3.1188 \(\int \frac{1-2 x}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=70 \[ \frac{1375}{3 x+2}+\frac{275}{2 (3 x+2)^2}+\frac{55}{3 (3 x+2)^3}+\frac{11}{4 (3 x+2)^4}+\frac{7}{15 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (5 x+3) \]

[Out]

7/(15*(2 + 3*x)^5) + 11/(4*(2 + 3*x)^4) + 55/(3*(2 + 3*x)^3) + 275/(2*(2 + 3*x)^
2) + 1375/(2 + 3*x) - 6875*Log[2 + 3*x] + 6875*Log[3 + 5*x]

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Rubi [A]  time = 0.0674979, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1375}{3 x+2}+\frac{275}{2 (3 x+2)^2}+\frac{55}{3 (3 x+2)^3}+\frac{11}{4 (3 x+2)^4}+\frac{7}{15 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (5 x+3) \]

Antiderivative was successfully verified.

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

[Out]

7/(15*(2 + 3*x)^5) + 11/(4*(2 + 3*x)^4) + 55/(3*(2 + 3*x)^3) + 275/(2*(2 + 3*x)^
2) + 1375/(2 + 3*x) - 6875*Log[2 + 3*x] + 6875*Log[3 + 5*x]

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Rubi in Sympy [A]  time = 9.97079, size = 63, normalized size = 0.9 \[ - 6875 \log{\left (3 x + 2 \right )} + 6875 \log{\left (5 x + 3 \right )} + \frac{1375}{3 x + 2} + \frac{275}{2 \left (3 x + 2\right )^{2}} + \frac{55}{3 \left (3 x + 2\right )^{3}} + \frac{11}{4 \left (3 x + 2\right )^{4}} + \frac{7}{15 \left (3 x + 2\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6875*log(3*x + 2) + 6875*log(5*x + 3) + 1375/(3*x + 2) + 275/(2*(3*x + 2)**2) +
55/(3*(3*x + 2)**3) + 11/(4*(3*x + 2)**4) + 7/(15*(3*x + 2)**5)

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Mathematica [A]  time = 0.0505967, size = 50, normalized size = 0.71 \[ \frac{2227500 x^4+6014250 x^3+6091800 x^2+2743565 x+463586}{20 (3 x+2)^5}-6875 \log (3 x+2)+6875 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

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

[Out]

(463586 + 2743565*x + 6091800*x^2 + 6014250*x^3 + 2227500*x^4)/(20*(2 + 3*x)^5)
- 6875*Log[2 + 3*x] + 6875*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.013, size = 63, normalized size = 0.9 \[{\frac{7}{15\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{11}{4\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{55}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{275}{2\, \left ( 2+3\,x \right ) ^{2}}}+1375\, \left ( 2+3\,x \right ) ^{-1}-6875\,\ln \left ( 2+3\,x \right ) +6875\,\ln \left ( 3+5\,x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7/15/(2+3*x)^5+11/4/(2+3*x)^4+55/3/(2+3*x)^3+275/2/(2+3*x)^2+1375/(2+3*x)-6875*l
n(2+3*x)+6875*ln(3+5*x)

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Maxima [A]  time = 1.35073, size = 89, normalized size = 1.27 \[ \frac{2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 2743565 \, x + 463586}{20 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + 6875 \, \log \left (5 \, x + 3\right ) - 6875 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 2743565*x + 463586)/(243*x^5 + 8
10*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 6875*log(5*x + 3) - 6875*log(3*x + 2
)

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Fricas [A]  time = 0.211704, size = 155, normalized size = 2.21 \[ \frac{2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 137500 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (5 \, x + 3\right ) - 137500 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 2743565 \, x + 463586}{20 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 137500*(243*x^5 + 810*x^4 + 1080
*x^3 + 720*x^2 + 240*x + 32)*log(5*x + 3) - 137500*(243*x^5 + 810*x^4 + 1080*x^3
+ 720*x^2 + 240*x + 32)*log(3*x + 2) + 2743565*x + 463586)/(243*x^5 + 810*x^4 +
1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.460215, size = 61, normalized size = 0.87 \[ \frac{2227500 x^{4} + 6014250 x^{3} + 6091800 x^{2} + 2743565 x + 463586}{4860 x^{5} + 16200 x^{4} + 21600 x^{3} + 14400 x^{2} + 4800 x + 640} + 6875 \log{\left (x + \frac{3}{5} \right )} - 6875 \log{\left (x + \frac{2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2227500*x**4 + 6014250*x**3 + 6091800*x**2 + 2743565*x + 463586)/(4860*x**5 + 1
6200*x**4 + 21600*x**3 + 14400*x**2 + 4800*x + 640) + 6875*log(x + 3/5) - 6875*l
og(x + 2/3)

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GIAC/XCAS [A]  time = 0.206477, size = 65, normalized size = 0.93 \[ \frac{2227500 \, x^{4} + 6014250 \, x^{3} + 6091800 \, x^{2} + 2743565 \, x + 463586}{20 \,{\left (3 \, x + 2\right )}^{5}} + 6875 \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - 6875 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(2227500*x^4 + 6014250*x^3 + 6091800*x^2 + 2743565*x + 463586)/(3*x + 2)^5
+ 6875*ln(abs(5*x + 3)) - 6875*ln(abs(3*x + 2))