3.1485 \(\int \frac{(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=68 \[ -\frac{6561 x^7}{70}-\frac{114453 x^6}{200}-\frac{8018271 x^5}{5000}-\frac{111146499 x^4}{40000}-\frac{345533877 x^3}{100000}-\frac{7136193339 x^2}{2000000}-\frac{40089855591 x}{10000000}-\frac{5764801 \log (1-2 x)}{2816}+\frac{\log (5 x+3)}{4296875} \]

[Out]

(-40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100000 - (111146499*x^4)/40000 - (80182
71*x^5)/5000 - (114453*x^6)/200 - (6561*x^7)/70 - (5764801*Log[1 - 2*x])/2816 + Log[3 + 5*x]/4296875

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Rubi [A]  time = 0.0301892, antiderivative size = 68, 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} \[ -\frac{6561 x^7}{70}-\frac{114453 x^6}{200}-\frac{8018271 x^5}{5000}-\frac{111146499 x^4}{40000}-\frac{345533877 x^3}{100000}-\frac{7136193339 x^2}{2000000}-\frac{40089855591 x}{10000000}-\frac{5764801 \log (1-2 x)}{2816}+\frac{\log (5 x+3)}{4296875} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^8/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-40089855591*x)/10000000 - (7136193339*x^2)/2000000 - (345533877*x^3)/100000 - (111146499*x^4)/40000 - (80182
71*x^5)/5000 - (114453*x^6)/200 - (6561*x^7)/70 - (5764801*Log[1 - 2*x])/2816 + Log[3 + 5*x]/4296875

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{(2+3 x)^8}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{40089855591}{10000000}-\frac{7136193339 x}{1000000}-\frac{1036601631 x^2}{100000}-\frac{111146499 x^3}{10000}-\frac{8018271 x^4}{1000}-\frac{343359 x^5}{100}-\frac{6561 x^6}{10}-\frac{5764801}{1408 (-1+2 x)}+\frac{1}{859375 (3+5 x)}\right ) \, dx\\ &=-\frac{40089855591 x}{10000000}-\frac{7136193339 x^2}{2000000}-\frac{345533877 x^3}{100000}-\frac{111146499 x^4}{40000}-\frac{8018271 x^5}{5000}-\frac{114453 x^6}{200}-\frac{6561 x^7}{70}-\frac{5764801 \log (1-2 x)}{2816}+\frac{\log (3+5 x)}{4296875}\\ \end{align*}

Mathematica [A]  time = 0.0259529, size = 62, normalized size = 0.91 \[ -\frac{3 \left (2187000000 x^7+13352850000 x^6+37418598000 x^5+64835457750 x^4+80624571300 x^3+83255588955 x^2+93542996379 x+40324556806\right )}{70000000}-\frac{5764801 \log (3-6 x)}{2816}+\frac{\log (-3 (5 x+3))}{4296875} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^8/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-3*(40324556806 + 93542996379*x + 83255588955*x^2 + 80624571300*x^3 + 64835457750*x^4 + 37418598000*x^5 + 133
52850000*x^6 + 2187000000*x^7))/70000000 - (5764801*Log[3 - 6*x])/2816 + Log[-3*(3 + 5*x)]/4296875

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Maple [A]  time = 0.006, size = 51, normalized size = 0.8 \begin{align*} -{\frac{6561\,{x}^{7}}{70}}-{\frac{114453\,{x}^{6}}{200}}-{\frac{8018271\,{x}^{5}}{5000}}-{\frac{111146499\,{x}^{4}}{40000}}-{\frac{345533877\,{x}^{3}}{100000}}-{\frac{7136193339\,{x}^{2}}{2000000}}-{\frac{40089855591\,x}{10000000}}-{\frac{5764801\,\ln \left ( 2\,x-1 \right ) }{2816}}+{\frac{\ln \left ( 3+5\,x \right ) }{4296875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^8/(1-2*x)/(3+5*x),x)

[Out]

-6561/70*x^7-114453/200*x^6-8018271/5000*x^5-111146499/40000*x^4-345533877/100000*x^3-7136193339/2000000*x^2-4
0089855591/10000000*x-5764801/2816*ln(2*x-1)+1/4296875*ln(3+5*x)

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Maxima [A]  time = 2.20996, size = 68, normalized size = 1. \begin{align*} -\frac{6561}{70} \, x^{7} - \frac{114453}{200} \, x^{6} - \frac{8018271}{5000} \, x^{5} - \frac{111146499}{40000} \, x^{4} - \frac{345533877}{100000} \, x^{3} - \frac{7136193339}{2000000} \, x^{2} - \frac{40089855591}{10000000} \, x + \frac{1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac{5764801}{2816} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)

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Fricas [A]  time = 1.3158, size = 267, normalized size = 3.93 \begin{align*} -\frac{6561}{70} \, x^{7} - \frac{114453}{200} \, x^{6} - \frac{8018271}{5000} \, x^{5} - \frac{111146499}{40000} \, x^{4} - \frac{345533877}{100000} \, x^{3} - \frac{7136193339}{2000000} \, x^{2} - \frac{40089855591}{10000000} \, x + \frac{1}{4296875} \, \log \left (5 \, x + 3\right ) - \frac{5764801}{2816} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(5*x + 3) - 5764801/2816*log(2*x - 1)

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Sympy [A]  time = 0.141375, size = 63, normalized size = 0.93 \begin{align*} - \frac{6561 x^{7}}{70} - \frac{114453 x^{6}}{200} - \frac{8018271 x^{5}}{5000} - \frac{111146499 x^{4}}{40000} - \frac{345533877 x^{3}}{100000} - \frac{7136193339 x^{2}}{2000000} - \frac{40089855591 x}{10000000} - \frac{5764801 \log{\left (x - \frac{1}{2} \right )}}{2816} + \frac{\log{\left (x + \frac{3}{5} \right )}}{4296875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**8/(1-2*x)/(3+5*x),x)

[Out]

-6561*x**7/70 - 114453*x**6/200 - 8018271*x**5/5000 - 111146499*x**4/40000 - 345533877*x**3/100000 - 713619333
9*x**2/2000000 - 40089855591*x/10000000 - 5764801*log(x - 1/2)/2816 + log(x + 3/5)/4296875

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Giac [A]  time = 1.68871, size = 70, normalized size = 1.03 \begin{align*} -\frac{6561}{70} \, x^{7} - \frac{114453}{200} \, x^{6} - \frac{8018271}{5000} \, x^{5} - \frac{111146499}{40000} \, x^{4} - \frac{345533877}{100000} \, x^{3} - \frac{7136193339}{2000000} \, x^{2} - \frac{40089855591}{10000000} \, x + \frac{1}{4296875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{5764801}{2816} \, \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)^8/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-6561/70*x^7 - 114453/200*x^6 - 8018271/5000*x^5 - 111146499/40000*x^4 - 345533877/100000*x^3 - 7136193339/200
0000*x^2 - 40089855591/10000000*x + 1/4296875*log(abs(5*x + 3)) - 5764801/2816*log(abs(2*x - 1))