3.1147 \(\int (1-2 x) (2+3 x)^4 (3+5 x) \, dx\)

Optimal. Leaf size=34 \[ -\frac{10}{189} (3 x+2)^7+\frac{37}{162} (3 x+2)^6-\frac{7}{135} (3 x+2)^5 \]

[Out]

(-7*(2 + 3*x)^5)/135 + (37*(2 + 3*x)^6)/162 - (10*(2 + 3*x)^7)/189

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Rubi [A]  time = 0.0165138, 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} \[ -\frac{10}{189} (3 x+2)^7+\frac{37}{162} (3 x+2)^6-\frac{7}{135} (3 x+2)^5 \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-7*(2 + 3*x)^5)/135 + (37*(2 + 3*x)^6)/162 - (10*(2 + 3*x)^7)/189

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)^4 (3+5 x) \, dx &=\int \left (-\frac{7}{9} (2+3 x)^4+\frac{37}{9} (2+3 x)^5-\frac{10}{9} (2+3 x)^6\right ) \, dx\\ &=-\frac{7}{135} (2+3 x)^5+\frac{37}{162} (2+3 x)^6-\frac{10}{189} (2+3 x)^7\\ \end{align*}

Mathematica [A]  time = 0.0010793, size = 42, normalized size = 1.24 \[ -\frac{810 x^7}{7}-\frac{747 x^6}{2}-\frac{2133 x^5}{5}-132 x^4+\frac{392 x^3}{3}+136 x^2+48 x \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

48*x + 136*x^2 + (392*x^3)/3 - 132*x^4 - (2133*x^5)/5 - (747*x^6)/2 - (810*x^7)/7

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Maple [A]  time = 0.002, size = 35, normalized size = 1. \begin{align*} -{\frac{810\,{x}^{7}}{7}}-{\frac{747\,{x}^{6}}{2}}-{\frac{2133\,{x}^{5}}{5}}-132\,{x}^{4}+{\frac{392\,{x}^{3}}{3}}+136\,{x}^{2}+48\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^4*(3+5*x),x)

[Out]

-810/7*x^7-747/2*x^6-2133/5*x^5-132*x^4+392/3*x^3+136*x^2+48*x

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Maxima [A]  time = 1.74737, size = 46, normalized size = 1.35 \begin{align*} -\frac{810}{7} \, x^{7} - \frac{747}{2} \, x^{6} - \frac{2133}{5} \, x^{5} - 132 \, x^{4} + \frac{392}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="maxima")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x

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Fricas [A]  time = 1.29736, size = 103, normalized size = 3.03 \begin{align*} -\frac{810}{7} x^{7} - \frac{747}{2} x^{6} - \frac{2133}{5} x^{5} - 132 x^{4} + \frac{392}{3} x^{3} + 136 x^{2} + 48 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="fricas")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x

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Sympy [A]  time = 0.060902, size = 39, normalized size = 1.15 \begin{align*} - \frac{810 x^{7}}{7} - \frac{747 x^{6}}{2} - \frac{2133 x^{5}}{5} - 132 x^{4} + \frac{392 x^{3}}{3} + 136 x^{2} + 48 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**4*(3+5*x),x)

[Out]

-810*x**7/7 - 747*x**6/2 - 2133*x**5/5 - 132*x**4 + 392*x**3/3 + 136*x**2 + 48*x

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Giac [A]  time = 1.98187, size = 46, normalized size = 1.35 \begin{align*} -\frac{810}{7} \, x^{7} - \frac{747}{2} \, x^{6} - \frac{2133}{5} \, x^{5} - 132 \, x^{4} + \frac{392}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="giac")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x