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

Optimal. Leaf size=45 \[ -\frac {20}{243 (3 x+2)^3}+\frac {4}{9 (3 x+2)^4}-\frac {91}{135 (3 x+2)^5}+\frac {49}{486 (3 x+2)^6} \]

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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 {20}{243 (3 x+2)^3}+\frac {4}{9 (3 x+2)^4}-\frac {91}{135 (3 x+2)^5}+\frac {49}{486 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(486*(2 + 3*x)^6) - 91/(135*(2 + 3*x)^5) + 4/(9*(2 + 3*x)^4) - 20/(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)^2 (3+5 x)}{(2+3 x)^7} \, dx &=\int \left (-\frac {49}{27 (2+3 x)^7}+\frac {91}{9 (2+3 x)^6}-\frac {16}{3 (2+3 x)^5}+\frac {20}{27 (2+3 x)^4}\right ) \, dx\\ &=\frac {49}{486 (2+3 x)^6}-\frac {91}{135 (2+3 x)^5}+\frac {4}{9 (2+3 x)^4}-\frac {20}{243 (2+3 x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.58 \begin {gather*} -\frac {5400 x^3+1080 x^2-846 x+311}{2430 (3 x+2)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2430*(311 - 846*x + 1080*x^2 + 5400*x^3)/(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)^2 (3+5 x)}{(2+3 x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.32, size = 49, normalized size = 1.09 \begin {gather*} -\frac {5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \, {\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)^2*(3+5*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(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 = 1.21, size = 24, normalized size = 0.53 \begin {gather*} -\frac {5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(3*x + 2)^6

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maple [A]  time = 0.01, size = 38, normalized size = 0.84 \begin {gather*} \frac {49}{486 \left (3 x +2\right )^{6}}-\frac {91}{135 \left (3 x +2\right )^{5}}+\frac {4}{9 \left (3 x +2\right )^{4}}-\frac {20}{243 \left (3 x +2\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/486/(3*x+2)^6-91/135/(3*x+2)^5+4/9/(3*x+2)^4-20/243/(3*x+2)^3

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maxima [A]  time = 0.68, size = 49, normalized size = 1.09 \begin {gather*} -\frac {5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \, {\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)^2*(3+5*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(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 = 0.03, size = 37, normalized size = 0.82 \begin {gather*} \frac {4}{9\,{\left (3\,x+2\right )}^4}-\frac {20}{243\,{\left (3\,x+2\right )}^3}-\frac {91}{135\,{\left (3\,x+2\right )}^5}+\frac {49}{486\,{\left (3\,x+2\right )}^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/(9*(3*x + 2)^4) - 20/(243*(3*x + 2)^3) - 91/(135*(3*x + 2)^5) + 49/(486*(3*x + 2)^6)

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sympy [A]  time = 0.16, size = 44, normalized size = 0.98 \begin {gather*} \frac {- 5400 x^{3} - 1080 x^{2} + 846 x - 311}{1771470 x^{6} + 7085880 x^{5} + 11809800 x^{4} + 10497600 x^{3} + 5248800 x^{2} + 1399680 x + 155520} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5400*x**3 - 1080*x**2 + 846*x - 311)/(1771470*x**6 + 7085880*x**5 + 11809800*x**4 + 10497600*x**3 + 5248800*
x**2 + 1399680*x + 155520)

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