3.1382 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{250 x^4}{27}+\frac{1700 x^3}{81}-\frac{1795 x^2}{81}+\frac{16253 x}{729}-\frac{1813}{729 (3 x+2)}+\frac{343}{4374 (3 x+2)^2}-\frac{10073}{729} \log (3 x+2) \]

[Out]

(16253*x)/729 - (1795*x^2)/81 + (1700*x^3)/81 - (250*x^4)/27 + 343/(4374*(2 + 3*x)^2) - 1813/(729*(2 + 3*x)) -
(10073*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0270603, antiderivative size = 59, normalized size of antiderivative = 1., 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} \[ -\frac{250 x^4}{27}+\frac{1700 x^3}{81}-\frac{1795 x^2}{81}+\frac{16253 x}{729}-\frac{1813}{729 (3 x+2)}+\frac{343}{4374 (3 x+2)^2}-\frac{10073}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16253*x)/729 - (1795*x^2)/81 + (1700*x^3)/81 - (250*x^4)/27 + 343/(4374*(2 + 3*x)^2) - 1813/(729*(2 + 3*x)) -
(10073*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)^3}{(2+3 x)^3} \, dx &=\int \left (\frac{16253}{729}-\frac{3590 x}{81}+\frac{1700 x^2}{27}-\frac{1000 x^3}{27}-\frac{343}{729 (2+3 x)^3}+\frac{1813}{243 (2+3 x)^2}-\frac{10073}{243 (2+3 x)}\right ) \, dx\\ &=\frac{16253 x}{729}-\frac{1795 x^2}{81}+\frac{1700 x^3}{81}-\frac{250 x^4}{27}+\frac{343}{4374 (2+3 x)^2}-\frac{1813}{729 (2+3 x)}-\frac{10073}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0172839, size = 56, normalized size = 0.95 \[ \frac{-364500 x^6+340200 x^5+67230 x^4+81702 x^3+2072124 x^2+2076942 x-60438 (3 x+2)^2 \log (3 x+2)+551755}{4374 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(551755 + 2076942*x + 2072124*x^2 + 81702*x^3 + 67230*x^4 + 340200*x^5 - 364500*x^6 - 60438*(2 + 3*x)^2*Log[2
+ 3*x])/(4374*(2 + 3*x)^2)

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*}{\frac{16253\,x}{729}}-{\frac{1795\,{x}^{2}}{81}}+{\frac{1700\,{x}^{3}}{81}}-{\frac{250\,{x}^{4}}{27}}+{\frac{343}{4374\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{1813}{1458+2187\,x}}-{\frac{10073\,\ln \left ( 2+3\,x \right ) }{729}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16253/729*x-1795/81*x^2+1700/81*x^3-250/27*x^4+343/4374/(2+3*x)^2-1813/729/(2+3*x)-10073/729*ln(2+3*x)

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Maxima [A]  time = 1.04099, size = 62, normalized size = 1.05 \begin{align*} -\frac{250}{27} \, x^{4} + \frac{1700}{81} \, x^{3} - \frac{1795}{81} \, x^{2} + \frac{16253}{729} \, x - \frac{49 \,{\left (666 \, x + 437\right )}}{4374 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{10073}{729} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250/27*x^4 + 1700/81*x^3 - 1795/81*x^2 + 16253/729*x - 49/4374*(666*x + 437)/(9*x^2 + 12*x + 4) - 10073/729*l
og(3*x + 2)

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Fricas [A]  time = 1.24419, size = 203, normalized size = 3.44 \begin{align*} -\frac{364500 \, x^{6} - 340200 \, x^{5} - 67230 \, x^{4} - 81702 \, x^{3} - 782496 \, x^{2} + 60438 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 357438 \, x + 21413}{4374 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4374*(364500*x^6 - 340200*x^5 - 67230*x^4 - 81702*x^3 - 782496*x^2 + 60438*(9*x^2 + 12*x + 4)*log(3*x + 2)
- 357438*x + 21413)/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 0.12366, size = 49, normalized size = 0.83 \begin{align*} - \frac{250 x^{4}}{27} + \frac{1700 x^{3}}{81} - \frac{1795 x^{2}}{81} + \frac{16253 x}{729} - \frac{32634 x + 21413}{39366 x^{2} + 52488 x + 17496} - \frac{10073 \log{\left (3 x + 2 \right )}}{729} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250*x**4/27 + 1700*x**3/81 - 1795*x**2/81 + 16253*x/729 - (32634*x + 21413)/(39366*x**2 + 52488*x + 17496) -
10073*log(3*x + 2)/729

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Giac [A]  time = 3.10414, size = 57, normalized size = 0.97 \begin{align*} -\frac{250}{27} \, x^{4} + \frac{1700}{81} \, x^{3} - \frac{1795}{81} \, x^{2} + \frac{16253}{729} \, x - \frac{49 \,{\left (666 \, x + 437\right )}}{4374 \,{\left (3 \, x + 2\right )}^{2}} - \frac{10073}{729} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250/27*x^4 + 1700/81*x^3 - 1795/81*x^2 + 16253/729*x - 49/4374*(666*x + 437)/(3*x + 2)^2 - 10073/729*log(abs(
3*x + 2))