3.1178 \(\int (1-2 x) (2+3 x)^5 (3+5 x)^3 \, dx\)

Optimal. Leaf size=56 \[ -\frac{25}{243} (3 x+2)^{10}+\frac{1025 (3 x+2)^9}{2187}-\frac{185}{648} (3 x+2)^8+\frac{107 (3 x+2)^7}{1701}-\frac{7 (3 x+2)^6}{1458} \]

[Out]

(-7*(2 + 3*x)^6)/1458 + (107*(2 + 3*x)^7)/1701 - (185*(2 + 3*x)^8)/648 + (1025*(2 + 3*x)^9)/2187 - (25*(2 + 3*
x)^10)/243

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Rubi [A]  time = 0.0277132, 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} \[ -\frac{25}{243} (3 x+2)^{10}+\frac{1025 (3 x+2)^9}{2187}-\frac{185}{648} (3 x+2)^8+\frac{107 (3 x+2)^7}{1701}-\frac{7 (3 x+2)^6}{1458} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 + 3*x)^6)/1458 + (107*(2 + 3*x)^7)/1701 - (185*(2 + 3*x)^8)/648 + (1025*(2 + 3*x)^9)/2187 - (25*(2 + 3*
x)^10)/243

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)^5 (3+5 x)^3 \, dx &=\int \left (-\frac{7}{81} (2+3 x)^5+\frac{107}{81} (2+3 x)^6-\frac{185}{27} (2+3 x)^7+\frac{1025}{81} (2+3 x)^8-\frac{250}{81} (2+3 x)^9\right ) \, dx\\ &=-\frac{7 (2+3 x)^6}{1458}+\frac{107 (2+3 x)^7}{1701}-\frac{185}{648} (2+3 x)^8+\frac{1025 (2+3 x)^9}{2187}-\frac{25}{243} (2+3 x)^{10}\\ \end{align*}

Mathematica [A]  time = 0.0021216, size = 55, normalized size = 0.98 \[ -6075 x^{10}-31275 x^9-\frac{544185 x^8}{8}-\frac{547767 x^7}{7}-\frac{90143 x^6}{2}-1810 x^5+16570 x^4+12480 x^3+4536 x^2+864 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

864*x + 4536*x^2 + 12480*x^3 + 16570*x^4 - 1810*x^5 - (90143*x^6)/2 - (547767*x^7)/7 - (544185*x^8)/8 - 31275*
x^9 - 6075*x^10

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Maple [A]  time = 0.001, size = 50, normalized size = 0.9 \begin{align*} -6075\,{x}^{10}-31275\,{x}^{9}-{\frac{544185\,{x}^{8}}{8}}-{\frac{547767\,{x}^{7}}{7}}-{\frac{90143\,{x}^{6}}{2}}-1810\,{x}^{5}+16570\,{x}^{4}+12480\,{x}^{3}+4536\,{x}^{2}+864\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x^10-31275*x^9-544185/8*x^8-547767/7*x^7-90143/2*x^6-1810*x^5+16570*x^4+12480*x^3+4536*x^2+864*x

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Maxima [A]  time = 1.258, size = 66, normalized size = 1.18 \begin{align*} -6075 \, x^{10} - 31275 \, x^{9} - \frac{544185}{8} \, x^{8} - \frac{547767}{7} \, x^{7} - \frac{90143}{2} \, x^{6} - 1810 \, x^{5} + 16570 \, x^{4} + 12480 \, x^{3} + 4536 \, x^{2} + 864 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x

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Fricas [A]  time = 1.54453, size = 165, normalized size = 2.95 \begin{align*} -6075 x^{10} - 31275 x^{9} - \frac{544185}{8} x^{8} - \frac{547767}{7} x^{7} - \frac{90143}{2} x^{6} - 1810 x^{5} + 16570 x^{4} + 12480 x^{3} + 4536 x^{2} + 864 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x

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Sympy [A]  time = 0.067523, size = 53, normalized size = 0.95 \begin{align*} - 6075 x^{10} - 31275 x^{9} - \frac{544185 x^{8}}{8} - \frac{547767 x^{7}}{7} - \frac{90143 x^{6}}{2} - 1810 x^{5} + 16570 x^{4} + 12480 x^{3} + 4536 x^{2} + 864 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x**10 - 31275*x**9 - 544185*x**8/8 - 547767*x**7/7 - 90143*x**6/2 - 1810*x**5 + 16570*x**4 + 12480*x**3
+ 4536*x**2 + 864*x

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Giac [A]  time = 2.58518, size = 66, normalized size = 1.18 \begin{align*} -6075 \, x^{10} - 31275 \, x^{9} - \frac{544185}{8} \, x^{8} - \frac{547767}{7} \, x^{7} - \frac{90143}{2} \, x^{6} - 1810 \, x^{5} + 16570 \, x^{4} + 12480 \, x^{3} + 4536 \, x^{2} + 864 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x^10 - 31275*x^9 - 544185/8*x^8 - 547767/7*x^7 - 90143/2*x^6 - 1810*x^5 + 16570*x^4 + 12480*x^3 + 4536*x
^2 + 864*x