3.1525 \(\int \frac{1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=53 \[ \frac{155}{121 (5 x+3)}-\frac{5}{22 (5 x+3)^2}-\frac{8 \log (1-2 x)}{9317}-\frac{27}{7} \log (3 x+2)+\frac{5135 \log (5 x+3)}{1331} \]

[Out]

-5/(22*(3 + 5*x)^2) + 155/(121*(3 + 5*x)) - (8*Log[1 - 2*x])/9317 - (27*Log[2 + 3*x])/7 + (5135*Log[3 + 5*x])/
1331

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Rubi [A]  time = 0.0231117, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {72} \[ \frac{155}{121 (5 x+3)}-\frac{5}{22 (5 x+3)^2}-\frac{8 \log (1-2 x)}{9317}-\frac{27}{7} \log (3 x+2)+\frac{5135 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-5/(22*(3 + 5*x)^2) + 155/(121*(3 + 5*x)) - (8*Log[1 - 2*x])/9317 - (27*Log[2 + 3*x])/7 + (5135*Log[3 + 5*x])/
1331

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x) (2+3 x) (3+5 x)^3} \, dx &=\int \left (-\frac{16}{9317 (-1+2 x)}-\frac{81}{7 (2+3 x)}+\frac{25}{11 (3+5 x)^3}-\frac{775}{121 (3+5 x)^2}+\frac{25675}{1331 (3+5 x)}\right ) \, dx\\ &=-\frac{5}{22 (3+5 x)^2}+\frac{155}{121 (3+5 x)}-\frac{8 \log (1-2 x)}{9317}-\frac{27}{7} \log (2+3 x)+\frac{5135 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0348979, size = 43, normalized size = 0.81 \[ \frac{\frac{1925 (62 x+35)}{(5 x+3)^2}-16 \log (1-2 x)-71874 \log (6 x+4)+71890 \log (10 x+6)}{18634} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1925*(35 + 62*x))/(3 + 5*x)^2 - 16*Log[1 - 2*x] - 71874*Log[4 + 6*x] + 71890*Log[6 + 10*x])/18634

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Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \begin{align*} -{\frac{8\,\ln \left ( 2\,x-1 \right ) }{9317}}-{\frac{27\,\ln \left ( 2+3\,x \right ) }{7}}-{\frac{5}{22\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{155}{363+605\,x}}+{\frac{5135\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/9317*ln(2*x-1)-27/7*ln(2+3*x)-5/22/(3+5*x)^2+155/121/(3+5*x)+5135/1331*ln(3+5*x)

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Maxima [A]  time = 1.15368, size = 59, normalized size = 1.11 \begin{align*} \frac{25 \,{\left (62 \, x + 35\right )}}{242 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{5135}{1331} \, \log \left (5 \, x + 3\right ) - \frac{27}{7} \, \log \left (3 \, x + 2\right ) - \frac{8}{9317} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/242*(62*x + 35)/(25*x^2 + 30*x + 9) + 5135/1331*log(5*x + 3) - 27/7*log(3*x + 2) - 8/9317*log(2*x - 1)

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Fricas [A]  time = 1.36417, size = 227, normalized size = 4.28 \begin{align*} \frac{71890 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 71874 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (3 \, x + 2\right ) - 16 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 119350 \, x + 67375}{18634 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18634*(71890*(25*x^2 + 30*x + 9)*log(5*x + 3) - 71874*(25*x^2 + 30*x + 9)*log(3*x + 2) - 16*(25*x^2 + 30*x +
9)*log(2*x - 1) + 119350*x + 67375)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.183126, size = 44, normalized size = 0.83 \begin{align*} \frac{1550 x + 875}{6050 x^{2} + 7260 x + 2178} - \frac{8 \log{\left (x - \frac{1}{2} \right )}}{9317} + \frac{5135 \log{\left (x + \frac{3}{5} \right )}}{1331} - \frac{27 \log{\left (x + \frac{2}{3} \right )}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1550*x + 875)/(6050*x**2 + 7260*x + 2178) - 8*log(x - 1/2)/9317 + 5135*log(x + 3/5)/1331 - 27*log(x + 2/3)/7

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Giac [A]  time = 1.24844, size = 57, normalized size = 1.08 \begin{align*} \frac{25 \,{\left (62 \, x + 35\right )}}{242 \,{\left (5 \, x + 3\right )}^{2}} + \frac{5135}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{27}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{8}{9317} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/242*(62*x + 35)/(5*x + 3)^2 + 5135/1331*log(abs(5*x + 3)) - 27/7*log(abs(3*x + 2)) - 8/9317*log(abs(2*x - 1
))