3.12.84 \(\int \frac {(1-2 x) (3+5 x)^3}{2+3 x} \, dx\) [1184]

Optimal. Leaf size=37 \[ \frac {1097 x}{81}+\frac {545 x^2}{54}-\frac {475 x^3}{27}-\frac {125 x^4}{6}-\frac {7}{243} \log (2+3 x) \]

[Out]

1097/81*x+545/54*x^2-475/27*x^3-125/6*x^4-7/243*ln(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 37, 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 = {78} \begin {gather*} -\frac {125 x^4}{6}-\frac {475 x^3}{27}+\frac {545 x^2}{54}+\frac {1097 x}{81}-\frac {7}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1097*x)/81 + (545*x^2)/54 - (475*x^3)/27 - (125*x^4)/6 - (7*Log[2 + 3*x])/243

Rule 78

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)^3}{2+3 x} \, dx &=\int \left (\frac {1097}{81}+\frac {545 x}{27}-\frac {475 x^2}{9}-\frac {250 x^3}{3}-\frac {7}{81 (2+3 x)}\right ) \, dx\\ &=\frac {1097 x}{81}+\frac {545 x^2}{54}-\frac {475 x^3}{27}-\frac {125 x^4}{6}-\frac {7}{243} \log (2+3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.86 \begin {gather*} \frac {5024+19746 x+14715 x^2-25650 x^3-30375 x^4-42 \log (2+3 x)}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5024 + 19746*x + 14715*x^2 - 25650*x^3 - 30375*x^4 - 42*Log[2 + 3*x])/1458

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Maple [A]
time = 0.09, size = 28, normalized size = 0.76

method	result	size
default	\(\frac {1097 x}{81}+\frac {545 x^{2}}{54}-\frac {475 x^{3}}{27}-\frac {125 x^{4}}{6}-\frac {7 \ln \left (2+3 x \right )}{243}\)	\(28\)
norman	\(\frac {1097 x}{81}+\frac {545 x^{2}}{54}-\frac {475 x^{3}}{27}-\frac {125 x^{4}}{6}-\frac {7 \ln \left (2+3 x \right )}{243}\)	\(28\)
risch	\(\frac {1097 x}{81}+\frac {545 x^{2}}{54}-\frac {475 x^{3}}{27}-\frac {125 x^{4}}{6}-\frac {7 \ln \left (2+3 x \right )}{243}\)	\(28\)
meijerg	\(-\frac {7 \ln \left (1+\frac {3 x}{2}\right )}{243}+27 x +\frac {5 x \left (-\frac {9 x}{2}+6\right )}{3}-\frac {325 x \left (9 x^{2}-9 x +12\right )}{81}+\frac {100 x \left (-\frac {405}{8} x^{3}+45 x^{2}-45 x +60\right )}{243}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1097/81*x+545/54*x^2-475/27*x^3-125/6*x^4-7/243*ln(2+3*x)

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Maxima [A]
time = 0.33, size = 27, normalized size = 0.73 \begin {gather*} -\frac {125}{6} \, x^{4} - \frac {475}{27} \, x^{3} + \frac {545}{54} \, x^{2} + \frac {1097}{81} \, x - \frac {7}{243} \, \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)^3/(2+3*x),x, algorithm="maxima")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(3*x + 2)

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Fricas [A]
time = 0.95, size = 27, normalized size = 0.73 \begin {gather*} -\frac {125}{6} \, x^{4} - \frac {475}{27} \, x^{3} + \frac {545}{54} \, x^{2} + \frac {1097}{81} \, x - \frac {7}{243} \, \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)^3/(2+3*x),x, algorithm="fricas")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(3*x + 2)

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Sympy [A]
time = 0.03, size = 34, normalized size = 0.92 \begin {gather*} - \frac {125 x^{4}}{6} - \frac {475 x^{3}}{27} + \frac {545 x^{2}}{54} + \frac {1097 x}{81} - \frac {7 \log {\left (3 x + 2 \right )}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*x**4/6 - 475*x**3/27 + 545*x**2/54 + 1097*x/81 - 7*log(3*x + 2)/243

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Giac [A]
time = 2.23, size = 28, normalized size = 0.76 \begin {gather*} -\frac {125}{6} \, x^{4} - \frac {475}{27} \, x^{3} + \frac {545}{54} \, x^{2} + \frac {1097}{81} \, x - \frac {7}{243} \, \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)^3/(2+3*x),x, algorithm="giac")

[Out]

-125/6*x^4 - 475/27*x^3 + 545/54*x^2 + 1097/81*x - 7/243*log(abs(3*x + 2))

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Mupad [B]
time = 0.02, size = 25, normalized size = 0.68 \begin {gather*} \frac {1097\,x}{81}-\frac {7\,\ln \left (x+\frac {2}{3}\right )}{243}+\frac {545\,x^2}{54}-\frac {475\,x^3}{27}-\frac {125\,x^4}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1097*x)/81 - (7*log(x + 2/3))/243 + (545*x^2)/54 - (475*x^3)/27 - (125*x^4)/6

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