3.1465 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^6} \, dx\)

Optimal. Leaf size=76 \[ -\frac{484}{16807 (3 x+2)}-\frac{121}{2401 (3 x+2)^2}-\frac{121}{1029 (3 x+2)^3}+\frac{17}{441 (3 x+2)^4}-\frac{1}{315 (3 x+2)^5}-\frac{968 \log (1-2 x)}{117649}+\frac{968 \log (3 x+2)}{117649} \]

[Out]

-1/(315*(2 + 3*x)^5) + 17/(441*(2 + 3*x)^4) - 121/(1029*(2 + 3*x)^3) - 121/(2401*(2 + 3*x)^2) - 484/(16807*(2
+ 3*x)) - (968*Log[1 - 2*x])/117649 + (968*Log[2 + 3*x])/117649

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Rubi [A]  time = 0.0282296, antiderivative size = 76, 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} \[ -\frac{484}{16807 (3 x+2)}-\frac{121}{2401 (3 x+2)^2}-\frac{121}{1029 (3 x+2)^3}+\frac{17}{441 (3 x+2)^4}-\frac{1}{315 (3 x+2)^5}-\frac{968 \log (1-2 x)}{117649}+\frac{968 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(315*(2 + 3*x)^5) + 17/(441*(2 + 3*x)^4) - 121/(1029*(2 + 3*x)^3) - 121/(2401*(2 + 3*x)^2) - 484/(16807*(2
+ 3*x)) - (968*Log[1 - 2*x])/117649 + (968*Log[2 + 3*x])/117649

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+3 x)^6} \, dx &=\int \left (-\frac{1936}{117649 (-1+2 x)}+\frac{1}{21 (2+3 x)^6}-\frac{68}{147 (2+3 x)^5}+\frac{363}{343 (2+3 x)^4}+\frac{726}{2401 (2+3 x)^3}+\frac{1452}{16807 (2+3 x)^2}+\frac{2904}{117649 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{315 (2+3 x)^5}+\frac{17}{441 (2+3 x)^4}-\frac{121}{1029 (2+3 x)^3}-\frac{121}{2401 (2+3 x)^2}-\frac{484}{16807 (2+3 x)}-\frac{968 \log (1-2 x)}{117649}+\frac{968 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.0438077, size = 52, normalized size = 0.68 \[ \frac{4 \left (-\frac{7 \left (1764180 x^4+5733585 x^3+7563105 x^2+4442775 x+953231\right )}{4 (3 x+2)^5}-10890 \log (1-2 x)+10890 \log (6 x+4)\right )}{5294205} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(953231 + 4442775*x + 7563105*x^2 + 5733585*x^3 + 1764180*x^4))/(4*(2 + 3*x)^5) - 10890*Log[1 - 2*x] +
10890*Log[4 + 6*x]))/5294205

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Maple [A]  time = 0.007, size = 63, normalized size = 0.8 \begin{align*} -{\frac{968\,\ln \left ( 2\,x-1 \right ) }{117649}}-{\frac{1}{315\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{17}{441\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{121}{1029\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{121}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{484}{33614+50421\,x}}+{\frac{968\,\ln \left ( 2+3\,x \right ) }{117649}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-968/117649*ln(2*x-1)-1/315/(2+3*x)^5+17/441/(2+3*x)^4-121/1029/(2+3*x)^3-121/2401/(2+3*x)^2-484/16807/(2+3*x)
+968/117649*ln(2+3*x)

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Maxima [A]  time = 1.0505, size = 89, normalized size = 1.17 \begin{align*} -\frac{1764180 \, x^{4} + 5733585 \, x^{3} + 7563105 \, x^{2} + 4442775 \, x + 953231}{756315 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{968}{117649} \, \log \left (3 \, x + 2\right ) - \frac{968}{117649} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/756315*(1764180*x^4 + 5733585*x^3 + 7563105*x^2 + 4442775*x + 953231)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32) + 968/117649*log(3*x + 2) - 968/117649*log(2*x - 1)

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Fricas [A]  time = 1.3213, size = 383, normalized size = 5.04 \begin{align*} -\frac{12349260 \, x^{4} + 40135095 \, x^{3} + 52941735 \, x^{2} - 43560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 43560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 31099425 \, x + 6672617}{5294205 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5294205*(12349260*x^4 + 40135095*x^3 + 52941735*x^2 - 43560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)*log(3*x + 2) + 43560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 31099425*x +
6672617)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.191804, size = 65, normalized size = 0.86 \begin{align*} - \frac{1764180 x^{4} + 5733585 x^{3} + 7563105 x^{2} + 4442775 x + 953231}{183784545 x^{5} + 612615150 x^{4} + 816820200 x^{3} + 544546800 x^{2} + 181515600 x + 24202080} - \frac{968 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{968 \log{\left (x + \frac{2}{3} \right )}}{117649} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1764180*x**4 + 5733585*x**3 + 7563105*x**2 + 4442775*x + 953231)/(183784545*x**5 + 612615150*x**4 + 81682020
0*x**3 + 544546800*x**2 + 181515600*x + 24202080) - 968*log(x - 1/2)/117649 + 968*log(x + 2/3)/117649

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Giac [A]  time = 1.30291, size = 65, normalized size = 0.86 \begin{align*} -\frac{1764180 \, x^{4} + 5733585 \, x^{3} + 7563105 \, x^{2} + 4442775 \, x + 953231}{756315 \,{\left (3 \, x + 2\right )}^{5}} + \frac{968}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{968}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/756315*(1764180*x^4 + 5733585*x^3 + 7563105*x^2 + 4442775*x + 953231)/(3*x + 2)^5 + 968/117649*log(abs(3*x
+ 2)) - 968/117649*log(abs(2*x - 1))