3.1269 \(\int \frac{(1-2 x)^2 (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=66 \[ \frac{3800}{729 (3 x+2)}-\frac{8285}{1458 (3 x+2)^2}+\frac{4099}{2187 (3 x+2)^3}-\frac{763}{2916 (3 x+2)^4}+\frac{49}{3645 (3 x+2)^5}+\frac{500}{729} \log (3 x+2) \]

[Out]

49/(3645*(2 + 3*x)^5) - 763/(2916*(2 + 3*x)^4) + 4099/(2187*(2 + 3*x)^3) - 8285/
(1458*(2 + 3*x)^2) + 3800/(729*(2 + 3*x)) + (500*Log[2 + 3*x])/729

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Rubi [A]  time = 0.0637013, antiderivative size = 66, 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 \[ \frac{3800}{729 (3 x+2)}-\frac{8285}{1458 (3 x+2)^2}+\frac{4099}{2187 (3 x+2)^3}-\frac{763}{2916 (3 x+2)^4}+\frac{49}{3645 (3 x+2)^5}+\frac{500}{729} \log (3 x+2) \]

Antiderivative was successfully verified.

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

[Out]

49/(3645*(2 + 3*x)^5) - 763/(2916*(2 + 3*x)^4) + 4099/(2187*(2 + 3*x)^3) - 8285/
(1458*(2 + 3*x)^2) + 3800/(729*(2 + 3*x)) + (500*Log[2 + 3*x])/729

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Rubi in Sympy [A]  time = 10.6348, size = 56, normalized size = 0.85 \[ \frac{500 \log{\left (3 x + 2 \right )}}{729} + \frac{3800}{729 \left (3 x + 2\right )} - \frac{8285}{1458 \left (3 x + 2\right )^{2}} + \frac{4099}{2187 \left (3 x + 2\right )^{3}} - \frac{763}{2916 \left (3 x + 2\right )^{4}} + \frac{49}{3645 \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**2*(3+5*x)**3/(2+3*x)**6,x)

[Out]

500*log(3*x + 2)/729 + 3800/(729*(3*x + 2)) - 8285/(1458*(3*x + 2)**2) + 4099/(2
187*(3*x + 2)**3) - 763/(2916*(3*x + 2)**4) + 49/(3645*(3*x + 2)**5)

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Mathematica [A]  time = 0.0530599, size = 46, normalized size = 0.7 \[ \frac{18468000 x^4+42537150 x^3+36564120 x^2+13889625 x+30000 (3 x+2)^5 \log (30 x+20)+1965218}{43740 (3 x+2)^5} \]

Antiderivative was successfully verified.

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

[Out]

(1965218 + 13889625*x + 36564120*x^2 + 42537150*x^3 + 18468000*x^4 + 30000*(2 +
3*x)^5*Log[20 + 30*x])/(43740*(2 + 3*x)^5)

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Maple [A]  time = 0.01, size = 55, normalized size = 0.8 \[{\frac{49}{3645\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{763}{2916\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{4099}{2187\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{8285}{1458\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{3800}{1458+2187\,x}}+{\frac{500\,\ln \left ( 2+3\,x \right ) }{729}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49/3645/(2+3*x)^5-763/2916/(2+3*x)^4+4099/2187/(2+3*x)^3-8285/1458/(2+3*x)^2+380
0/729/(2+3*x)+500/729*ln(2+3*x)

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Maxima [A]  time = 1.32288, size = 78, normalized size = 1.18 \[ \frac{18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{500}{729} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(2*x - 1)^2/(3*x + 2)^6,x, algorithm="maxima")

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218)/(243
*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 500/729*log(3*x + 2)

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Fricas [A]  time = 0.21926, size = 111, normalized size = 1.68 \[ \frac{18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 30000 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 13889625 \, x + 1965218}{43740 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(2*x - 1)^2/(3*x + 2)^6,x, algorithm="fricas")

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 30000*(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 13889625*x + 1965218)/(243*x^5
+ 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.433378, size = 54, normalized size = 0.82 \[ \frac{18468000 x^{4} + 42537150 x^{3} + 36564120 x^{2} + 13889625 x + 1965218}{10628820 x^{5} + 35429400 x^{4} + 47239200 x^{3} + 31492800 x^{2} + 10497600 x + 1399680} + \frac{500 \log{\left (3 x + 2 \right )}}{729} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)**2*(3+5*x)**3/(2+3*x)**6,x)

[Out]

(18468000*x**4 + 42537150*x**3 + 36564120*x**2 + 13889625*x + 1965218)/(10628820
*x**5 + 35429400*x**4 + 47239200*x**3 + 31492800*x**2 + 10497600*x + 1399680) +
500*log(3*x + 2)/729

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GIAC/XCAS [A]  time = 0.219183, size = 53, normalized size = 0.8 \[ \frac{18468000 \, x^{4} + 42537150 \, x^{3} + 36564120 \, x^{2} + 13889625 \, x + 1965218}{43740 \,{\left (3 \, x + 2\right )}^{5}} + \frac{500}{729} \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^3*(2*x - 1)^2/(3*x + 2)^6,x, algorithm="giac")

[Out]

1/43740*(18468000*x^4 + 42537150*x^3 + 36564120*x^2 + 13889625*x + 1965218)/(3*x
+ 2)^5 + 500/729*ln(abs(3*x + 2))