 x^{11}}{11} - \frac{1703673 x^{10}}{2} - 1398447 x^{9} - \frac{11183805 x^{8}}{8} - 788238 x^{7} - 98966 x^{6} + 219224 x^{5} + 199012 x^{4} + 88800 x^{3} + 23328 x^{2} + 3456 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)**3,x)

[Out]

-91125*x**12/2 - 3262275*x**11/11 - 1703673*x**10/2 - 1398447*x**9 - 11183805*x**8/8 - 788238*x**7 - 98966*x**
6 + 219224*x**5 + 199012*x**4 + 88800*x**3 + 23328*x**2 + 3456*x

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Giac [A]  time = 2.39141, size = 80, normalized size = 1.43 \begin{align*} -\frac{91125}{2} \, x^{12} - \frac{3262275}{11} \, x^{11} - \frac{1703673}{2} \, x^{10} - 1398447 \, x^{9} - \frac{11183805}{8} \, x^{8} - 788238 \, x^{7} - 98966 \, x^{6} + 219224 \, x^{5} + 199012 \, x^{4} + 88800 \, x^{3} + 23328 \, x^{2} + 3456 \, 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)^3,x, algorithm="giac")

[Out]

-91125/2*x^12 - 3262275/11*x^11 - 1703673/2*x^10 - 1398447*x^9 - 11183805/8*x^8 - 788238*x^7 - 98966*x^6 + 219
224*x^5 + 199012*x^4 + 88800*x^3 + 23328*x^2 + 3456*x