3.1606 \(\int \frac{(2+3 x)^5}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=55 \[ \frac{243 x^2}{200}+\frac{3807 x}{500}+\frac{16807}{1936 (1-2 x)}-\frac{1}{75625 (5 x+3)}+\frac{228095 \log (1-2 x)}{21296}+\frac{169 \log (5 x+3)}{831875} \]

[Out]

16807/(1936*(1 - 2*x)) + (3807*x)/500 + (243*x^2)/200 - 1/(75625*(3 + 5*x)) + (228095*Log[1 - 2*x])/21296 + (1
69*Log[3 + 5*x])/831875

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Rubi [A]  time = 0.0260584, antiderivative size = 55, 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 = {88} \[ \frac{243 x^2}{200}+\frac{3807 x}{500}+\frac{16807}{1936 (1-2 x)}-\frac{1}{75625 (5 x+3)}+\frac{228095 \log (1-2 x)}{21296}+\frac{169 \log (5 x+3)}{831875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(1936*(1 - 2*x)) + (3807*x)/500 + (243*x^2)/200 - 1/(75625*(3 + 5*x)) + (228095*Log[1 - 2*x])/21296 + (1
69*Log[3 + 5*x])/831875

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^5}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{3807}{500}+\frac{243 x}{100}+\frac{16807}{968 (-1+2 x)^2}+\frac{228095}{10648 (-1+2 x)}+\frac{1}{15125 (3+5 x)^2}+\frac{169}{166375 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{1936 (1-2 x)}+\frac{3807 x}{500}+\frac{243 x^2}{200}-\frac{1}{75625 (3+5 x)}+\frac{228095 \log (1-2 x)}{21296}+\frac{169 \log (3+5 x)}{831875}\\ \end{align*}

Mathematica [A]  time = 0.0269572, size = 56, normalized size = 1.02 \[ \frac{-\frac{11 (52521907 x+31513109)}{10 x^2+x-3}+1796850 (3 x+2)^2+26593380 (3 x+2)[phone] \log (3-6 x)+2704 \log (-3 (5 x+3))}{13310000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(26593380*(2 + 3*x) + 1796850*(2 + 3*x)^2 - (11*(31513109 + 52521907*x))/(-3 + x + 10*x^2) + 142559375*Log[3 -
6*x] + 2704*Log[-3*(3 + 5*x)])/13310000

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Maple [A]  time = 0.008, size = 44, normalized size = 0.8 \begin{align*}{\frac{243\,{x}^{2}}{200}}+{\frac{3807\,x}{500}}-{\frac{16807}{3872\,x-1936}}+{\frac{228095\,\ln \left ( 2\,x-1 \right ) }{21296}}-{\frac{1}{226875+378125\,x}}+{\frac{169\,\ln \left ( 3+5\,x \right ) }{831875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/200*x^2+3807/500*x-16807/1936/(2*x-1)+228095/21296*ln(2*x-1)-1/75625/(3+5*x)+169/831875*ln(3+5*x)

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Maxima [A]  time = 2.55677, size = 57, normalized size = 1.04 \begin{align*} \frac{243}{200} \, x^{2} + \frac{3807}{500} \, x - \frac{52521907 \, x + 31513109}{1210000 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{169}{831875} \, \log \left (5 \, x + 3\right ) + \frac{228095}{21296} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/200*x^2 + 3807/500*x - 1/1210000*(52521907*x + 31513109)/(10*x^2 + x - 3) + 169/831875*log(5*x + 3) + 2280
95/21296*log(2*x - 1)

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Fricas [A]  time = 1.39118, size = 246, normalized size = 4.47 \begin{align*} \frac{161716500 \, x^{4} + 1029595050 \, x^{3} + 52827390 \, x^{2} + 2704 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 142559375 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 881767997 \, x - 346644199}{13310000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13310000*(161716500*x^4 + 1029595050*x^3 + 52827390*x^2 + 2704*(10*x^2 + x - 3)*log(5*x + 3) + 142559375*(10
*x^2 + x - 3)*log(2*x - 1) - 881767997*x - 346644199)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.157936, size = 46, normalized size = 0.84 \begin{align*} \frac{243 x^{2}}{200} + \frac{3807 x}{500} - \frac{52521907 x + 31513109}{12100000 x^{2} + 1210000 x - 3630000} + \frac{228095 \log{\left (x - \frac{1}{2} \right )}}{21296} + \frac{169 \log{\left (x + \frac{3}{5} \right )}}{831875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243*x**2/200 + 3807*x/500 - (52521907*x + 31513109)/(12100000*x**2 + 1210000*x - 3630000) + 228095*log(x - 1/2
)/21296 + 169*log(x + 3/5)/831875

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Giac [A]  time = 2.79911, size = 115, normalized size = 2.09 \begin{align*} -\frac{{\left (5 \, x + 3\right )}^{2}{\left (\frac{12829509}{5 \, x + 3} - \frac{142651871}{{\left (5 \, x + 3\right )}^{2}} + 646866\right )}}{6655000 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{1}{75625 \,{\left (5 \, x + 3\right )}} - \frac{107109}{10000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{228095}{21296} \, \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)^5/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/6655000*(5*x + 3)^2*(12829509/(5*x + 3) - 142651871/(5*x + 3)^2 + 646866)/(11/(5*x + 3) - 2) - 1/75625/(5*x
+ 3) - 107109/10000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 228095/21296*log(abs(-11/(5*x + 3) + 2))