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

Optimal. Leaf size=37 \[ \frac {25 x^4}{3}-\frac {140 x^3}{27}-\frac {251 x^2}{54}+\frac {340 x}{81}+\frac {49}{243} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} \frac {25 x^4}{3}-\frac {140 x^3}{27}-\frac {251 x^2}{54}+\frac {340 x}{81}+\frac {49}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(340*x)/81 - (251*x^2)/54 - (140*x^3)/27 + (25*x^4)/3 + (49*Log[2 + 3*x])/243

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(1-2 x)^2 (3+5 x)^2}{2+3 x} \, dx &=\int \left (\frac {340}{81}-\frac {251 x}{27}-\frac {140 x^2}{9}+\frac {100 x^3}{3}+\frac {49}{81 (2+3 x)}\right ) \, dx\\ &=\frac {340 x}{81}-\frac {251 x^2}{54}-\frac {140 x^3}{27}+\frac {25 x^4}{3}+\frac {49}{243} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 0.86 \begin {gather*} \frac {12150 x^4-7560 x^3-6777 x^2+6120 x+294 \log (3 x+2)+2452}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2452 + 6120*x - 6777*x^2 - 7560*x^3 + 12150*x^4 + 294*Log[2 + 3*x])/1458

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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}{2+3 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.14, size = 27, normalized size = 0.73 \begin {gather*} \frac {25}{3} \, x^{4} - \frac {140}{27} \, x^{3} - \frac {251}{54} \, x^{2} + \frac {340}{81} \, x + \frac {49}{243} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/3*x^4 - 140/27*x^3 - 251/54*x^2 + 340/81*x + 49/243*log(3*x + 2)

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giac [A]  time = 1.54, size = 28, normalized size = 0.76 \begin {gather*} \frac {25}{3} \, x^{4} - \frac {140}{27} \, x^{3} - \frac {251}{54} \, x^{2} + \frac {340}{81} \, x + \frac {49}{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)^2*(3+5*x)^2/(2+3*x),x, algorithm="giac")

[Out]

25/3*x^4 - 140/27*x^3 - 251/54*x^2 + 340/81*x + 49/243*log(abs(3*x + 2))

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maple [A]  time = 0.00, size = 28, normalized size = 0.76 \begin {gather*} \frac {25 x^{4}}{3}-\frac {140 x^{3}}{27}-\frac {251 x^{2}}{54}+\frac {340 x}{81}+\frac {49 \ln \left (3 x +2\right )}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

340/81*x-251/54*x^2-140/27*x^3+25/3*x^4+49/243*ln(3*x+2)

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maxima [A]  time = 0.61, size = 27, normalized size = 0.73 \begin {gather*} \frac {25}{3} \, x^{4} - \frac {140}{27} \, x^{3} - \frac {251}{54} \, x^{2} + \frac {340}{81} \, x + \frac {49}{243} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/3*x^4 - 140/27*x^3 - 251/54*x^2 + 340/81*x + 49/243*log(3*x + 2)

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mupad [B]  time = 0.03, size = 25, normalized size = 0.68 \begin {gather*} \frac {340\,x}{81}+\frac {49\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {251\,x^2}{54}-\frac {140\,x^3}{27}+\frac {25\,x^4}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(340*x)/81 + (49*log(x + 2/3))/243 - (251*x^2)/54 - (140*x^3)/27 + (25*x^4)/3

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sympy [A]  time = 0.10, size = 34, normalized size = 0.92 \begin {gather*} \frac {25 x^{4}}{3} - \frac {140 x^{3}}{27} - \frac {251 x^{2}}{54} + \frac {340 x}{81} + \frac {49 \log {\left (3 x + 2 \right )}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*x**4/3 - 140*x**3/27 - 251*x**2/54 + 340*x/81 + 49*log(3*x + 2)/243

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