3.1189 \(\int \frac{1-2 x}{(2+3 x)^7 (3+5 x)} \, dx\)

Optimal. Leaf size=81 \[ \frac{6875}{3 x+2}+\frac{1375}{2 (3 x+2)^2}+\frac{275}{3 (3 x+2)^3}+\frac{55}{4 (3 x+2)^4}+\frac{11}{5 (3 x+2)^5}+\frac{7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3) \]

[Out]

7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^
3) + 1375/(2*(2 + 3*x)^2) + 6875/(2 + 3*x) - 34375*Log[2 + 3*x] + 34375*Log[3 +
5*x]

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Rubi [A]  time = 0.0746975, antiderivative size = 81, 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 \[ \frac{6875}{3 x+2}+\frac{1375}{2 (3 x+2)^2}+\frac{275}{3 (3 x+2)^3}+\frac{55}{4 (3 x+2)^4}+\frac{11}{5 (3 x+2)^5}+\frac{7}{18 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

7/(18*(2 + 3*x)^6) + 11/(5*(2 + 3*x)^5) + 55/(4*(2 + 3*x)^4) + 275/(3*(2 + 3*x)^
3) + 1375/(2*(2 + 3*x)^2) + 6875/(2 + 3*x) - 34375*Log[2 + 3*x] + 34375*Log[3 +
5*x]

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Rubi in Sympy [A]  time = 11.1584, size = 73, normalized size = 0.9 \[ - 34375 \log{\left (3 x + 2 \right )} + 34375 \log{\left (5 x + 3 \right )} + \frac{6875}{3 x + 2} + \frac{1375}{2 \left (3 x + 2\right )^{2}} + \frac{275}{3 \left (3 x + 2\right )^{3}} + \frac{55}{4 \left (3 x + 2\right )^{4}} + \frac{11}{5 \left (3 x + 2\right )^{5}} + \frac{7}{18 \left (3 x + 2\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)/(2+3*x)**7/(3+5*x),x)

[Out]

-34375*log(3*x + 2) + 34375*log(5*x + 3) + 6875/(3*x + 2) + 1375/(2*(3*x + 2)**2
) + 275/(3*(3*x + 2)**3) + 55/(4*(3*x + 2)**4) + 11/(5*(3*x + 2)**5) + 7/(18*(3*
x + 2)**6)

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Mathematica [A]  time = 0.0744008, size = 75, normalized size = 0.93 \[ \frac{1237500 (3 x+2)^5+123750 (3 x+2)^4+16500 (3 x+2)^3+2475 (3 x+2)^2+396 (3 x+2)+70}{180 (3 x+2)^6}-34375 \log (3 x+2)+34375 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

(70 + 396*(2 + 3*x) + 2475*(2 + 3*x)^2 + 16500*(2 + 3*x)^3 + 123750*(2 + 3*x)^4
+ 1237500*(2 + 3*x)^5)/(180*(2 + 3*x)^6) - 34375*Log[2 + 3*x] + 34375*Log[-3*(3
+ 5*x)]

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Maple [A]  time = 0.012, size = 72, normalized size = 0.9 \[{\frac{7}{18\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{11}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{55}{4\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{275}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1375}{2\, \left ( 2+3\,x \right ) ^{2}}}+6875\, \left ( 2+3\,x \right ) ^{-1}-34375\,\ln \left ( 2+3\,x \right ) +34375\,\ln \left ( 3+5\,x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)/(2+3*x)^7/(3+5*x),x)

[Out]

7/18/(2+3*x)^6+11/5/(2+3*x)^5+55/4/(2+3*x)^4+275/3/(2+3*x)^3+1375/2/(2+3*x)^2+68
75/(2+3*x)-34375*ln(2+3*x)+34375*ln(3+5*x)

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Maxima [A]  time = 1.35008, size = 103, normalized size = 1.27 \[ \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 34375 \, \log \left (5 \, x + 3\right ) - 34375 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x - 1)/((5*x + 3)*(3*x + 2)^7),x, algorithm="maxima")

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 3095048
88*x + 41722762)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x +
64) + 34375*log(5*x + 3) - 34375*log(3*x + 2)

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Fricas [A]  time = 0.208175, size = 182, normalized size = 2.25 \[ \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 6187500 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 6187500 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 309504888 \, x + 41722762}{180 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x - 1)/((5*x + 3)*(3*x + 2)^7),x, algorithm="fricas")

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 6187500
*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3)
- 6187500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*lo
g(3*x + 2) + 309504888*x + 41722762)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

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Sympy [A]  time = 0.505178, size = 71, normalized size = 0.88 \[ \frac{300712500 x^{5} + 1012398750 x^{4} + 1363675500 x^{3} + 918643275 x^{2} + 309504888 x + 41722762}{131220 x^{6} + 524880 x^{5} + 874800 x^{4} + 777600 x^{3} + 388800 x^{2} + 103680 x + 11520} + 34375 \log{\left (x + \frac{3}{5} \right )} - 34375 \log{\left (x + \frac{2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1-2*x)/(2+3*x)**7/(3+5*x),x)

[Out]

(300712500*x**5 + 1012398750*x**4 + 1363675500*x**3 + 918643275*x**2 + 309504888
*x + 41722762)/(131220*x**6 + 524880*x**5 + 874800*x**4 + 777600*x**3 + 388800*x
**2 + 103680*x + 11520) + 34375*log(x + 3/5) - 34375*log(x + 2/3)

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GIAC/XCAS [A]  time = 0.211074, size = 72, normalized size = 0.89 \[ \frac{300712500 \, x^{5} + 1012398750 \, x^{4} + 1363675500 \, x^{3} + 918643275 \, x^{2} + 309504888 \, x + 41722762}{180 \,{\left (3 \, x + 2\right )}^{6}} + 34375 \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - 34375 \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x - 1)/((5*x + 3)*(3*x + 2)^7),x, algorithm="giac")

[Out]

1/180*(300712500*x^5 + 1012398750*x^4 + 1363675500*x^3 + 918643275*x^2 + 3095048
88*x + 41722762)/(3*x + 2)^6 + 34375*ln(abs(5*x + 3)) - 34375*ln(abs(3*x + 2))