3.13.30 \(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=66 \[ -\frac {2180}{729 (3 x+2)}+\frac {4099}{729 (3 x+2)^2}-\frac {11599}{2187 (3 x+2)^3}+\frac {931}{729 (3 x+2)^4}-\frac {343}{3645 (3 x+2)^5}-\frac {200}{729} \log (3 x+2) \]

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Rubi [A]  time = 0.03, antiderivative size = 66, 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 {2180}{729 (3 x+2)}+\frac {4099}{729 (3 x+2)^2}-\frac {11599}{2187 (3 x+2)^3}+\frac {931}{729 (3 x+2)^4}-\frac {343}{3645 (3 x+2)^5}-\frac {200}{729} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-343/(3645*(2 + 3*x)^5) + 931/(729*(2 + 3*x)^4) - 11599/(2187*(2 + 3*x)^3) + 4099/(729*(2 + 3*x)^2) - 2180/(72
9*(2 + 3*x)) - (200*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)^6} \, dx &=\int \left (\frac {343}{243 (2+3 x)^6}-\frac {3724}{243 (2+3 x)^5}+\frac {11599}{243 (2+3 x)^4}-\frac {8198}{243 (2+3 x)^3}+\frac {2180}{243 (2+3 x)^2}-\frac {200}{243 (2+3 x)}\right ) \, dx\\ &=-\frac {343}{3645 (2+3 x)^5}+\frac {931}{729 (2+3 x)^4}-\frac {11599}{2187 (2+3 x)^3}+\frac {4099}{729 (2+3 x)^2}-\frac {2180}{729 (2+3 x)}-\frac {200}{729} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 46, normalized size = 0.70 \begin {gather*} -\frac {2648700 x^4+5403105 x^3+4264965 x^2+1579785 x+3000 (3 x+2)^5 \log (30 x+20)+236399}{10935 (3 x+2)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

-1/10935*(236399 + 1579785*x + 4264965*x^2 + 5403105*x^3 + 2648700*x^4 + 3000*(2 + 3*x)^5*Log[20 + 30*x])/(2 +
3*x)^5

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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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^6,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)^3*(3 + 5*x)^2)/(2 + 3*x)^6, x]

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fricas [A]  time = 1.30, size = 82, normalized size = 1.24 \begin {gather*} -\frac {2648700 \, x^{4} + 5403105 \, x^{3} + 4264965 \, x^{2} + 3000 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 1579785 \, x + 236399}{10935 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/10935*(2648700*x^4 + 5403105*x^3 + 4264965*x^2 + 3000*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)
*log(3*x + 2) + 1579785*x + 236399)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 0.80, size = 39, normalized size = 0.59 \begin {gather*} -\frac {2648700 \, x^{4} + 5403105 \, x^{3} + 4264965 \, x^{2} + 1579785 \, x + 236399}{10935 \, {\left (3 \, x + 2\right )}^{5}} - \frac {200}{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)^6,x, algorithm="giac")

[Out]

-1/10935*(2648700*x^4 + 5403105*x^3 + 4264965*x^2 + 1579785*x + 236399)/(3*x + 2)^5 - 200/729*log(abs(3*x + 2)
)

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maple [A]  time = 0.01, size = 55, normalized size = 0.83 \begin {gather*} -\frac {200 \ln \left (3 x +2\right )}{729}-\frac {343}{3645 \left (3 x +2\right )^{5}}+\frac {931}{729 \left (3 x +2\right )^{4}}-\frac {11599}{2187 \left (3 x +2\right )^{3}}+\frac {4099}{729 \left (3 x +2\right )^{2}}-\frac {2180}{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)^6,x)

[Out]

-343/3645/(3*x+2)^5+931/729/(3*x+2)^4-11599/2187/(3*x+2)^3+4099/729/(3*x+2)^2-2180/729/(3*x+2)-200/729*ln(3*x+
2)

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maxima [A]  time = 0.54, size = 58, normalized size = 0.88 \begin {gather*} -\frac {2648700 \, x^{4} + 5403105 \, x^{3} + 4264965 \, x^{2} + 1579785 \, x + 236399}{10935 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {200}{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)^6,x, algorithm="maxima")

[Out]

-1/10935*(2648700*x^4 + 5403105*x^3 + 4264965*x^2 + 1579785*x + 236399)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^
2 + 240*x + 32) - 200/729*log(3*x + 2)

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mupad [B]  time = 0.04, size = 54, normalized size = 0.82 \begin {gather*} -\frac {200\,\ln \left (x+\frac {2}{3}\right )}{729}-\frac {\frac {2180\,x^4}{2187}+\frac {4447\,x^3}{2187}+\frac {94777\,x^2}{59049}+\frac {105319\,x}{177147}+\frac {236399}{2657205}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \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)^6,x)

[Out]

- (200*log(x + 2/3))/729 - ((105319*x)/177147 + (94777*x^2)/59049 + (4447*x^3)/2187 + (2180*x^4)/2187 + 236399
/2657205)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)

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sympy [A]  time = 0.17, size = 56, normalized size = 0.85 \begin {gather*} - \frac {2648700 x^{4} + 5403105 x^{3} + 4264965 x^{2} + 1579785 x + 236399}{2657205 x^{5} + 8857350 x^{4} + 11809800 x^{3} + 7873200 x^{2} + 2624400 x + 349920} - \frac {200 \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)**6,x)

[Out]

-(2648700*x**4 + 5403105*x**3 + 4264965*x**2 + 1579785*x + 236399)/(2657205*x**5 + 8857350*x**4 + 11809800*x**
3 + 7873200*x**2 + 2624400*x + 349920) - 200*log(3*x + 2)/729

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