3.12.45 \(\int \frac {(1-2 x)^2 (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=56 \[ \frac {250 x^2}{81}-\frac {2800 x}{243}+\frac {4099}{729 (3 x+2)}-\frac {763}{1458 (3 x+2)^2}+\frac {49}{2187 (3 x+2)^3}+\frac {8285}{729} \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 56, 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 {250 x^2}{81}-\frac {2800 x}{243}+\frac {4099}{729 (3 x+2)}-\frac {763}{1458 (3 x+2)^2}+\frac {49}{2187 (3 x+2)^3}+\frac {8285}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(-2800*x)/243 + (250*x^2)/81 + 49/(2187*(2 + 3*x)^3) - 763/(1458*(2 + 3*x)^2) + 4099/(729*(2 + 3*x)) + (8285*L
og[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)^2 (3+5 x)^3}{(2+3 x)^4} \, dx &=\int \left (-\frac {2800}{243}+\frac {500 x}{81}-\frac {49}{243 (2+3 x)^4}+\frac {763}{243 (2+3 x)^3}-\frac {4099}{243 (2+3 x)^2}+\frac {8285}{243 (2+3 x)}\right ) \, dx\\ &=-\frac {2800 x}{243}+\frac {250 x^2}{81}+\frac {49}{2187 (2+3 x)^3}-\frac {763}{1458 (2+3 x)^2}+\frac {4099}{729 (2+3 x)}+\frac {8285}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 51, normalized size = 0.91 \begin {gather*} -\frac {-364500 x^5+631800 x^4+3304800 x^3+3623454 x^2+1540539 x-49710 (3 x+2)^3 \log (30 x+20)+222904}{4374 (3 x+2)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

-1/4374*(222904 + 1540539*x + 3623454*x^2 + 3304800*x^3 + 631800*x^4 - 364500*x^5 - 49710*(2 + 3*x)^3*Log[20 +
30*x])/(2 + 3*x)^3

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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)^3}{(2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^2*(3 + 5*x)^3)/(2 + 3*x)^4, x]

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fricas [A]  time = 1.38, size = 67, normalized size = 1.20 \begin {gather*} \frac {364500 \, x^{5} - 631800 \, x^{4} - 2235600 \, x^{3} - 1485054 \, x^{2} + 49710 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 114939 \, x + 93896}{4374 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/4374*(364500*x^5 - 631800*x^4 - 2235600*x^3 - 1485054*x^2 + 49710*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2)
- 114939*x + 93896)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.97, size = 37, normalized size = 0.66 \begin {gather*} \frac {250}{81} \, x^{2} - \frac {2800}{243} \, x + \frac {221346 \, x^{2} + 288261 \, x + 93896}{4374 \, {\left (3 \, x + 2\right )}^{3}} + \frac {8285}{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)^2*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

250/81*x^2 - 2800/243*x + 1/4374*(221346*x^2 + 288261*x + 93896)/(3*x + 2)^3 + 8285/729*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 45, normalized size = 0.80 \begin {gather*} \frac {250 x^{2}}{81}-\frac {2800 x}{243}+\frac {8285 \ln \left (3 x +2\right )}{729}+\frac {49}{2187 \left (3 x +2\right )^{3}}-\frac {763}{1458 \left (3 x +2\right )^{2}}+\frac {4099}{729 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(5*x+3)^3/(3*x+2)^4,x)

[Out]

-2800/243*x+250/81*x^2+49/2187/(3*x+2)^3-763/1458/(3*x+2)^2+4099/729/(3*x+2)+8285/729*ln(3*x+2)

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maxima [A]  time = 0.55, size = 46, normalized size = 0.82 \begin {gather*} \frac {250}{81} \, x^{2} - \frac {2800}{243} \, x + \frac {221346 \, x^{2} + 288261 \, x + 93896}{4374 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {8285}{729} \, \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)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

250/81*x^2 - 2800/243*x + 1/4374*(221346*x^2 + 288261*x + 93896)/(27*x^3 + 54*x^2 + 36*x + 8) + 8285/729*log(3
*x + 2)

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mupad [B]  time = 0.04, size = 41, normalized size = 0.73 \begin {gather*} \frac {8285\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {2800\,x}{243}+\frac {\frac {4099\,x^2}{2187}+\frac {32029\,x}{13122}+\frac {46948}{59049}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}+\frac {250\,x^2}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x - 1)^2*(5*x + 3)^3)/(3*x + 2)^4,x)

[Out]

(8285*log(x + 2/3))/729 - (2800*x)/243 + ((32029*x)/13122 + (4099*x^2)/2187 + 46948/59049)/((4*x)/3 + 2*x^2 +
x^3 + 8/27) + (250*x^2)/81

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sympy [A]  time = 0.14, size = 46, normalized size = 0.82 \begin {gather*} \frac {250 x^{2}}{81} - \frac {2800 x}{243} + \frac {221346 x^{2} + 288261 x + 93896}{118098 x^{3} + 236196 x^{2} + 157464 x + 34992} + \frac {8285 \log {\left (3 x + 2 \right )}}{729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)**3/(2+3*x)**4,x)

[Out]

250*x**2/81 - 2800*x/243 + (221346*x**2 + 288261*x + 93896)/(118098*x**3 + 236196*x**2 + 157464*x + 34992) + 8
285*log(3*x + 2)/729

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