3.1691 \(\int \frac{2+3 x}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{37}{1331 (1-2 x)}-\frac{5}{1331 (5 x+3)}+\frac{7}{242 (1-2 x)^2}-\frac{195 \log (1-2 x)}{14641}+\frac{195 \log (5 x+3)}{14641} \]

[Out]

7/(242*(1 - 2*x)^2) + 37/(1331*(1 - 2*x)) - 5/(1331*(3 + 5*x)) - (195*Log[1 - 2*x])/14641 + (195*Log[3 + 5*x])
/14641

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Rubi [A]  time = 0.0235353, antiderivative size = 54, 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} \[ \frac{37}{1331 (1-2 x)}-\frac{5}{1331 (5 x+3)}+\frac{7}{242 (1-2 x)^2}-\frac{195 \log (1-2 x)}{14641}+\frac{195 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(242*(1 - 2*x)^2) + 37/(1331*(1 - 2*x)) - 5/(1331*(3 + 5*x)) - (195*Log[1 - 2*x])/14641 + (195*Log[3 + 5*x])
/14641

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 (3+5 x)^2} \, dx &=\int \left (-\frac{14}{121 (-1+2 x)^3}+\frac{74}{1331 (-1+2 x)^2}-\frac{390}{14641 (-1+2 x)}+\frac{25}{1331 (3+5 x)^2}+\frac{975}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{7}{242 (1-2 x)^2}+\frac{37}{1331 (1-2 x)}-\frac{5}{1331 (3+5 x)}-\frac{195 \log (1-2 x)}{14641}+\frac{195 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0180874, size = 62, normalized size = 1.15 \[ \frac{10}{1331 (5 (1-2 x)-11)}+\frac{37}{1331 (1-2 x)}+\frac{7}{242 (1-2 x)^2}+\frac{195 \log (11-5 (1-2 x))}{14641}-\frac{195 \log (1-2 x)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

10/(1331*(-11 + 5*(1 - 2*x))) + 7/(242*(1 - 2*x)^2) + 37/(1331*(1 - 2*x)) + (195*Log[11 - 5*(1 - 2*x)])/14641
- (195*Log[1 - 2*x])/14641

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Maple [A]  time = 0.009, size = 45, normalized size = 0.8 \begin{align*}{\frac{7}{242\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{37}{2662\,x-1331}}-{\frac{195\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{5}{3993+6655\,x}}+{\frac{195\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/242/(2*x-1)^2-37/1331/(2*x-1)-195/14641*ln(2*x-1)-5/1331/(3+5*x)+195/14641*ln(3+5*x)

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Maxima [A]  time = 1.04547, size = 62, normalized size = 1.15 \begin{align*} -\frac{780 \, x^{2} - 351 \, x - 443}{2662 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{195}{14641} \, \log \left (5 \, x + 3\right ) - \frac{195}{14641} \, \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/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/2662*(780*x^2 - 351*x - 443)/(20*x^3 - 8*x^2 - 7*x + 3) + 195/14641*log(5*x + 3) - 195/14641*log(2*x - 1)

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Fricas [A]  time = 1.49958, size = 211, normalized size = 3.91 \begin{align*} -\frac{8580 \, x^{2} - 390 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 390 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 3861 \, x - 4873}{29282 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(8580*x^2 - 390*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 390*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1)
- 3861*x - 4873)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.152774, size = 44, normalized size = 0.81 \begin{align*} - \frac{780 x^{2} - 351 x - 443}{53240 x^{3} - 21296 x^{2} - 18634 x + 7986} - \frac{195 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{195 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(780*x**2 - 351*x - 443)/(53240*x**3 - 21296*x**2 - 18634*x + 7986) - 195*log(x - 1/2)/14641 + 195*log(x + 3/
5)/14641

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Giac [A]  time = 3.62931, size = 69, normalized size = 1.28 \begin{align*} -\frac{5}{1331 \,{\left (5 \, x + 3\right )}} + \frac{10 \,{\left (\frac{792}{5 \, x + 3} - 109\right )}}{14641 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{195}{14641} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/1331/(5*x + 3) + 10/14641*(792/(5*x + 3) - 109)/(11/(5*x + 3) - 2)^2 - 195/14641*log(abs(-11/(5*x + 3) + 2)
)