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

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

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Rubi [A]  time = 0.02, antiderivative size = 54, 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 = {88} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 48, normalized size = 0.89 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^4}{(1-2 x)^2 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.22, size = 75, normalized size = 1.39 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.13, size = 69, normalized size = 1.28 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \begin {gather*} \frac {9261 \ln \left (2 x -1\right )}{58564}+\frac {7074 \ln \left (5 x +3\right )}{1830125}-\frac {1}{30250 \left (5 x +3\right )^{2}}-\frac {136}{166375 \left (5 x +3\right )}-\frac {2401}{5324 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 46, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.12, size = 41, normalized size = 0.76 \begin {gather*} \frac {9261\,\ln \left (x-\frac {1}{2}\right )}{58564}+\frac {7074\,\ln \left (x+\frac {3}{5}\right )}{1830125}+\frac {\frac {1501713\,x^2}{6655000}+\frac {4502169\,x}{16637500}+\frac {2699471}{33275000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9261*log(x - 1/2))/58564 + (7074*log(x + 3/5))/1830125 + ((4502169*x)/16637500 + (1501713*x^2)/6655000 + 2699
471/33275000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50)

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sympy [A]  time = 0.18, size = 46, normalized size = 0.85 \begin {gather*} \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 {gather*}

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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