3.1146 \(\int (1-2 x) (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=34 \[ -\frac{5}{108} (3 x+2)^8+\frac{37}{189} (3 x+2)^7-\frac{7}{162} (3 x+2)^6 \]

[Out]

(-7*(2 + 3*x)^6)/162 + (37*(2 + 3*x)^7)/189 - (5*(2 + 3*x)^8)/108

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Rubi [A]  time = 0.0176127, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ -\frac{5}{108} (3 x+2)^8+\frac{37}{189} (3 x+2)^7-\frac{7}{162} (3 x+2)^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(2 + 3*x)^6)/162 + (37*(2 + 3*x)^7)/189 - (5*(2 + 3*x)^8)/108

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) \, dx &=\int \left (-\frac{7}{9} (2+3 x)^5+\frac{37}{9} (2+3 x)^6-\frac{10}{9} (2+3 x)^7\right ) \, dx\\ &=-\frac{7}{162} (2+3 x)^6+\frac{37}{189} (2+3 x)^7-\frac{5}{108} (2+3 x)^8\\ \end{align*}

Mathematica [A]  time = 0.0020148, size = 47, normalized size = 1.38 \[ -\frac{1215 x^8}{4}-\frac{8343 x^7}{7}-\frac{3627 x^6}{2}-1170 x^5+30 x^4+\frac{1600 x^3}{3}+344 x^2+96 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

96*x + 344*x^2 + (1600*x^3)/3 + 30*x^4 - 1170*x^5 - (3627*x^6)/2 - (8343*x^7)/7 - (1215*x^8)/4

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Maple [A]  time = 0.001, size = 40, normalized size = 1.2 \begin{align*} -{\frac{1215\,{x}^{8}}{4}}-{\frac{8343\,{x}^{7}}{7}}-{\frac{3627\,{x}^{6}}{2}}-1170\,{x}^{5}+30\,{x}^{4}+{\frac{1600\,{x}^{3}}{3}}+344\,{x}^{2}+96\,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),x)

[Out]

-1215/4*x^8-8343/7*x^7-3627/2*x^6-1170*x^5+30*x^4+1600/3*x^3+344*x^2+96*x

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Maxima [A]  time = 1.21518, size = 53, normalized size = 1.56 \begin{align*} -\frac{1215}{4} \, x^{8} - \frac{8343}{7} \, x^{7} - \frac{3627}{2} \, x^{6} - 1170 \, x^{5} + 30 \, x^{4} + \frac{1600}{3} \, x^{3} + 344 \, x^{2} + 96 \, 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),x, algorithm="maxima")

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x

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Fricas [A]  time = 1.36712, size = 120, normalized size = 3.53 \begin{align*} -\frac{1215}{4} x^{8} - \frac{8343}{7} x^{7} - \frac{3627}{2} x^{6} - 1170 x^{5} + 30 x^{4} + \frac{1600}{3} x^{3} + 344 x^{2} + 96 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),x, algorithm="fricas")

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x

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Sympy [A]  time = 0.062024, size = 44, normalized size = 1.29 \begin{align*} - \frac{1215 x^{8}}{4} - \frac{8343 x^{7}}{7} - \frac{3627 x^{6}}{2} - 1170 x^{5} + 30 x^{4} + \frac{1600 x^{3}}{3} + 344 x^{2} + 96 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),x)

[Out]

-1215*x**8/4 - 8343*x**7/7 - 3627*x**6/2 - 1170*x**5 + 30*x**4 + 1600*x**3/3 + 344*x**2 + 96*x

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Giac [A]  time = 3.75204, size = 53, normalized size = 1.56 \begin{align*} -\frac{1215}{4} \, x^{8} - \frac{8343}{7} \, x^{7} - \frac{3627}{2} \, x^{6} - 1170 \, x^{5} + 30 \, x^{4} + \frac{1600}{3} \, x^{3} + 344 \, x^{2} + 96 \, 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),x, algorithm="giac")

[Out]

-1215/4*x^8 - 8343/7*x^7 - 3627/2*x^6 - 1170*x^5 + 30*x^4 + 1600/3*x^3 + 344*x^2 + 96*x