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

Optimal. Leaf size=52 \[ -\frac {200 x^3}{81}+\frac {230 x^2}{27}-\frac {1546 x}{81}+\frac {3724}{729 (3 x+2)}-\frac {343}{1458 (3 x+2)^2}+\frac {11599}{729} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 52, 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 {200 x^3}{81}+\frac {230 x^2}{27}-\frac {1546 x}{81}+\frac {3724}{729 (3 x+2)}-\frac {343}{1458 (3 x+2)^2}+\frac {11599}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1546*x)/81 + (230*x^2)/27 - (200*x^3)/81 - 343/(1458*(2 + 3*x)^2) + 3724/(729*(2 + 3*x)) + (11599*Log[2 + 3*
x])/729

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)^3 (3+5 x)^2}{(2+3 x)^3} \, dx &=\int \left (-\frac {1546}{81}+\frac {460 x}{27}-\frac {200 x^2}{27}+\frac {343}{243 (2+3 x)^3}-\frac {3724}{243 (2+3 x)^2}+\frac {11599}{243 (2+3 x)}\right ) \, dx\\ &=-\frac {1546 x}{81}+\frac {230 x^2}{27}-\frac {200 x^3}{81}-\frac {343}{1458 (2+3 x)^2}+\frac {3724}{729 (2+3 x)}+\frac {11599}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.98 \begin {gather*} -\frac {97200 x^5-205740 x^4+347436 x^3+1531512 x^2+1171896 x-69594 (3 x+2)^2 \log (30 x+20)+258005}{4374 (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4374*(258005 + 1171896*x + 1531512*x^2 + 347436*x^3 - 205740*x^4 + 97200*x^5 - 69594*(2 + 3*x)^2*Log[20 + 3
0*x])/(2 + 3*x)^2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 57, normalized size = 1.10 \begin {gather*} -\frac {32400 \, x^{5} - 68580 \, x^{4} + 115812 \, x^{3} + 284256 \, x^{2} - 23198 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 88968 \, x - 14553}{1458 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1458*(32400*x^5 - 68580*x^4 + 115812*x^3 + 284256*x^2 - 23198*(9*x^2 + 12*x + 4)*log(3*x + 2) + 88968*x - 1
4553)/(9*x^2 + 12*x + 4)

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giac [A]  time = 0.84, size = 37, normalized size = 0.71 \begin {gather*} -\frac {200}{81} \, x^{3} + \frac {230}{27} \, x^{2} - \frac {1546}{81} \, x + \frac {49 \, {\left (152 \, x + 99\right )}}{486 \, {\left (3 \, x + 2\right )}^{2}} + \frac {11599}{729} \, \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*(3+5*x)^2/(2+3*x)^3,x, algorithm="giac")

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(3*x + 2)^2 + 11599/729*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 41, normalized size = 0.79 \begin {gather*} -\frac {200 x^{3}}{81}+\frac {230 x^{2}}{27}-\frac {1546 x}{81}+\frac {11599 \ln \left (3 x +2\right )}{729}-\frac {343}{1458 \left (3 x +2\right )^{2}}+\frac {3724}{729 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1546/81*x+230/27*x^2-200/81*x^3-343/1458/(3*x+2)^2+3724/729/(3*x+2)+11599/729*ln(3*x+2)

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maxima [A]  time = 0.46, size = 41, normalized size = 0.79 \begin {gather*} -\frac {200}{81} \, x^{3} + \frac {230}{27} \, x^{2} - \frac {1546}{81} \, x + \frac {49 \, {\left (152 \, x + 99\right )}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {11599}{729} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200/81*x^3 + 230/27*x^2 - 1546/81*x + 49/486*(152*x + 99)/(9*x^2 + 12*x + 4) + 11599/729*log(3*x + 2)

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mupad [B]  time = 0.03, size = 36, normalized size = 0.69 \begin {gather*} \frac {11599\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {1546\,x}{81}+\frac {\frac {3724\,x}{2187}+\frac {539}{486}}{x^2+\frac {4\,x}{3}+\frac {4}{9}}+\frac {230\,x^2}{27}-\frac {200\,x^3}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11599*log(x + 2/3))/729 - (1546*x)/81 + ((3724*x)/2187 + 539/486)/((4*x)/3 + x^2 + 4/9) + (230*x^2)/27 - (200
*x^3)/81

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sympy [A]  time = 0.13, size = 44, normalized size = 0.85 \begin {gather*} - \frac {200 x^{3}}{81} + \frac {230 x^{2}}{27} - \frac {1546 x}{81} - \frac {- 7448 x - 4851}{4374 x^{2} + 5832 x + 1944} + \frac {11599 \log {\left (3 x + 2 \right )}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-200*x**3/81 + 230*x**2/27 - 1546*x/81 - (-7448*x - 4851)/(4374*x**2 + 5832*x + 1944) + 11599*log(3*x + 2)/729

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