3.14.69 \(\int \frac {(1-2 x)^3 (3+5 x)^2}{(2+3 x)^6} \, dx\) [1369]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {90} \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 90

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 = 0.03, size = 46, normalized size = 0.70 \begin {gather*} -\frac {236399+1579785 x+4264965 x^2+5403105 x^3+2648700 x^4+3000 (2+3 x)^5 \log (20+30 x)}{10935 (2+3 x)^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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Maple [A]
time = 0.12, size = 55, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {2180}{9} x^{4}-\frac {4447}{9} x^{3}-\frac {94777}{243} x^{2}-\frac {105319}{729} x -\frac {236399}{10935}}{\left (2+3 x \right )^{5}}-\frac {200 \ln \left (2+3 x \right )}{729}\)	\(39\)
norman	\(\frac {\frac {76307}{324} x^{3}+\frac {131759}{432} x^{4}+\frac {8587}{486} x +\frac {1735}{18} x^{2}+\frac {236399}{1440} x^{5}}{\left (2+3 x \right )^{5}}-\frac {200 \ln \left (2+3 x \right )}{729}\)	\(42\)
default	\(-\frac {343}{3645 \left (2+3 x \right )^{5}}+\frac {931}{729 \left (2+3 x \right )^{4}}-\frac {11599}{2187 \left (2+3 x \right )^{3}}+\frac {4099}{729 \left (2+3 x \right )^{2}}-\frac {2180}{729 \left (2+3 x \right )}-\frac {200 \ln \left (2+3 x \right )}{729}\)	\(55\)
meijerg	\(\frac {9 x \left (\frac {81}{16} x^{4}+\frac {135}{8} x^{3}+\frac {45}{2} x^{2}+15 x +5\right )}{320 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {3 x^{2} \left (\frac {27}{8} x^{3}+\frac {45}{4} x^{2}+15 x +10\right )}{160 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {47 x^{3} \left (\frac {9}{4} x^{2}+\frac {15}{2} x +10\right )}{1920 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {69 x^{4} \left (\frac {3 x}{2}+5\right )}{640 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {3 x^{5}}{16 \left (1+\frac {3 x}{2}\right )^{5}}+\frac {5 x \left (\frac {11097}{16} x^{4}+\frac {10395}{8} x^{3}+\frac {2115}{2} x^{2}+405 x +60\right )}{729 \left (1+\frac {3 x}{2}\right )^{5}}-\frac {200 \ln \left (1+\frac {3 x}{2}\right )}{729}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3*(3+5*x)^2/(2+3*x)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, 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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Fricas [A]
time = 0.41, 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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Sympy [A]
time = 0.06, 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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Giac [A]
time = 1.19, 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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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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