3.1414 \(\int \frac{(1-2 x)^3}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{16698}{3 x+2}-\frac{6655}{5 x+3}-\frac{2541}{2 (3 x+2)^2}-\frac{3136}{27 (3 x+2)^3}-\frac{343}{36 (3 x+2)^4}+103455 \log (3 x+2)-103455 \log (5 x+3) \]

[Out]

-343/(36*(2 + 3*x)^4) - 3136/(27*(2 + 3*x)^3) - 2541/(2*(2 + 3*x)^2) - 16698/(2 + 3*x) - 6655/(3 + 5*x) + 1034
55*Log[2 + 3*x] - 103455*Log[3 + 5*x]

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Rubi [A]  time = 0.0370769, antiderivative size = 68, 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} \[ -\frac{16698}{3 x+2}-\frac{6655}{5 x+3}-\frac{2541}{2 (3 x+2)^2}-\frac{3136}{27 (3 x+2)^3}-\frac{343}{36 (3 x+2)^4}+103455 \log (3 x+2)-103455 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-343/(36*(2 + 3*x)^4) - 3136/(27*(2 + 3*x)^3) - 2541/(2*(2 + 3*x)^2) - 16698/(2 + 3*x) - 6655/(3 + 5*x) + 1034
55*Log[2 + 3*x] - 103455*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)^5 (3+5 x)^2} \, dx &=\int \left (\frac{343}{3 (2+3 x)^5}+\frac{3136}{3 (2+3 x)^4}+\frac{7623}{(2+3 x)^3}+\frac{50094}{(2+3 x)^2}+\frac{310365}{2+3 x}+\frac{33275}{(3+5 x)^2}-\frac{517275}{3+5 x}\right ) \, dx\\ &=-\frac{343}{36 (2+3 x)^4}-\frac{3136}{27 (2+3 x)^3}-\frac{2541}{2 (2+3 x)^2}-\frac{16698}{2+3 x}-\frac{6655}{3+5 x}+103455 \log (2+3 x)-103455 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0307689, size = 70, normalized size = 1.03 \[ -\frac{16698}{3 x+2}-\frac{6655}{5 x+3}-\frac{2541}{2 (3 x+2)^2}-\frac{3136}{27 (3 x+2)^3}-\frac{343}{36 (3 x+2)^4}+103455 \log (5 (3 x+2))-103455 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-343/(36*(2 + 3*x)^4) - 3136/(27*(2 + 3*x)^3) - 2541/(2*(2 + 3*x)^2) - 16698/(2 + 3*x) - 6655/(3 + 5*x) + 1034
55*Log[5*(2 + 3*x)] - 103455*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \begin{align*} -{\frac{343}{36\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{3136}{27\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{2541}{2\, \left ( 2+3\,x \right ) ^{2}}}-16698\, \left ( 2+3\,x \right ) ^{-1}-6655\, \left ( 3+5\,x \right ) ^{-1}+103455\,\ln \left ( 2+3\,x \right ) -103455\,\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)^5/(3+5*x)^2,x)

[Out]

-343/36/(2+3*x)^4-3136/27/(2+3*x)^3-2541/2/(2+3*x)^2-16698/(2+3*x)-6655/(3+5*x)+103455*ln(2+3*x)-103455*ln(3+5
*x)

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Maxima [A]  time = 1.02909, size = 89, normalized size = 1.31 \begin{align*} -\frac{301674780 \, x^{4} + 794410254 \, x^{3} + 784130946 \, x^{2} + 343827337 \, x + 56505975}{108 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - 103455 \, \log \left (5 \, x + 3\right ) + 103455 \, \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)^5/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/108*(301674780*x^4 + 794410254*x^3 + 784130946*x^2 + 343827337*x + 56505975)/(405*x^5 + 1323*x^4 + 1728*x^3
+ 1128*x^2 + 368*x + 48) - 103455*log(5*x + 3) + 103455*log(3*x + 2)

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Fricas [A]  time = 1.28398, size = 401, normalized size = 5.9 \begin{align*} -\frac{301674780 \, x^{4} + 794410254 \, x^{3} + 784130946 \, x^{2} + 11173140 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 11173140 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 343827337 \, x + 56505975}{108 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*(301674780*x^4 + 794410254*x^3 + 784130946*x^2 + 11173140*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 3
68*x + 48)*log(5*x + 3) - 11173140*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(3*x + 2) + 3438
27337*x + 56505975)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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Sympy [A]  time = 0.18778, size = 61, normalized size = 0.9 \begin{align*} - \frac{301674780 x^{4} + 794410254 x^{3} + 784130946 x^{2} + 343827337 x + 56505975}{43740 x^{5} + 142884 x^{4} + 186624 x^{3} + 121824 x^{2} + 39744 x + 5184} - 103455 \log{\left (x + \frac{3}{5} \right )} + 103455 \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)**5/(3+5*x)**2,x)

[Out]

-(301674780*x**4 + 794410254*x**3 + 784130946*x**2 + 343827337*x + 56505975)/(43740*x**5 + 142884*x**4 + 18662
4*x**3 + 121824*x**2 + 39744*x + 5184) - 103455*log(x + 3/5) + 103455*log(x + 2/3)

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Giac [A]  time = 1.53836, size = 90, normalized size = 1.32 \begin{align*} -\frac{6655}{5 \, x + 3} + \frac{5 \,{\left (\frac{9923332}{5 \, x + 3} + \frac{3831284}{{\left (5 \, x + 3\right )}^{2}} + \frac{514536}{{\left (5 \, x + 3\right )}^{3}} + 8795037\right )}}{4 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{4}} + 103455 \, \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)^3/(2+3*x)^5/(3+5*x)^2,x, algorithm="giac")

[Out]

-6655/(5*x + 3) + 5/4*(9923332/(5*x + 3) + 3831284/(5*x + 3)^2 + 514536/(5*x + 3)^3 + 8795037)/(1/(5*x + 3) +
3)^4 + 103455*log(abs(-1/(5*x + 3) - 3))