3.12.13 \(\int \frac {(1-2 x)^2 (3+5 x)}{(2+3 x)^8} \, dx\)

Optimal. Leaf size=45 \[ -\frac {5}{81 (3 x+2)^4}+\frac {16}{45 (3 x+2)^5}-\frac {91}{162 (3 x+2)^6}+\frac {7}{81 (3 x+2)^7} \]

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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 {5}{81 (3 x+2)^4}+\frac {16}{45 (3 x+2)^5}-\frac {91}{162 (3 x+2)^6}+\frac {7}{81 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^8,x]

[Out]

7/(81*(2 + 3*x)^7) - 91/(162*(2 + 3*x)^6) + 16/(45*(2 + 3*x)^5) - 5/(81*(2 + 3*x)^4)

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)^8} \, dx &=\int \left (-\frac {49}{27 (2+3 x)^8}+\frac {91}{9 (2+3 x)^7}-\frac {16}{3 (2+3 x)^6}+\frac {20}{27 (2+3 x)^5}\right ) \, dx\\ &=\frac {7}{81 (2+3 x)^7}-\frac {91}{162 (2+3 x)^6}+\frac {16}{45 (2+3 x)^5}-\frac {5}{81 (2+3 x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 26, normalized size = 0.58 \begin {gather*} -\frac {1350 x^3+108 x^2-291 x+88}{810 (3 x+2)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^8,x]

[Out]

-1/810*(88 - 291*x + 108*x^2 + 1350*x^3)/(2 + 3*x)^7

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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)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^8,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^8, x]

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fricas [A]  time = 1.34, size = 54, normalized size = 1.20 \begin {gather*} -\frac {1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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)^8,x, algorithm="fricas")

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2
+ 1344*x + 128)

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giac [A]  time = 1.16, size = 24, normalized size = 0.53 \begin {gather*} -\frac {1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \, {\left (3 \, x + 2\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(3*x + 2)^7

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maple [A]  time = 0.00, size = 38, normalized size = 0.84 \begin {gather*} \frac {7}{81 \left (3 x +2\right )^{7}}-\frac {91}{162 \left (3 x +2\right )^{6}}+\frac {16}{45 \left (3 x +2\right )^{5}}-\frac {5}{81 \left (3 x +2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3)/(3*x+2)^8,x)

[Out]

7/81/(3*x+2)^7-91/162/(3*x+2)^6+16/45/(3*x+2)^5-5/81/(3*x+2)^4

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maxima [A]  time = 0.47, size = 54, normalized size = 1.20 \begin {gather*} -\frac {1350 \, x^{3} + 108 \, x^{2} - 291 \, x + 88}{810 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\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)^8,x, algorithm="maxima")

[Out]

-1/810*(1350*x^3 + 108*x^2 - 291*x + 88)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2
+ 1344*x + 128)

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mupad [B]  time = 1.11, size = 37, normalized size = 0.82 \begin {gather*} \frac {16}{45\,{\left (3\,x+2\right )}^5}-\frac {5}{81\,{\left (3\,x+2\right )}^4}-\frac {91}{162\,{\left (3\,x+2\right )}^6}+\frac {7}{81\,{\left (3\,x+2\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(5*x + 3))/(3*x + 2)^8,x)

[Out]

16/(45*(3*x + 2)^5) - 5/(81*(3*x + 2)^4) - 91/(162*(3*x + 2)^6) + 7/(81*(3*x + 2)^7)

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sympy [A]  time = 0.17, size = 49, normalized size = 1.09 \begin {gather*} \frac {- 1350 x^{3} - 108 x^{2} + 291 x - 88}{1771470 x^{7} + 8266860 x^{6} + 16533720 x^{5} + 18370800 x^{4} + 12247200 x^{3} + 4898880 x^{2} + 1088640 x + 103680} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)/(2+3*x)**8,x)

[Out]

(-1350*x**3 - 108*x**2 + 291*x - 88)/(1771470*x**7 + 8266860*x**6 + 16533720*x**5 + 18370800*x**4 + 12247200*x
**3 + 4898880*x**2 + 1088640*x + 103680)

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