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

Optimal. Leaf size=65 \[ \frac {242}{2401 (1-2 x)}-\frac {319}{2401 (3 x+2)}+\frac {11}{343 (3 x+2)^2}-\frac {1}{441 (3 x+2)^3}-\frac {1364 \log (1-2 x)}{16807}+\frac {1364 \log (3 x+2)}{16807} \]

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Rubi [A]  time = 0.03, antiderivative size = 65, 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 {242}{2401 (1-2 x)}-\frac {319}{2401 (3 x+2)}+\frac {11}{343 (3 x+2)^2}-\frac {1}{441 (3 x+2)^3}-\frac {1364 \log (1-2 x)}{16807}+\frac {1364 \log (3 x+2)}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

242/(2401*(1 - 2*x)) - 1/(441*(2 + 3*x)^3) + 11/(343*(2 + 3*x)^2) - 319/(2401*(2 + 3*x)) - (1364*Log[1 - 2*x])
/16807 + (1364*Log[2 + 3*x])/16807

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 {(3+5 x)^2}{(1-2 x)^2 (2+3 x)^4} \, dx &=\int \left (\frac {484}{2401 (-1+2 x)^2}-\frac {2728}{16807 (-1+2 x)}+\frac {1}{49 (2+3 x)^4}-\frac {66}{343 (2+3 x)^3}+\frac {957}{2401 (2+3 x)^2}+\frac {4092}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {242}{2401 (1-2 x)}-\frac {1}{441 (2+3 x)^3}+\frac {11}{343 (2+3 x)^2}-\frac {319}{2401 (2+3 x)}-\frac {1364 \log (1-2 x)}{16807}+\frac {1364 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 54, normalized size = 0.83 \begin {gather*} \frac {2 \left (-\frac {7 \left (110484 x^3+156519 x^2+66329 x+7277\right )}{2 (2 x-1) (3 x+2)^3}-6138 \log (1-2 x)+6138 \log (6 x+4)\right )}{151263} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(7277 + 66329*x + 156519*x^2 + 110484*x^3))/(2*(-1 + 2*x)*(2 + 3*x)^3) - 6138*Log[1 - 2*x] + 6138*Log[
4 + 6*x]))/151263

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.38, size = 95, normalized size = 1.46 \begin {gather*} -\frac {773388 \, x^{3} + 1095633 \, x^{2} - 12276 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) + 12276 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) + 464303 \, x + 50939}{151263 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/151263*(773388*x^3 + 1095633*x^2 - 12276*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) + 12276*(54*x^4
+ 81*x^3 + 18*x^2 - 20*x - 8)*log(2*x - 1) + 464303*x + 50939)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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giac [A]  time = 0.97, size = 60, normalized size = 0.92 \begin {gather*} -\frac {242}{2401 \, {\left (2 \, x - 1\right )}} + \frac {2 \, {\left (\frac {36120}{2 \, x - 1} + \frac {40621}{{\left (2 \, x - 1\right )}^{2}} + 8031\right )}}{16807 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{3}} + \frac {1364}{16807} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-242/2401/(2*x - 1) + 2/16807*(36120/(2*x - 1) + 40621/(2*x - 1)^2 + 8031)/(7/(2*x - 1) + 3)^3 + 1364/16807*lo
g(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \begin {gather*} -\frac {1364 \ln \left (2 x -1\right )}{16807}+\frac {1364 \ln \left (3 x +2\right )}{16807}-\frac {1}{441 \left (3 x +2\right )^{3}}+\frac {11}{343 \left (3 x +2\right )^{2}}-\frac {319}{2401 \left (3 x +2\right )}-\frac {242}{2401 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/441/(3*x+2)^3+11/343/(3*x+2)^2-319/2401/(3*x+2)+1364/16807*ln(3*x+2)-242/2401/(2*x-1)-1364/16807*ln(2*x-1)

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maxima [A]  time = 0.61, size = 56, normalized size = 0.86 \begin {gather*} -\frac {110484 \, x^{3} + 156519 \, x^{2} + 66329 \, x + 7277}{21609 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac {1364}{16807} \, \log \left (3 \, x + 2\right ) - \frac {1364}{16807} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21609*(110484*x^3 + 156519*x^2 + 66329*x + 7277)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 1364/16807*log(3*x
+ 2) - 1364/16807*log(2*x - 1)

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mupad [B]  time = 0.05, size = 46, normalized size = 0.71 \begin {gather*} \frac {2728\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {682\,x^3}{7203}+\frac {5797\,x^2}{43218}+\frac {66329\,x}{1166886}+\frac {7277}{1166886}}{x^4+\frac {3\,x^3}{2}+\frac {x^2}{3}-\frac {10\,x}{27}-\frac {4}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2728*atanh((12*x)/7 + 1/7))/16807 - ((66329*x)/1166886 + (5797*x^2)/43218 + (682*x^3)/7203 + 7277/1166886)/(x
^2/3 - (10*x)/27 + (3*x^3)/2 + x^4 - 4/27)

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sympy [A]  time = 0.17, size = 56, normalized size = 0.86 \begin {gather*} \frac {- 110484 x^{3} - 156519 x^{2} - 66329 x - 7277}{1166886 x^{4} + 1750329 x^{3} + 388962 x^{2} - 432180 x - 172872} - \frac {1364 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {1364 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-110484*x**3 - 156519*x**2 - 66329*x - 7277)/(1166886*x**4 + 1750329*x**3 + 388962*x**2 - 432180*x - 172872)
- 1364*log(x - 1/2)/16807 + 1364*log(x + 2/3)/16807

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