3.11.34 \(\int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=45 \[ \frac {50}{243 (3 x+2)^3}-\frac {65}{108 (3 x+2)^4}+\frac {8}{45 (3 x+2)^5}-\frac {7}{486 (3 x+2)^6} \]

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {50}{243 (3 x+2)^3}-\frac {65}{108 (3 x+2)^4}+\frac {8}{45 (3 x+2)^5}-\frac {7}{486 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(486*(2 + 3*x)^6) + 8/(45*(2 + 3*x)^5) - 65/(108*(2 + 3*x)^4) + 50/(243*(2 + 3*x)^3)

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 \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^7} \, dx &=\int \left (\frac {7}{27 (2+3 x)^7}-\frac {8}{3 (2+3 x)^6}+\frac {65}{9 (2+3 x)^5}-\frac {50}{27 (2+3 x)^4}\right ) \, dx\\ &=-\frac {7}{486 (2+3 x)^6}+\frac {8}{45 (2+3 x)^5}-\frac {65}{108 (2+3 x)^4}+\frac {50}{243 (2+3 x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.58 \begin {gather*} \frac {27000 x^3+27675 x^2+3492 x-2042}{4860 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2042 + 3492*x + 27675*x^2 + 27000*x^3)/(4860*(2 + 3*x)^6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x) (3+5 x)^2}{(2+3 x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.44, size = 49, normalized size = 1.09 \begin {gather*} \frac {27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x +
64)

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giac [A]  time = 1.23, size = 24, normalized size = 0.53 \begin {gather*} \frac {27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^7,x, algorithm="giac")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(3*x + 2)^6

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maple [A]  time = 0.01, size = 38, normalized size = 0.84 \begin {gather*} -\frac {7}{486 \left (3 x +2\right )^{6}}+\frac {8}{45 \left (3 x +2\right )^{5}}-\frac {65}{108 \left (3 x +2\right )^{4}}+\frac {50}{243 \left (3 x +2\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(5*x+3)^2/(3*x+2)^7,x)

[Out]

-7/486/(3*x+2)^6+8/45/(3*x+2)^5-65/108/(3*x+2)^4+50/243/(3*x+2)^3

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maxima [A]  time = 0.53, size = 49, normalized size = 1.09 \begin {gather*} \frac {27000 \, x^{3} + 27675 \, x^{2} + 3492 \, x - 2042}{4860 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^2/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/4860*(27000*x^3 + 27675*x^2 + 3492*x - 2042)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x +
64)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.82 \begin {gather*} \frac {50}{243\,{\left (3\,x+2\right )}^3}-\frac {65}{108\,{\left (3\,x+2\right )}^4}+\frac {8}{45\,{\left (3\,x+2\right )}^5}-\frac {7}{486\,{\left (3\,x+2\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)*(5*x + 3)^2)/(3*x + 2)^7,x)

[Out]

50/(243*(3*x + 2)^3) - 65/(108*(3*x + 2)^4) + 8/(45*(3*x + 2)^5) - 7/(486*(3*x + 2)^6)

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sympy [A]  time = 0.16, size = 46, normalized size = 1.02 \begin {gather*} - \frac {- 27000 x^{3} - 27675 x^{2} - 3492 x + 2042}{3542940 x^{6} + 14171760 x^{5} + 23619600 x^{4} + 20995200 x^{3} + 10497600 x^{2} + 2799360 x + 311040} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**2/(2+3*x)**7,x)

[Out]

-(-27000*x**3 - 27675*x**2 - 3492*x + 2042)/(3542940*x**6 + 14171760*x**5 + 23619600*x**4 + 20995200*x**3 + 10
497600*x**2 + 2799360*x + 311040)

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