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]

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

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 1.01, size = 45, normalized size = 0.74 \[ -\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left (5 \, x + 3\right ) - \frac {823543}{1408} \, \log \left (2 \, x - 1\right ) \]

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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giac [A]  time = 0.98, size = 47, normalized size = 0.77 \[ -\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {823543}{1408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 46, normalized size = 0.75 \[ -\frac {729 x^{6}}{20}-\frac {99873 x^{5}}{500}-\frac {2006937 x^{4}}{4000}-\frac {7889751 x^{3}}{10000}-\frac {187738857 x^{2}}{200000}-\frac {1127138733 x}{1000000}-\frac {823543 \ln \left (2 x -1\right )}{1408}+\frac {\ln \left (5 x +3\right )}{859375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^7/(1-2*x)/(5*x+3),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+1/85937
5*ln(5*x+3)-823543/1408*ln(2*x-1)

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maxima [A]  time = 0.57, size = 45, normalized size = 0.74 \[ -\frac {729}{20} \, x^{6} - \frac {99873}{500} \, x^{5} - \frac {2006937}{4000} \, x^{4} - \frac {7889751}{10000} \, x^{3} - \frac {187738857}{200000} \, x^{2} - \frac {1127138733}{1000000} \, x + \frac {1}{859375} \, \log \left (5 \, x + 3\right ) - \frac {823543}{1408} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.05, size = 41, normalized size = 0.67 \[ \frac {\ln \left (x+\frac {3}{5}\right )}{859375}-\frac {823543\,\ln \left (x-\frac {1}{2}\right )}{1408}-\frac {1127138733\,x}{1000000}-\frac {187738857\,x^2}{200000}-\frac {7889751\,x^3}{10000}-\frac {2006937\,x^4}{4000}-\frac {99873\,x^5}{500}-\frac {729\,x^6}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 56, normalized size = 0.92 \[ - \frac {729 x^{6}}{20} - \frac {99873 x^{5}}{500} - \frac {2006937 x^{4}}{4000} - \frac {7889751 x^{3}}{10000} - \frac {187738857 x^{2}}{200000} - \frac {1127138733 x}{1000000} - \frac {823543 \log {\left (x - \frac {1}{2} \right )}}{1408} + \frac {\log {\left (x + \frac {3}{5} \right )}}{859375} \]

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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