3.1506 \(\int \frac{2+3 x}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{7}{121 (5 x+3)}-\frac{1}{110 (5 x+3)^2}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (5 x+3)}{1331} \]

[Out]

-1/(110*(3 + 5*x)^2) - 7/(121*(3 + 5*x)) - (14*Log[1 - 2*x])/1331 + (14*Log[3 +
5*x])/1331

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Rubi [A]  time = 0.0437471, antiderivative size = 43, 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 \[ -\frac{7}{121 (5 x+3)}-\frac{1}{110 (5 x+3)^2}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

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

[Out]

-1/(110*(3 + 5*x)^2) - 7/(121*(3 + 5*x)) - (14*Log[1 - 2*x])/1331 + (14*Log[3 +
5*x])/1331

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Rubi in Sympy [A]  time = 7.3005, size = 36, normalized size = 0.84 \[ - \frac{14 \log{\left (- 2 x + 1 \right )}}{1331} + \frac{14 \log{\left (5 x + 3 \right )}}{1331} - \frac{7}{121 \left (5 x + 3\right )} - \frac{1}{110 \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-14*log(-2*x + 1)/1331 + 14*log(5*x + 3)/1331 - 7/(121*(5*x + 3)) - 1/(110*(5*x
+ 3)**2)

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Mathematica [A]  time = 0.0294183, size = 35, normalized size = 0.81 \[ \frac{-\frac{11 (350 x+221)}{(5 x+3)^2}-140 \log (5-10 x)+140 \log (5 x+3)}{13310} \]

Antiderivative was successfully verified.

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

[Out]

((-11*(221 + 350*x))/(3 + 5*x)^2 - 140*Log[5 - 10*x] + 140*Log[3 + 5*x])/13310

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Maple [A]  time = 0.01, size = 36, normalized size = 0.8 \[ -{\frac{1}{110\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{7}{363+605\,x}}+{\frac{14\,\ln \left ( 3+5\,x \right ) }{1331}}-{\frac{14\,\ln \left ( -1+2\,x \right ) }{1331}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/110/(3+5*x)^2-7/121/(3+5*x)+14/1331*ln(3+5*x)-14/1331*ln(-1+2*x)

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Maxima [A]  time = 1.34092, size = 49, normalized size = 1.14 \[ -\frac{350 \, x + 221}{1210 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{14}{1331} \, \log \left (5 \, x + 3\right ) - \frac{14}{1331} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1210*(350*x + 221)/(25*x^2 + 30*x + 9) + 14/1331*log(5*x + 3) - 14/1331*log(2
*x - 1)

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Fricas [A]  time = 0.210773, size = 74, normalized size = 1.72 \[ \frac{140 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 140 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 3850 \, x - 2431}{13310 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13310*(140*(25*x^2 + 30*x + 9)*log(5*x + 3) - 140*(25*x^2 + 30*x + 9)*log(2*x
- 1) - 3850*x - 2431)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.335433, size = 34, normalized size = 0.79 \[ - \frac{350 x + 221}{30250 x^{2} + 36300 x + 10890} - \frac{14 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{14 \log{\left (x + \frac{3}{5} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(350*x + 221)/(30250*x**2 + 36300*x + 10890) - 14*log(x - 1/2)/1331 + 14*log(x
+ 3/5)/1331

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GIAC/XCAS [A]  time = 0.205512, size = 45, normalized size = 1.05 \[ -\frac{350 \, x + 221}{1210 \,{\left (5 \, x + 3\right )}^{2}} + \frac{14}{1331} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ) - \frac{14}{1331} \,{\rm ln}\left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1210*(350*x + 221)/(5*x + 3)^2 + 14/1331*ln(abs(5*x + 3)) - 14/1331*ln(abs(2*
x - 1))