3.1523 \(\int \frac{2+3 x}{(1-2 x) (3+5 x)^3} \, dx\)

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 = 0.0157576, 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, Rules used = {77} \[ -\frac{7}{121 (5 x+3)}-\frac{1}{110 (5 x+3)^2}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (5 x+3)}{1331} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{2+3 x}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac{28}{1331 (-1+2 x)}+\frac{1}{11 (3+5 x)^3}+\frac{35}{121 (3+5 x)^2}+\frac{70}{1331 (3+5 x)}\right ) \, dx\\ &=-\frac{1}{110 (3+5 x)^2}-\frac{7}{121 (3+5 x)}-\frac{14 \log (1-2 x)}{1331}+\frac{14 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.018053, 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 = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{14\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{110\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{7}{363+605\,x}}+{\frac{14\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.20983, size = 49, normalized size = 1.14 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(1-2*x)/(3+5*x)^3,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 = 1.31476, size = 166, normalized size = 3.86 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)/(1-2*x)/(3+5*x)^3,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 = 0.135476, size = 34, normalized size = 0.79 \begin{align*} - \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} \end{align*}

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 [A]  time = 1.2838, size = 45, normalized size = 1.05 \begin{align*} -\frac{350 \, x + 221}{1210 \,{\left (5 \, x + 3\right )}^{2}} + \frac{14}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{14}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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