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

Optimal. Leaf size=30 \[ -\frac {50 x^3}{9}-\frac {5 x^2}{18}+\frac {118 x}{27}+\frac {7}{81} \log (3 x+2) \]

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Rubi [A]  time = 0.01, antiderivative size = 30, 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 x^3}{9}-\frac {5 x^2}{18}+\frac {118 x}{27}+\frac {7}{81} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(118*x)/27 - (5*x^2)/18 - (50*x^3)/9 + (7*Log[2 + 3*x])/81

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} \, dx &=\int \left (\frac {118}{27}-\frac {5 x}{9}-\frac {50 x^2}{3}+\frac {7}{27 (2+3 x)}\right ) \, dx\\ &=\frac {118 x}{27}-\frac {5 x^2}{18}-\frac {50 x^3}{9}+\frac {7}{81} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.90 \begin {gather*} \frac {1}{486} \left (-2700 x^3-135 x^2+2124 x+42 \log (3 x+2)+676\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(676 + 2124*x - 135*x^2 - 2700*x^3 + 42*Log[2 + 3*x])/486

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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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 22, normalized size = 0.73 \begin {gather*} -\frac {50}{9} \, x^{3} - \frac {5}{18} \, x^{2} + \frac {118}{27} \, x + \frac {7}{81} \, \log \left (3 \, x + 2\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),x, algorithm="fricas")

[Out]

-50/9*x^3 - 5/18*x^2 + 118/27*x + 7/81*log(3*x + 2)

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giac [A]  time = 1.22, size = 23, normalized size = 0.77 \begin {gather*} -\frac {50}{9} \, x^{3} - \frac {5}{18} \, x^{2} + \frac {118}{27} \, x + \frac {7}{81} \, \log \left ({\left | 3 \, x + 2 \right |}\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),x, algorithm="giac")

[Out]

-50/9*x^3 - 5/18*x^2 + 118/27*x + 7/81*log(abs(3*x + 2))

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maple [A]  time = 0.00, size = 23, normalized size = 0.77 \begin {gather*} -\frac {50 x^{3}}{9}-\frac {5 x^{2}}{18}+\frac {118 x}{27}+\frac {7 \ln \left (3 x +2\right )}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

118/27*x-5/18*x^2-50/9*x^3+7/81*ln(3*x+2)

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maxima [A]  time = 0.50, size = 22, normalized size = 0.73 \begin {gather*} -\frac {50}{9} \, x^{3} - \frac {5}{18} \, x^{2} + \frac {118}{27} \, x + \frac {7}{81} \, \log \left (3 \, x + 2\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),x, algorithm="maxima")

[Out]

-50/9*x^3 - 5/18*x^2 + 118/27*x + 7/81*log(3*x + 2)

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mupad [B]  time = 0.03, size = 20, normalized size = 0.67 \begin {gather*} \frac {118\,x}{27}+\frac {7\,\ln \left (x+\frac {2}{3}\right )}{81}-\frac {5\,x^2}{18}-\frac {50\,x^3}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(118*x)/27 + (7*log(x + 2/3))/81 - (5*x^2)/18 - (50*x^3)/9

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sympy [A]  time = 0.09, size = 27, normalized size = 0.90 \begin {gather*} - \frac {50 x^{3}}{9} - \frac {5 x^{2}}{18} + \frac {118 x}{27} + \frac {7 \log {\left (3 x + 2 \right )}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50*x**3/9 - 5*x**2/18 + 118*x/27 + 7*log(3*x + 2)/81

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