3.1385 \(\int \frac{(1-2 x)^3 (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{1000 x}{729}+\frac{14390}{729 (3 x+2)}-\frac{66193}{4374 (3 x+2)^2}+\frac{10073}{2187 (3 x+2)^3}-\frac{1813}{2916 (3 x+2)^4}+\frac{343}{10935 (3 x+2)^5}+\frac{3700}{729} \log (3 x+2) \]

[Out]

(-1000*x)/729 + 343/(10935*(2 + 3*x)^5) - 1813/(2916*(2 + 3*x)^4) + 10073/(2187*(2 + 3*x)^3) - 66193/(4374*(2
+ 3*x)^2) + 14390/(729*(2 + 3*x)) + (3700*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0274804, antiderivative size = 71, 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} \[ -\frac{1000 x}{729}+\frac{14390}{729 (3 x+2)}-\frac{66193}{4374 (3 x+2)^2}+\frac{10073}{2187 (3 x+2)^3}-\frac{1813}{2916 (3 x+2)^4}+\frac{343}{10935 (3 x+2)^5}+\frac{3700}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1000*x)/729 + 343/(10935*(2 + 3*x)^5) - 1813/(2916*(2 + 3*x)^4) + 10073/(2187*(2 + 3*x)^3) - 66193/(4374*(2
+ 3*x)^2) + 14390/(729*(2 + 3*x)) + (3700*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)^6} \, dx &=\int \left (-\frac{1000}{729}-\frac{343}{729 (2+3 x)^6}+\frac{1813}{243 (2+3 x)^5}-\frac{10073}{243 (2+3 x)^4}+\frac{66193}{729 (2+3 x)^3}-\frac{14390}{243 (2+3 x)^2}+\frac{3700}{243 (2+3 x)}\right ) \, dx\\ &=-\frac{1000 x}{729}+\frac{343}{10935 (2+3 x)^5}-\frac{1813}{2916 (2+3 x)^4}+\frac{10073}{2187 (2+3 x)^3}-\frac{66193}{4374 (2+3 x)^2}+\frac{14390}{729 (2+3 x)}+\frac{3700}{729} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0203597, size = 56, normalized size = 0.79 \[ \frac{-14580000 x^6-58320000 x^5-27264600 x^4+82222290 x^3+109363320 x^2+49872855 x+222000 (3 x+2)^5 \log (3 x+2)+7991782}{43740 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7991782 + 49872855*x + 109363320*x^2 + 82222290*x^3 - 27264600*x^4 - 58320000*x^5 - 14580000*x^6 + 222000*(2
+ 3*x)^5*Log[2 + 3*x])/(43740*(2 + 3*x)^5)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{1000\,x}{729}}+{\frac{343}{10935\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{1813}{2916\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{10073}{2187\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{66193}{4374\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{14390}{1458+2187\,x}}+{\frac{3700\,\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)^6,x)

[Out]

-1000/729*x+343/10935/(2+3*x)^5-1813/2916/(2+3*x)^4+10073/2187/(2+3*x)^3-66193/4374/(2+3*x)^2+14390/729/(2+3*x
)+3700/729*ln(2+3*x)

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Maxima [A]  time = 1.04159, size = 82, normalized size = 1.15 \begin{align*} -\frac{1000}{729} \, x + \frac{23311800 \, x^{4} + 56207430 \, x^{3} + 50854440 \, x^{2} + 20464285 \, x + 3090594}{14580 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{3700}{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)^6,x, algorithm="maxima")

[Out]

-1000/729*x + 1/14580*(23311800*x^4 + 56207430*x^3 + 50854440*x^2 + 20464285*x + 3090594)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32) + 3700/729*log(3*x + 2)

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Fricas [A]  time = 1.2938, size = 317, normalized size = 4.46 \begin{align*} -\frac{4860000 \, x^{6} + 16200000 \, x^{5} - 1711800 \, x^{4} - 41807430 \, x^{3} - 46054440 \, x^{2} - 74000 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) - 19824285 \, x - 3090594}{14580 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/14580*(4860000*x^6 + 16200000*x^5 - 1711800*x^4 - 41807430*x^3 - 46054440*x^2 - 74000*(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) - 19824285*x - 3090594)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2
+ 240*x + 32)

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Sympy [A]  time = 0.161613, size = 60, normalized size = 0.85 \begin{align*} - \frac{1000 x}{729} + \frac{23311800 x^{4} + 56207430 x^{3} + 50854440 x^{2} + 20464285 x + 3090594}{3542940 x^{5} + 11809800 x^{4} + 15746400 x^{3} + 10497600 x^{2} + 3499200 x + 466560} + \frac{3700 \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)**6,x)

[Out]

-1000*x/729 + (23311800*x**4 + 56207430*x**3 + 50854440*x**2 + 20464285*x + 3090594)/(3542940*x**5 + 11809800*
x**4 + 15746400*x**3 + 10497600*x**2 + 3499200*x + 466560) + 3700*log(3*x + 2)/729

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Giac [A]  time = 2.34738, size = 57, normalized size = 0.8 \begin{align*} -\frac{1000}{729} \, x + \frac{23311800 \, x^{4} + 56207430 \, x^{3} + 50854440 \, x^{2} + 20464285 \, x + 3090594}{14580 \,{\left (3 \, x + 2\right )}^{5}} + \frac{3700}{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)^6,x, algorithm="giac")

[Out]

-1000/729*x + 1/14580*(23311800*x^4 + 56207430*x^3 + 50854440*x^2 + 20464285*x + 3090594)/(3*x + 2)^5 + 3700/7
29*log(abs(3*x + 2))