3.1448 \(\int \frac{3+5 x}{(1-2 x) (2+3 x)^6} \, dx\)

Optimal. Leaf size=76 \[ -\frac{88}{16807 (3 x+2)}-\frac{22}{2401 (3 x+2)^2}-\frac{22}{1029 (3 x+2)^3}-\frac{11}{196 (3 x+2)^4}+\frac{1}{105 (3 x+2)^5}-\frac{176 \log (1-2 x)}{117649}+\frac{176 \log (3 x+2)}{117649} \]

[Out]

1/(105*(2 + 3*x)^5) - 11/(196*(2 + 3*x)^4) - 22/(1029*(2 + 3*x)^3) - 22/(2401*(2 + 3*x)^2) - 88/(16807*(2 + 3*
x)) - (176*Log[1 - 2*x])/117649 + (176*Log[2 + 3*x])/117649

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Rubi [A]  time = 0.0277983, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{88}{16807 (3 x+2)}-\frac{22}{2401 (3 x+2)^2}-\frac{22}{1029 (3 x+2)^3}-\frac{11}{196 (3 x+2)^4}+\frac{1}{105 (3 x+2)^5}-\frac{176 \log (1-2 x)}{117649}+\frac{176 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(105*(2 + 3*x)^5) - 11/(196*(2 + 3*x)^4) - 22/(1029*(2 + 3*x)^3) - 22/(2401*(2 + 3*x)^2) - 88/(16807*(2 + 3*
x)) - (176*Log[1 - 2*x])/117649 + (176*Log[2 + 3*x])/117649

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{3+5 x}{(1-2 x) (2+3 x)^6} \, dx &=\int \left (-\frac{352}{117649 (-1+2 x)}-\frac{1}{7 (2+3 x)^6}+\frac{33}{49 (2+3 x)^5}+\frac{66}{343 (2+3 x)^4}+\frac{132}{2401 (2+3 x)^3}+\frac{264}{16807 (2+3 x)^2}+\frac{528}{117649 (2+3 x)}\right ) \, dx\\ &=\frac{1}{105 (2+3 x)^5}-\frac{11}{196 (2+3 x)^4}-\frac{22}{1029 (2+3 x)^3}-\frac{22}{2401 (2+3 x)^2}-\frac{88}{16807 (2+3 x)}-\frac{176 \log (1-2 x)}{117649}+\frac{176 \log (2+3 x)}{117649}\\ \end{align*}

Mathematica [A]  time = 0.0336811, size = 50, normalized size = 0.66 \[ \frac{-\frac{7 \left (427680 x^4+1389960 x^3+1833480 x^2+1268025 x+348226\right )}{(3 x+2)^5}-10560 \log (3-6 x)+10560 \log (3 x+2)}{7058940} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(348226 + 1268025*x + 1833480*x^2 + 1389960*x^3 + 427680*x^4))/(2 + 3*x)^5 - 10560*Log[3 - 6*x] + 10560*L
og[2 + 3*x])/7058940

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Maple [A]  time = 0.007, size = 63, normalized size = 0.8 \begin{align*} -{\frac{176\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{105\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{11}{196\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{22}{1029\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{22}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{88}{33614+50421\,x}}+{\frac{176\,\ln \left ( 2+3\,x \right ) }{117649}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-176/117649*ln(2*x-1)+1/105/(2+3*x)^5-11/196/(2+3*x)^4-22/1029/(2+3*x)^3-22/2401/(2+3*x)^2-88/16807/(2+3*x)+17
6/117649*ln(2+3*x)

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Maxima [A]  time = 1.01386, size = 89, normalized size = 1.17 \begin{align*} -\frac{427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{176}{117649} \, \log \left (3 \, x + 2\right ) - \frac{176}{117649} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32) + 176/117649*log(3*x + 2) - 176/117649*log(2*x - 1)

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Fricas [A]  time = 1.36529, size = 379, normalized size = 4.99 \begin{align*} -\frac{2993760 \, x^{4} + 9729720 \, x^{3} + 12834360 \, x^{2} - 10560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 10560 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 8876175 \, x + 2437582}{7058940 \,{\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)/(1-2*x)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/7058940*(2993760*x^4 + 9729720*x^3 + 12834360*x^2 - 10560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x +
32)*log(3*x + 2) + 10560*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 8876175*x + 243
7582)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.177532, size = 65, normalized size = 0.86 \begin{align*} - \frac{427680 x^{4} + 1389960 x^{3} + 1833480 x^{2} + 1268025 x + 348226}{245046060 x^{5} + 816820200 x^{4} + 1089093600 x^{3} + 726062400 x^{2} + 242020800 x + 32269440} - \frac{176 \log{\left (x - \frac{1}{2} \right )}}{117649} + \frac{176 \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)/(1-2*x)/(2+3*x)**6,x)

[Out]

-(427680*x**4 + 1389960*x**3 + 1833480*x**2 + 1268025*x + 348226)/(245046060*x**5 + 816820200*x**4 + 108909360
0*x**3 + 726062400*x**2 + 242020800*x + 32269440) - 176*log(x - 1/2)/117649 + 176*log(x + 2/3)/117649

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Giac [A]  time = 2.41032, size = 65, normalized size = 0.86 \begin{align*} -\frac{427680 \, x^{4} + 1389960 \, x^{3} + 1833480 \, x^{2} + 1268025 \, x + 348226}{1008420 \,{\left (3 \, x + 2\right )}^{5}} + \frac{176}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{176}{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)/(1-2*x)/(2+3*x)^6,x, algorithm="giac")

[Out]

-1/1008420*(427680*x^4 + 1389960*x^3 + 1833480*x^2 + 1268025*x + 348226)/(3*x + 2)^5 + 176/117649*log(abs(3*x
+ 2)) - 176/117649*log(abs(2*x - 1))