3.1220 \(\int \frac{1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{20875}{3 x+2}-\frac{6875}{5 x+3}-\frac{1675}{(3 x+2)^2}-\frac{505}{3 (3 x+2)^3}-\frac{17}{(3 x+2)^4}-\frac{7}{5 (3 x+2)^5}+125000 \log (3 x+2)-125000 \log (5 x+3) \]

[Out]

-7/(5*(2 + 3*x)^5) - 17/(2 + 3*x)^4 - 505/(3*(2 + 3*x)^3) - 1675/(2 + 3*x)^2 - 20875/(2 + 3*x) - 6875/(3 + 5*x
) + 125000*Log[2 + 3*x] - 125000*Log[3 + 5*x]

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Rubi [A]  time = 0.036878, antiderivative size = 75, 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} \[ -\frac{20875}{3 x+2}-\frac{6875}{5 x+3}-\frac{1675}{(3 x+2)^2}-\frac{505}{3 (3 x+2)^3}-\frac{17}{(3 x+2)^4}-\frac{7}{5 (3 x+2)^5}+125000 \log (3 x+2)-125000 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(5*(2 + 3*x)^5) - 17/(2 + 3*x)^4 - 505/(3*(2 + 3*x)^3) - 1675/(2 + 3*x)^2 - 20875/(2 + 3*x) - 6875/(3 + 5*x
) + 125000*Log[2 + 3*x] - 125000*Log[3 + 5*x]

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{1-2 x}{(2+3 x)^6 (3+5 x)^2} \, dx &=\int \left (\frac{21}{(2+3 x)^6}+\frac{204}{(2+3 x)^5}+\frac{1515}{(2+3 x)^4}+\frac{10050}{(2+3 x)^3}+\frac{62625}{(2+3 x)^2}+\frac{375000}{2+3 x}+\frac{34375}{(3+5 x)^2}-\frac{625000}{3+5 x}\right ) \, dx\\ &=-\frac{7}{5 (2+3 x)^5}-\frac{17}{(2+3 x)^4}-\frac{505}{3 (2+3 x)^3}-\frac{1675}{(2+3 x)^2}-\frac{20875}{2+3 x}-\frac{6875}{3+5 x}+125000 \log (2+3 x)-125000 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0267492, size = 77, normalized size = 1.03 \[ -\frac{20875}{3 x+2}-\frac{6875}{5 x+3}-\frac{1675}{(3 x+2)^2}-\frac{505}{3 (3 x+2)^3}-\frac{17}{(3 x+2)^4}-\frac{7}{5 (3 x+2)^5}+125000 \log (3 x+2)-125000 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(5*(2 + 3*x)^5) - 17/(2 + 3*x)^4 - 505/(3*(2 + 3*x)^3) - 1675/(2 + 3*x)^2 - 20875/(2 + 3*x) - 6875/(3 + 5*x
) + 125000*Log[2 + 3*x] - 125000*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.009, size = 72, normalized size = 1. \begin{align*} -{\frac{7}{5\, \left ( 2+3\,x \right ) ^{5}}}-17\, \left ( 2+3\,x \right ) ^{-4}-{\frac{505}{3\, \left ( 2+3\,x \right ) ^{3}}}-1675\, \left ( 2+3\,x \right ) ^{-2}-20875\, \left ( 2+3\,x \right ) ^{-1}-6875\, \left ( 3+5\,x \right ) ^{-1}+125000\,\ln \left ( 2+3\,x \right ) -125000\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/5/(2+3*x)^5-17/(2+3*x)^4-505/3/(2+3*x)^3-1675/(2+3*x)^2-20875/(2+3*x)-6875/(3+5*x)+125000*ln(2+3*x)-125000*
ln(3+5*x)

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Maxima [A]  time = 1.02158, size = 103, normalized size = 1.37 \begin{align*} -\frac{151875000 \, x^{5} + 501187500 \, x^{4} + 661387500 \, x^{3} + 436271250 \, x^{2} + 143844850 \, x + 18964893}{15 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - 125000 \, \log \left (5 \, x + 3\right ) + 125000 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(151875000*x^5 + 501187500*x^4 + 661387500*x^3 + 436271250*x^2 + 143844850*x + 18964893)/(1215*x^6 + 477
9*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 125000*log(5*x + 3) + 125000*log(3*x + 2)

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Fricas [A]  time = 1.55063, size = 467, normalized size = 6.23 \begin{align*} -\frac{151875000 \, x^{5} + 501187500 \, x^{4} + 661387500 \, x^{3} + 436271250 \, x^{2} + 1875000 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 1875000 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 143844850 \, x + 18964893}{15 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(151875000*x^5 + 501187500*x^4 + 661387500*x^3 + 436271250*x^2 + 1875000*(1215*x^6 + 4779*x^5 + 7830*x^4
+ 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 1875000*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*
x^2 + 880*x + 96)*log(3*x + 2) + 143844850*x + 18964893)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2
+ 880*x + 96)

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Sympy [A]  time = 0.190404, size = 71, normalized size = 0.95 \begin{align*} - \frac{151875000 x^{5} + 501187500 x^{4} + 661387500 x^{3} + 436271250 x^{2} + 143844850 x + 18964893}{18225 x^{6} + 71685 x^{5} + 117450 x^{4} + 102600 x^{3} + 50400 x^{2} + 13200 x + 1440} - 125000 \log{\left (x + \frac{3}{5} \right )} + 125000 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(151875000*x**5 + 501187500*x**4 + 661387500*x**3 + 436271250*x**2 + 143844850*x + 18964893)/(18225*x**6 + 71
685*x**5 + 117450*x**4 + 102600*x**3 + 50400*x**2 + 13200*x + 1440) - 125000*log(x + 3/5) + 125000*log(x + 2/3
)

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Giac [A]  time = 2.09233, size = 103, normalized size = 1.37 \begin{align*} -\frac{6875}{5 \, x + 3} + \frac{1875 \,{\left (\frac{34866}{5 \, x + 3} + \frac{19635}{{\left (5 \, x + 3\right )}^{2}} + \frac{5040}{{\left (5 \, x + 3\right )}^{3}} + \frac{505}{{\left (5 \, x + 3\right )}^{4}} + 23625\right )}}{{\left (\frac{1}{5 \, x + 3} + 3\right )}^{5}} + 125000 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6875/(5*x + 3) + 1875*(34866/(5*x + 3) + 19635/(5*x + 3)^2 + 5040/(5*x + 3)^3 + 505/(5*x + 3)^4 + 23625)/(1/(
5*x + 3) + 3)^5 + 125000*log(abs(-1/(5*x + 3) - 3))