3.1486 \(\int \frac{(2+3 x)^7}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=61 \[ -\frac{729 x^6}{20}-\frac{99873 x^5}{500}-\frac{2006937 x^4}{4000}-\frac{7889751 x^3}{10000}-\frac{187738857 x^2}{200000}-\frac{1127138733 x}{1000000}-\frac{823543 \log (1-2 x)}{1408}+\frac{\log (5 x+3)}{859375} \]

[Out]

(-1127138733*x)/1000000 - (187738857*x^2)/200000 - (7889751*x^3)/10000 - (2006937*x^4)/4000 - (99873*x^5)/500
- (729*x^6)/20 - (823543*Log[1 - 2*x])/1408 + Log[3 + 5*x]/859375

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Rubi [A]  time = 0.0254426, antiderivative size = 61, 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} \[ -\frac{729 x^6}{20}-\frac{99873 x^5}{500}-\frac{2006937 x^4}{4000}-\frac{7889751 x^3}{10000}-\frac{187738857 x^2}{200000}-\frac{1127138733 x}{1000000}-\frac{823543 \log (1-2 x)}{1408}+\frac{\log (5 x+3)}{859375} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^7/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-1127138733*x)/1000000 - (187738857*x^2)/200000 - (7889751*x^3)/10000 - (2006937*x^4)/4000 - (99873*x^5)/500
- (729*x^6)/20 - (823543*Log[1 - 2*x])/1408 + Log[3 + 5*x]/859375

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)^7}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{1127138733}{1000000}-\frac{187738857 x}{100000}-\frac{23669253 x^2}{10000}-\frac{2006937 x^3}{1000}-\frac{99873 x^4}{100}-\frac{2187 x^5}{10}-\frac{823543}{704 (-1+2 x)}+\frac{1}{171875 (3+5 x)}\right ) \, dx\\ &=-\frac{1127138733 x}{1000000}-\frac{187738857 x^2}{200000}-\frac{7889751 x^3}{10000}-\frac{2006937 x^4}{4000}-\frac{99873 x^5}{500}-\frac{729 x^6}{20}-\frac{823543 \log (1-2 x)}{1408}+\frac{\log (3+5 x)}{859375}\\ \end{align*}

Mathematica [A]  time = 0.0241124, size = 58, normalized size = 0.95 \[ \frac{64 \log (-3 (5 x+3))-165 \left (12150000 x^6+66582000 x^5+167244750 x^4+262991700 x^3+312898095 x^2+375712911 x+163998254\right )}{55000000}-\frac{823543 \log (3-6 x)}{1408} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^7/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-823543*Log[3 - 6*x])/1408 + (-165*(163998254 + 375712911*x + 312898095*x^2 + 262991700*x^3 + 167244750*x^4 +
66582000*x^5 + 12150000*x^6) + 64*Log[-3*(3 + 5*x)])/55000000

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*} -{\frac{729\,{x}^{6}}{20}}-{\frac{99873\,{x}^{5}}{500}}-{\frac{2006937\,{x}^{4}}{4000}}-{\frac{7889751\,{x}^{3}}{10000}}-{\frac{187738857\,{x}^{2}}{200000}}-{\frac{1127138733\,x}{1000000}}-{\frac{823543\,\ln \left ( 2\,x-1 \right ) }{1408}}+{\frac{\ln \left ( 3+5\,x \right ) }{859375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^7/(1-2*x)/(3+5*x),x)

[Out]

-729/20*x^6-99873/500*x^5-2006937/4000*x^4-7889751/10000*x^3-187738857/200000*x^2-1127138733/1000000*x-823543/
1408*ln(2*x-1)+1/859375*ln(3+5*x)

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Maxima [A]  time = 1.12, size = 61, normalized size = 1. \begin{align*} -\frac{729}{20} \, x^{6} - \frac{99873}{500} \, x^{5} - \frac{2006937}{4000} \, x^{4} - \frac{7889751}{10000} \, x^{3} - \frac{187738857}{200000} \, x^{2} - \frac{1127138733}{1000000} \, x + \frac{1}{859375} \, \log \left (5 \, x + 3\right ) - \frac{823543}{1408} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 187738857/200000*x^2 - 1127138733/1000000
*x + 1/859375*log(5*x + 3) - 823543/1408*log(2*x - 1)

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Fricas [A]  time = 1.2069, size = 223, normalized size = 3.66 \begin{align*} -\frac{729}{20} \, x^{6} - \frac{99873}{500} \, x^{5} - \frac{2006937}{4000} \, x^{4} - \frac{7889751}{10000} \, x^{3} - \frac{187738857}{200000} \, x^{2} - \frac{1127138733}{1000000} \, x + \frac{1}{859375} \, \log \left (5 \, x + 3\right ) - \frac{823543}{1408} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 187738857/200000*x^2 - 1127138733/1000000
*x + 1/859375*log(5*x + 3) - 823543/1408*log(2*x - 1)

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Sympy [A]  time = 0.133645, size = 56, normalized size = 0.92 \begin{align*} - \frac{729 x^{6}}{20} - \frac{99873 x^{5}}{500} - \frac{2006937 x^{4}}{4000} - \frac{7889751 x^{3}}{10000} - \frac{187738857 x^{2}}{200000} - \frac{1127138733 x}{1000000} - \frac{823543 \log{\left (x - \frac{1}{2} \right )}}{1408} + \frac{\log{\left (x + \frac{3}{5} \right )}}{859375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**7/(1-2*x)/(3+5*x),x)

[Out]

-729*x**6/20 - 99873*x**5/500 - 2006937*x**4/4000 - 7889751*x**3/10000 - 187738857*x**2/200000 - 1127138733*x/
1000000 - 823543*log(x - 1/2)/1408 + log(x + 3/5)/859375

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Giac [A]  time = 2.64739, size = 63, normalized size = 1.03 \begin{align*} -\frac{729}{20} \, x^{6} - \frac{99873}{500} \, x^{5} - \frac{2006937}{4000} \, x^{4} - \frac{7889751}{10000} \, x^{3} - \frac{187738857}{200000} \, x^{2} - \frac{1127138733}{1000000} \, x + \frac{1}{859375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{823543}{1408} \, \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)^7/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-729/20*x^6 - 99873/500*x^5 - 2006937/4000*x^4 - 7889751/10000*x^3 - 187738857/200000*x^2 - 1127138733/1000000
*x + 1/859375*log(abs(5*x + 3)) - 823543/1408*log(abs(2*x - 1))