3.1152 \(\int \frac {(1-2 x) (3+5 x)}{2+3 x} \, dx\)

Optimal. Leaf size=23 \[ -\frac {5 x^2}{3}+\frac {17 x}{9}-\frac {7}{27} \log (3 x+2) \]

[Out]

17/9*x-5/3*x^2-7/27*ln(2+3*x)

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Rubi [A]  time = 0.01, antiderivative size = 23, 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} \[ -\frac {5 x^2}{3}+\frac {17 x}{9}-\frac {7}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(17*x)/9 - (5*x^2)/3 - (7*Log[2 + 3*x])/27

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} \, dx &=\int \left (\frac {17}{9}-\frac {10 x}{3}-\frac {7}{9 (2+3 x)}\right ) \, dx\\ &=\frac {17 x}{9}-\frac {5 x^2}{3}-\frac {7}{27} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.96 \[ \frac {1}{27} \left (-45 x^2+51 x-7 \log (3 x+2)+54\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(54 + 51*x - 45*x^2 - 7*Log[2 + 3*x])/27

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fricas [A]  time = 0.91, size = 17, normalized size = 0.74 \[ -\frac {5}{3} \, x^{2} + \frac {17}{9} \, x - \frac {7}{27} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(3*x + 2)

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giac [A]  time = 1.12, size = 18, normalized size = 0.78 \[ -\frac {5}{3} \, x^{2} + \frac {17}{9} \, x - \frac {7}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(abs(3*x + 2))

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maple [A]  time = 0.00, size = 18, normalized size = 0.78 \[ -\frac {5 x^{2}}{3}+\frac {17 x}{9}-\frac {7 \ln \left (3 x +2\right )}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17/9*x-5/3*x^2-7/27*ln(3*x+2)

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maxima [A]  time = 0.78, size = 17, normalized size = 0.74 \[ -\frac {5}{3} \, x^{2} + \frac {17}{9} \, x - \frac {7}{27} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*x^2 + 17/9*x - 7/27*log(3*x + 2)

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mupad [B]  time = 0.03, size = 15, normalized size = 0.65 \[ \frac {17\,x}{9}-\frac {7\,\ln \left (x+\frac {2}{3}\right )}{27}-\frac {5\,x^2}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(17*x)/9 - (7*log(x + 2/3))/27 - (5*x^2)/3

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sympy [A]  time = 0.08, size = 20, normalized size = 0.87 \[ - \frac {5 x^{2}}{3} + \frac {17 x}{9} - \frac {7 \log {\left (3 x + 2 \right )}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*x**2/3 + 17*x/9 - 7*log(3*x + 2)/27

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