3.12.11 \(\int \frac {(1-2 x)^2 (3+5 x)}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=45 \[ -\frac {10}{81 (3 x+2)^2}+\frac {16}{27 (3 x+2)^3}-\frac {91}{108 (3 x+2)^4}+\frac {49}{405 (3 x+2)^5} \]

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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 {10}{81 (3 x+2)^2}+\frac {16}{27 (3 x+2)^3}-\frac {91}{108 (3 x+2)^4}+\frac {49}{405 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

49/(405*(2 + 3*x)^5) - 91/(108*(2 + 3*x)^4) + 16/(27*(2 + 3*x)^3) - 10/(81*(2 + 3*x)^2)

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)^6} \, dx &=\int \left (-\frac {49}{27 (2+3 x)^6}+\frac {91}{9 (2+3 x)^5}-\frac {16}{3 (2+3 x)^4}+\frac {20}{27 (2+3 x)^3}\right ) \, dx\\ &=\frac {49}{405 (2+3 x)^5}-\frac {91}{108 (2+3 x)^4}+\frac {16}{27 (2+3 x)^3}-\frac {10}{81 (2+3 x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.58 \begin {gather*} -\frac {1800 x^3+720 x^2-75 x+98}{540 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

-1/540*(98 - 75*x + 720*x^2 + 1800*x^3)/(2 + 3*x)^5

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^6,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^6, x]

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fricas [A]  time = 1.33, size = 44, normalized size = 0.98 \begin {gather*} -\frac {1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 1.20, size = 24, normalized size = 0.53 \begin {gather*} -\frac {1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \, {\left (3 \, x + 2\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^6,x, algorithm="giac")

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(3*x + 2)^5

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maple [A]  time = 0.00, size = 38, normalized size = 0.84 \begin {gather*} \frac {49}{405 \left (3 x +2\right )^{5}}-\frac {91}{108 \left (3 x +2\right )^{4}}+\frac {16}{27 \left (3 x +2\right )^{3}}-\frac {10}{81 \left (3 x +2\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3)/(3*x+2)^6,x)

[Out]

49/405/(3*x+2)^5-91/108/(3*x+2)^4+16/27/(3*x+2)^3-10/81/(3*x+2)^2

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maxima [A]  time = 0.59, size = 44, normalized size = 0.98 \begin {gather*} -\frac {1800 \, x^{3} + 720 \, x^{2} - 75 \, x + 98}{540 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="maxima")

[Out]

-1/540*(1800*x^3 + 720*x^2 - 75*x + 98)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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mupad [B]  time = 1.10, size = 37, normalized size = 0.82 \begin {gather*} \frac {16}{27\,{\left (3\,x+2\right )}^3}-\frac {10}{81\,{\left (3\,x+2\right )}^2}-\frac {91}{108\,{\left (3\,x+2\right )}^4}+\frac {49}{405\,{\left (3\,x+2\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(5*x + 3))/(3*x + 2)^6,x)

[Out]

16/(27*(3*x + 2)^3) - 10/(81*(3*x + 2)^2) - 91/(108*(3*x + 2)^4) + 49/(405*(3*x + 2)^5)

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sympy [A]  time = 0.15, size = 39, normalized size = 0.87 \begin {gather*} \frac {- 1800 x^{3} - 720 x^{2} + 75 x - 98}{131220 x^{5} + 437400 x^{4} + 583200 x^{3} + 388800 x^{2} + 129600 x + 17280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)/(2+3*x)**6,x)

[Out]

(-1800*x**3 - 720*x**2 + 75*x - 98)/(131220*x**5 + 437400*x**4 + 583200*x**3 + 388800*x**2 + 129600*x + 17280)

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