3.1400 \(\int \frac{(1-2 x)^3}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=70 \[ \frac{6655}{3 x+2}+\frac{1331}{2 (3 x+2)^2}+\frac{7189}{81 (3 x+2)^3}+\frac{1421}{108 (3 x+2)^4}+\frac{343}{135 (3 x+2)^5}-33275 \log (3 x+2)+33275 \log (5 x+3) \]

[Out]

343/(135*(2 + 3*x)^5) + 1421/(108*(2 + 3*x)^4) + 7189/(81*(2 + 3*x)^3) + 1331/(2*(2 + 3*x)^2) + 6655/(2 + 3*x)
- 33275*Log[2 + 3*x] + 33275*Log[3 + 5*x]

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Rubi [A]  time = 0.0273518, antiderivative size = 70, 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} \[ \frac{6655}{3 x+2}+\frac{1331}{2 (3 x+2)^2}+\frac{7189}{81 (3 x+2)^3}+\frac{1421}{108 (3 x+2)^4}+\frac{343}{135 (3 x+2)^5}-33275 \log (3 x+2)+33275 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(135*(2 + 3*x)^5) + 1421/(108*(2 + 3*x)^4) + 7189/(81*(2 + 3*x)^3) + 1331/(2*(2 + 3*x)^2) + 6655/(2 + 3*x)
- 33275*Log[2 + 3*x] + 33275*Log[3 + 5*x]

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{(1-2 x)^3}{(2+3 x)^6 (3+5 x)} \, dx &=\int \left (-\frac{343}{9 (2+3 x)^6}-\frac{1421}{9 (2+3 x)^5}-\frac{7189}{9 (2+3 x)^4}-\frac{3993}{(2+3 x)^3}-\frac{19965}{(2+3 x)^2}-\frac{99825}{2+3 x}+\frac{166375}{3+5 x}\right ) \, dx\\ &=\frac{343}{135 (2+3 x)^5}+\frac{1421}{108 (2+3 x)^4}+\frac{7189}{81 (2+3 x)^3}+\frac{1331}{2 (2+3 x)^2}+\frac{6655}{2+3 x}-33275 \log (2+3 x)+33275 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0398071, size = 50, normalized size = 0.71 \[ \frac{873269100 x^4+2357826570 x^3+2388229560 x^2+1075586865 x+181744346}{1620 (3 x+2)^5}-33275 \log (5 (3 x+2))+33275 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(181744346 + 1075586865*x + 2388229560*x^2 + 2357826570*x^3 + 873269100*x^4)/(1620*(2 + 3*x)^5) - 33275*Log[5*
(2 + 3*x)] + 33275*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \begin{align*}{\frac{343}{135\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{1421}{108\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{7189}{81\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1331}{2\, \left ( 2+3\,x \right ) ^{2}}}+6655\, \left ( 2+3\,x \right ) ^{-1}-33275\,\ln \left ( 2+3\,x \right ) +33275\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/135/(2+3*x)^5+1421/108/(2+3*x)^4+7189/81/(2+3*x)^3+1331/2/(2+3*x)^2+6655/(2+3*x)-33275*ln(2+3*x)+33275*ln(
3+5*x)

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Maxima [A]  time = 1.04573, size = 89, normalized size = 1.27 \begin{align*} \frac{873269100 \, x^{4} + 2357826570 \, x^{3} + 2388229560 \, x^{2} + 1075586865 \, x + 181744346}{1620 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + 33275 \, \log \left (5 \, x + 3\right ) - 33275 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1620*(873269100*x^4 + 2357826570*x^3 + 2388229560*x^2 + 1075586865*x + 181744346)/(243*x^5 + 810*x^4 + 1080*
x^3 + 720*x^2 + 240*x + 32) + 33275*log(5*x + 3) - 33275*log(3*x + 2)

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Fricas [A]  time = 1.39854, size = 398, normalized size = 5.69 \begin{align*} \frac{873269100 \, x^{4} + 2357826570 \, x^{3} + 2388229560 \, x^{2} + 53905500 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (5 \, x + 3\right ) - 53905500 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 1075586865 \, x + 181744346}{1620 \,{\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((1-2*x)^3/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/1620*(873269100*x^4 + 2357826570*x^3 + 2388229560*x^2 + 53905500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 2
40*x + 32)*log(5*x + 3) - 53905500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 107558
6865*x + 181744346)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [A]  time = 0.181181, size = 61, normalized size = 0.87 \begin{align*} \frac{873269100 x^{4} + 2357826570 x^{3} + 2388229560 x^{2} + 1075586865 x + 181744346}{393660 x^{5} + 1312200 x^{4} + 1749600 x^{3} + 1166400 x^{2} + 388800 x + 51840} + 33275 \log{\left (x + \frac{3}{5} \right )} - 33275 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(873269100*x**4 + 2357826570*x**3 + 2388229560*x**2 + 1075586865*x + 181744346)/(393660*x**5 + 1312200*x**4 +
1749600*x**3 + 1166400*x**2 + 388800*x + 51840) + 33275*log(x + 3/5) - 33275*log(x + 2/3)

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Giac [A]  time = 2.19264, size = 65, normalized size = 0.93 \begin{align*} \frac{873269100 \, x^{4} + 2357826570 \, x^{3} + 2388229560 \, x^{2} + 1075586865 \, x + 181744346}{1620 \,{\left (3 \, x + 2\right )}^{5}} + 33275 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 33275 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1620*(873269100*x^4 + 2357826570*x^3 + 2388229560*x^2 + 1075586865*x + 181744346)/(3*x + 2)^5 + 33275*log(ab
s(5*x + 3)) - 33275*log(abs(3*x + 2))