3.1621 \(\int \frac{(2+3 x)^4}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{2401}{5324 (1-2 x)}-\frac{136}{166375 (5 x+3)}-\frac{1}{30250 (5 x+3)^2}+\frac{9261 \log (1-2 x)}{58564}+\frac{7074 \log (5 x+3)}{1830125} \]

[Out]

2401/(5324*(1 - 2*x)) - 1/(30250*(3 + 5*x)^2) - 136/(166375*(3 + 5*x)) + (9261*Log[1 - 2*x])/58564 + (7074*Log
[3 + 5*x])/1830125

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Rubi [A]  time = 0.0236523, 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 = {88} \[ \frac{2401}{5324 (1-2 x)}-\frac{136}{166375 (5 x+3)}-\frac{1}{30250 (5 x+3)^2}+\frac{9261 \log (1-2 x)}{58564}+\frac{7074 \log (5 x+3)}{1830125} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^2*(3 + 5*x)^3),x]

[Out]

2401/(5324*(1 - 2*x)) - 1/(30250*(3 + 5*x)^2) - 136/(166375*(3 + 5*x)) + (9261*Log[1 - 2*x])/58564 + (7074*Log
[3 + 5*x])/1830125

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)^4}{(1-2 x)^2 (3+5 x)^3} \, dx &=\int \left (\frac{2401}{2662 (-1+2 x)^2}+\frac{9261}{29282 (-1+2 x)}+\frac{1}{3025 (3+5 x)^3}+\frac{136}{33275 (3+5 x)^2}+\frac{7074}{366025 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{5324 (1-2 x)}-\frac{1}{30250 (3+5 x)^2}-\frac{136}{166375 (3+5 x)}+\frac{9261 \log (1-2 x)}{58564}+\frac{7074 \log (3+5 x)}{1830125}\\ \end{align*}

Mathematica [A]  time = 0.0299197, size = 48, normalized size = 0.89 \[ \frac{\frac{3301375}{1-2 x}-\frac{5984}{5 x+3}-\frac{242}{(5 x+3)^2}+1157625 \log (1-2 x)+28296 \log (10 x+6)}{7320500} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^2*(3 + 5*x)^3),x]

[Out]

(3301375/(1 - 2*x) - 242/(3 + 5*x)^2 - 5984/(3 + 5*x) + 1157625*Log[1 - 2*x] + 28296*Log[6 + 10*x])/7320500

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*} -{\frac{2401}{10648\,x-5324}}+{\frac{9261\,\ln \left ( 2\,x-1 \right ) }{58564}}-{\frac{1}{30250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{136}{499125+831875\,x}}+{\frac{7074\,\ln \left ( 3+5\,x \right ) }{1830125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^2/(3+5*x)^3,x)

[Out]

-2401/5324/(2*x-1)+9261/58564*ln(2*x-1)-1/30250/(3+5*x)^2-136/166375/(3+5*x)+7074/1830125*ln(3+5*x)

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Maxima [A]  time = 1.44037, size = 62, normalized size = 1.15 \begin{align*} -\frac{7508565 \, x^{2} + 9004338 \, x + 2699471}{665500 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{7074}{1830125} \, \log \left (5 \, x + 3\right ) + \frac{9261}{58564} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

-1/665500*(7508565*x^2 + 9004338*x + 2699471)/(50*x^3 + 35*x^2 - 12*x - 9) + 7074/1830125*log(5*x + 3) + 9261/
58564*log(2*x - 1)

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Fricas [A]  time = 1.27277, size = 246, normalized size = 4.56 \begin{align*} -\frac{82594215 \, x^{2} - 28296 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) - 1157625 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 99047718 \, x + 29694181}{7320500 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^2/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/7320500*(82594215*x^2 - 28296*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) - 1157625*(50*x^3 + 35*x^2 - 12*x -
9)*log(2*x - 1) + 99047718*x + 29694181)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.168958, size = 44, normalized size = 0.81 \begin{align*} - \frac{7508565 x^{2} + 9004338 x + 2699471}{33275000 x^{3} + 23292500 x^{2} - 7986000 x - 5989500} + \frac{9261 \log{\left (x - \frac{1}{2} \right )}}{58564} + \frac{7074 \log{\left (x + \frac{3}{5} \right )}}{1830125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**2/(3+5*x)**3,x)

[Out]

-(7508565*x**2 + 9004338*x + 2699471)/(33275000*x**3 + 23292500*x**2 - 7986000*x - 5989500) + 9261*log(x - 1/2
)/58564 + 7074*log(x + 3/5)/1830125

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Giac [A]  time = 1.83478, size = 93, normalized size = 1.72 \begin{align*} -\frac{2401}{5324 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{1518}{2 \, x - 1} + 685\right )}}{366025 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} - \frac{81}{500} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{7074}{1830125} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

-2401/5324/(2*x - 1) + 2/366025*(1518/(2*x - 1) + 685)/(11/(2*x - 1) + 5)^2 - 81/500*log(1/2*abs(2*x - 1)/(2*x
- 1)^2) + 7074/1830125*log(abs(-11/(2*x - 1) - 5))