3.1487 \(\int \frac{(2+3 x)^6}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=54 \[ -\frac{729 x^5}{50}-\frac{28431 x^4}{400}-\frac{159813 x^3}{1000}-\frac{4693491 x^2}{20000}-\frac{31289679 x}{100000}-\frac{117649}{704} \log (1-2 x)+\frac{\log (5 x+3)}{171875} \]

[Out]

(-31289679*x)/100000 - (4693491*x^2)/20000 - (159813*x^3)/1000 - (28431*x^4)/400 - (729*x^5)/50 - (117649*Log[
1 - 2*x])/704 + Log[3 + 5*x]/171875

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Rubi [A]  time = 0.0227725, antiderivative size = 54, 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} \[ -\frac{729 x^5}{50}-\frac{28431 x^4}{400}-\frac{159813 x^3}{1000}-\frac{4693491 x^2}{20000}-\frac{31289679 x}{100000}-\frac{117649}{704} \log (1-2 x)+\frac{\log (5 x+3)}{171875} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-31289679*x)/100000 - (4693491*x^2)/20000 - (159813*x^3)/1000 - (28431*x^4)/400 - (729*x^5)/50 - (117649*Log[
1 - 2*x])/704 + Log[3 + 5*x]/171875

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)^6}{(1-2 x) (3+5 x)} \, dx &=\int \left (-\frac{31289679}{100000}-\frac{4693491 x}{10000}-\frac{479439 x^2}{1000}-\frac{28431 x^3}{100}-\frac{729 x^4}{10}-\frac{117649}{352 (-1+2 x)}+\frac{1}{34375 (3+5 x)}\right ) \, dx\\ &=-\frac{31289679 x}{100000}-\frac{4693491 x^2}{20000}-\frac{159813 x^3}{1000}-\frac{28431 x^4}{400}-\frac{729 x^5}{50}-\frac{117649}{704} \log (1-2 x)+\frac{\log (3+5 x)}{171875}\\ \end{align*}

Mathematica [A]  time = 0.0181967, size = 50, normalized size = 0.93 \[ \frac{-2970 \left (54000 x^5+263250 x^4+591900 x^3+869165 x^2+1158877 x+516778\right )-1838265625 \log (3-6 x)+64 \log (-3 (5 x+3))}{11000000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)*(3 + 5*x)),x]

[Out]

(-2970*(516778 + 1158877*x + 869165*x^2 + 591900*x^3 + 263250*x^4 + 54000*x^5) - 1838265625*Log[3 - 6*x] + 64*
Log[-3*(3 + 5*x)])/11000000

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Maple [A]  time = 0.005, size = 41, normalized size = 0.8 \begin{align*} -{\frac{729\,{x}^{5}}{50}}-{\frac{28431\,{x}^{4}}{400}}-{\frac{159813\,{x}^{3}}{1000}}-{\frac{4693491\,{x}^{2}}{20000}}-{\frac{31289679\,x}{100000}}-{\frac{117649\,\ln \left ( 2\,x-1 \right ) }{704}}+{\frac{\ln \left ( 3+5\,x \right ) }{171875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)/(3+5*x),x)

[Out]

-729/50*x^5-28431/400*x^4-159813/1000*x^3-4693491/20000*x^2-31289679/100000*x-117649/704*ln(2*x-1)+1/171875*ln
(3+5*x)

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Maxima [A]  time = 1.11384, size = 54, normalized size = 1. \begin{align*} -\frac{729}{50} \, x^{5} - \frac{28431}{400} \, x^{4} - \frac{159813}{1000} \, x^{3} - \frac{4693491}{20000} \, x^{2} - \frac{31289679}{100000} \, x + \frac{1}{171875} \, \log \left (5 \, x + 3\right ) - \frac{117649}{704} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)/(3+5*x),x, algorithm="maxima")

[Out]

-729/50*x^5 - 28431/400*x^4 - 159813/1000*x^3 - 4693491/20000*x^2 - 31289679/100000*x + 1/171875*log(5*x + 3)
- 117649/704*log(2*x - 1)

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Fricas [A]  time = 1.36757, size = 185, normalized size = 3.43 \begin{align*} -\frac{729}{50} \, x^{5} - \frac{28431}{400} \, x^{4} - \frac{159813}{1000} \, x^{3} - \frac{4693491}{20000} \, x^{2} - \frac{31289679}{100000} \, x + \frac{1}{171875} \, \log \left (5 \, x + 3\right ) - \frac{117649}{704} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)/(3+5*x),x, algorithm="fricas")

[Out]

-729/50*x^5 - 28431/400*x^4 - 159813/1000*x^3 - 4693491/20000*x^2 - 31289679/100000*x + 1/171875*log(5*x + 3)
- 117649/704*log(2*x - 1)

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Sympy [A]  time = 0.129243, size = 49, normalized size = 0.91 \begin{align*} - \frac{729 x^{5}}{50} - \frac{28431 x^{4}}{400} - \frac{159813 x^{3}}{1000} - \frac{4693491 x^{2}}{20000} - \frac{31289679 x}{100000} - \frac{117649 \log{\left (x - \frac{1}{2} \right )}}{704} + \frac{\log{\left (x + \frac{3}{5} \right )}}{171875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6/(1-2*x)/(3+5*x),x)

[Out]

-729*x**5/50 - 28431*x**4/400 - 159813*x**3/1000 - 4693491*x**2/20000 - 31289679*x/100000 - 117649*log(x - 1/2
)/704 + log(x + 3/5)/171875

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Giac [A]  time = 2.43554, size = 57, normalized size = 1.06 \begin{align*} -\frac{729}{50} \, x^{5} - \frac{28431}{400} \, x^{4} - \frac{159813}{1000} \, x^{3} - \frac{4693491}{20000} \, x^{2} - \frac{31289679}{100000} \, x + \frac{1}{171875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{117649}{704} \, \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)^6/(1-2*x)/(3+5*x),x, algorithm="giac")

[Out]

-729/50*x^5 - 28431/400*x^4 - 159813/1000*x^3 - 4693491/20000*x^2 - 31289679/100000*x + 1/171875*log(abs(5*x +
3)) - 117649/704*log(abs(2*x - 1))