3.1519 \(\int \frac{(2+3 x)^5}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac{243 x^2}{500}-\frac{4941 x}{2500}-\frac{167}{378125 (5 x+3)}-\frac{1}{68750 (5 x+3)^2}-\frac{16807 \log (1-2 x)}{10648}+\frac{11224 \log (5 x+3)}{4159375} \]

[Out]

(-4941*x)/2500 - (243*x^2)/500 - 1/(68750*(3 + 5*x)^2) - 167/(378125*(3 + 5*x)) - (16807*Log[1 - 2*x])/10648 +
(11224*Log[3 + 5*x])/4159375

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Rubi [A]  time = 0.0239801, 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} \[ -\frac{243 x^2}{500}-\frac{4941 x}{2500}-\frac{167}{378125 (5 x+3)}-\frac{1}{68750 (5 x+3)^2}-\frac{16807 \log (1-2 x)}{10648}+\frac{11224 \log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4941*x)/2500 - (243*x^2)/500 - 1/(68750*(3 + 5*x)^2) - 167/(378125*(3 + 5*x)) - (16807*Log[1 - 2*x])/10648 +
(11224*Log[3 + 5*x])/4159375

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) (3+5 x)^3} \, dx &=\int \left (-\frac{4941}{2500}-\frac{243 x}{250}-\frac{16807}{5324 (-1+2 x)}+\frac{1}{6875 (3+5 x)^3}+\frac{167}{75625 (3+5 x)^2}+\frac{11224}{831875 (3+5 x)}\right ) \, dx\\ &=-\frac{4941 x}{2500}-\frac{243 x^2}{500}-\frac{1}{68750 (3+5 x)^2}-\frac{167}{378125 (3+5 x)}-\frac{16807 \log (1-2 x)}{10648}+\frac{11224 \log (3+5 x)}{4159375}\\ \end{align*}

Mathematica [A]  time = 0.0259825, size = 50, normalized size = 0.91 \[ \frac{-\frac{11 \left (73507500 x^4+387139500 x^3+217337175 x^2-93782210 x-60415061\right )}{(5 x+3)^2}-105043750 \log (1-2 x)+179584 \log (10 x+6)}{66550000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-60415061 - 93782210*x + 217337175*x^2 + 387139500*x^3 + 73507500*x^4))/(3 + 5*x)^2 - 105043750*Log[1 -
2*x] + 179584*Log[6 + 10*x])/66550000

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Maple [A]  time = 0.009, size = 44, normalized size = 0.8 \begin{align*} -{\frac{243\,{x}^{2}}{500}}-{\frac{4941\,x}{2500}}-{\frac{16807\,\ln \left ( 2\,x-1 \right ) }{10648}}-{\frac{1}{68750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{167}{1134375+1890625\,x}}+{\frac{11224\,\ln \left ( 3+5\,x \right ) }{4159375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/500*x^2-4941/2500*x-16807/10648*ln(2*x-1)-1/68750/(3+5*x)^2-167/378125/(3+5*x)+11224/4159375*ln(3+5*x)

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Maxima [A]  time = 1.21372, size = 59, normalized size = 1.07 \begin{align*} -\frac{243}{500} \, x^{2} - \frac{4941}{2500} \, x - \frac{1670 \, x + 1013}{756250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{11224}{4159375} \, \log \left (5 \, x + 3\right ) - \frac{16807}{10648} \, \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)/(3+5*x)^3,x, algorithm="maxima")

[Out]

-243/500*x^2 - 4941/2500*x - 1/756250*(1670*x + 1013)/(25*x^2 + 30*x + 9) + 11224/4159375*log(5*x + 3) - 16807
/10648*log(2*x - 1)

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Fricas [A]  time = 1.29283, size = 257, normalized size = 4.67 \begin{align*} -\frac{404291250 \, x^{4} + 2129267250 \, x^{3} + 2118486150 \, x^{2} - 89792 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 52521875 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 591955870 \, x + 44572}{33275000 \,{\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)^5/(1-2*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/33275000*(404291250*x^4 + 2129267250*x^3 + 2118486150*x^2 - 89792*(25*x^2 + 30*x + 9)*log(5*x + 3) + 525218
75*(25*x^2 + 30*x + 9)*log(2*x - 1) + 591955870*x + 44572)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.16732, size = 46, normalized size = 0.84 \begin{align*} - \frac{243 x^{2}}{500} - \frac{4941 x}{2500} - \frac{1670 x + 1013}{18906250 x^{2} + 22687500 x + 6806250} - \frac{16807 \log{\left (x - \frac{1}{2} \right )}}{10648} + \frac{11224 \log{\left (x + \frac{3}{5} \right )}}{4159375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**2/500 - 4941*x/2500 - (1670*x + 1013)/(18906250*x**2 + 22687500*x + 6806250) - 16807*log(x - 1/2)/1064
8 + 11224*log(x + 3/5)/4159375

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Giac [A]  time = 1.24815, size = 55, normalized size = 1. \begin{align*} -\frac{243}{500} \, x^{2} - \frac{4941}{2500} \, x - \frac{1670 \, x + 1013}{756250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{11224}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{16807}{10648} \, \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)^5/(1-2*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-243/500*x^2 - 4941/2500*x - 1/756250*(1670*x + 1013)/(5*x + 3)^2 + 11224/4159375*log(abs(5*x + 3)) - 16807/10
648*log(abs(2*x - 1))