3.11.16 \(\int \frac {(1-2 x) (3+5 x)}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=34 \[ \frac {10}{27 (3 x+2)}-\frac {37}{54 (3 x+2)^2}+\frac {7}{81 (3 x+2)^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {10}{27 (3 x+2)}-\frac {37}{54 (3 x+2)^2}+\frac {7}{81 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(81*(2 + 3*x)^3) - 37/(54*(2 + 3*x)^2) + 10/(27*(2 + 3*x))

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+3 x)^4} \, dx &=\int \left (-\frac {7}{9 (2+3 x)^4}+\frac {37}{9 (2+3 x)^3}-\frac {10}{9 (2+3 x)^2}\right ) \, dx\\ &=\frac {7}{81 (2+3 x)^3}-\frac {37}{54 (2+3 x)^2}+\frac {10}{27 (2+3 x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.62 \begin {gather*} \frac {540 x^2+387 x+32}{162 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32 + 387*x + 540*x^2)/(162*(2 + 3*x)^3)

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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+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.24, size = 29, normalized size = 0.85 \begin {gather*} \frac {540 \, x^{2} + 387 \, x + 32}{162 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/162*(540*x^2 + 387*x + 32)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 1.32, size = 19, normalized size = 0.56 \begin {gather*} \frac {540 \, x^{2} + 387 \, x + 32}{162 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/162*(540*x^2 + 387*x + 32)/(3*x + 2)^3

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maple [A]  time = 0.00, size = 29, normalized size = 0.85 \begin {gather*} \frac {7}{81 \left (3 x +2\right )^{3}}-\frac {37}{54 \left (3 x +2\right )^{2}}+\frac {10}{27 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/81/(3*x+2)^3-37/54/(3*x+2)^2+10/27/(3*x+2)

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maxima [A]  time = 0.57, size = 29, normalized size = 0.85 \begin {gather*} \frac {540 \, x^{2} + 387 \, x + 32}{162 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/162*(540*x^2 + 387*x + 32)/(27*x^3 + 54*x^2 + 36*x + 8)

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mupad [B]  time = 1.10, size = 28, normalized size = 0.82 \begin {gather*} \frac {10}{27\,\left (3\,x+2\right )}-\frac {37}{54\,{\left (3\,x+2\right )}^2}+\frac {7}{81\,{\left (3\,x+2\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/(27*(3*x + 2)) - 37/(54*(3*x + 2)^2) + 7/(81*(3*x + 2)^3)

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sympy [A]  time = 0.12, size = 27, normalized size = 0.79 \begin {gather*} - \frac {- 540 x^{2} - 387 x - 32}{4374 x^{3} + 8748 x^{2} + 5832 x + 1296} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-540*x**2 - 387*x - 32)/(4374*x**3 + 8748*x**2 + 5832*x + 1296)

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