3.1497 \(\int \frac{1}{(1-2 x) (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=64 \[ \frac{3897}{343 (3 x+2)}+\frac{111}{98 (3 x+2)^2}+\frac{1}{7 (3 x+2)^3}-\frac{16 \log (1-2 x)}{26411}-\frac{136419 \log (3 x+2)}{2401}+\frac{625}{11} \log (5 x+3) \]

[Out]

1/(7*(2 + 3*x)^3) + 111/(98*(2 + 3*x)^2) + 3897/(343*(2 + 3*x)) - (16*Log[1 - 2*x])/26411 - (136419*Log[2 + 3*
x])/2401 + (625*Log[3 + 5*x])/11

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Rubi [A]  time = 0.0281747, antiderivative size = 64, 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} \[ \frac{3897}{343 (3 x+2)}+\frac{111}{98 (3 x+2)^2}+\frac{1}{7 (3 x+2)^3}-\frac{16 \log (1-2 x)}{26411}-\frac{136419 \log (3 x+2)}{2401}+\frac{625}{11} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

1/(7*(2 + 3*x)^3) + 111/(98*(2 + 3*x)^2) + 3897/(343*(2 + 3*x)) - (16*Log[1 - 2*x])/26411 - (136419*Log[2 + 3*
x])/2401 + (625*Log[3 + 5*x])/11

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)^4 (3+5 x)} \, dx &=\int \left (-\frac{32}{26411 (-1+2 x)}-\frac{9}{7 (2+3 x)^4}-\frac{333}{49 (2+3 x)^3}-\frac{11691}{343 (2+3 x)^2}-\frac{409257}{2401 (2+3 x)}+\frac{3125}{11 (3+5 x)}\right ) \, dx\\ &=\frac{1}{7 (2+3 x)^3}+\frac{111}{98 (2+3 x)^2}+\frac{3897}{343 (2+3 x)}-\frac{16 \log (1-2 x)}{26411}-\frac{136419 \log (2+3 x)}{2401}+\frac{625}{11} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0477001, size = 50, normalized size = 0.78 \[ \frac{\frac{77 \left (70146 x^2+95859 x+32828\right )}{2 (3 x+2)^3}-16 \log (1-2 x)-1500609 \log (6 x+4)+1500625 \log (10 x+6)}{26411} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)),x]

[Out]

((77*(32828 + 95859*x + 70146*x^2))/(2*(2 + 3*x)^3) - 16*Log[1 - 2*x] - 1500609*Log[4 + 6*x] + 1500625*Log[6 +
10*x])/26411

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Maple [A]  time = 0.008, size = 53, normalized size = 0.8 \begin{align*} -{\frac{16\,\ln \left ( 2\,x-1 \right ) }{26411}}+{\frac{1}{7\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{111}{98\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{3897}{686+1029\,x}}-{\frac{136419\,\ln \left ( 2+3\,x \right ) }{2401}}+{\frac{625\,\ln \left ( 3+5\,x \right ) }{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)/(2+3*x)^4/(3+5*x),x)

[Out]

-16/26411*ln(2*x-1)+1/7/(2+3*x)^3+111/98/(2+3*x)^2+3897/343/(2+3*x)-136419/2401*ln(2+3*x)+625/11*ln(3+5*x)

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Maxima [A]  time = 1.17172, size = 73, normalized size = 1.14 \begin{align*} \frac{70146 \, x^{2} + 95859 \, x + 32828}{686 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{625}{11} \, \log \left (5 \, x + 3\right ) - \frac{136419}{2401} \, \log \left (3 \, x + 2\right ) - \frac{16}{26411} \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

1/686*(70146*x^2 + 95859*x + 32828)/(27*x^3 + 54*x^2 + 36*x + 8) + 625/11*log(5*x + 3) - 136419/2401*log(3*x +
2) - 16/26411*log(2*x - 1)

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Fricas [A]  time = 1.31921, size = 304, normalized size = 4.75 \begin{align*} \frac{5401242 \, x^{2} + 3001250 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 3001218 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 32 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 7381143 \, x + 2527756}{52822 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/52822*(5401242*x^2 + 3001250*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 3001218*(27*x^3 + 54*x^2 + 36*x + 8
)*log(3*x + 2) - 32*(27*x^3 + 54*x^2 + 36*x + 8)*log(2*x - 1) + 7381143*x + 2527756)/(27*x^3 + 54*x^2 + 36*x +
8)

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Sympy [A]  time = 0.201466, size = 54, normalized size = 0.84 \begin{align*} \frac{70146 x^{2} + 95859 x + 32828}{18522 x^{3} + 37044 x^{2} + 24696 x + 5488} - \frac{16 \log{\left (x - \frac{1}{2} \right )}}{26411} + \frac{625 \log{\left (x + \frac{3}{5} \right )}}{11} - \frac{136419 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)**4/(3+5*x),x)

[Out]

(70146*x**2 + 95859*x + 32828)/(18522*x**3 + 37044*x**2 + 24696*x + 5488) - 16*log(x - 1/2)/26411 + 625*log(x
+ 3/5)/11 - 136419*log(x + 2/3)/2401

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Giac [A]  time = 1.5211, size = 63, normalized size = 0.98 \begin{align*} \frac{70146 \, x^{2} + 95859 \, x + 32828}{686 \,{\left (3 \, x + 2\right )}^{3}} + \frac{625}{11} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{136419}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{16}{26411} \, \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)^4/(3+5*x),x, algorithm="giac")

[Out]

1/686*(70146*x^2 + 95859*x + 32828)/(3*x + 2)^3 + 625/11*log(abs(5*x + 3)) - 136419/2401*log(abs(3*x + 2)) - 1
6/26411*log(abs(2*x - 1))