3.1317 \(\int \frac{(1-2 x)^2}{(2+3 x)^5 (3+5 x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{7480}{3 x+2}-\frac{3025}{5 x+3}-\frac{1133}{2 (3 x+2)^2}-\frac{154}{3 (3 x+2)^3}-\frac{49}{12 (3 x+2)^4}+46475 \log (3 x+2)-46475 \log (5 x+3) \]

[Out]

-49/(12*(2 + 3*x)^4) - 154/(3*(2 + 3*x)^3) - 1133/(2*(2 + 3*x)^2) - 7480/(2 + 3*x) - 3025/(3 + 5*x) + 46475*Lo
g[2 + 3*x] - 46475*Log[3 + 5*x]

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Rubi [A]  time = 0.0332528, 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} \[ -\frac{7480}{3 x+2}-\frac{3025}{5 x+3}-\frac{1133}{2 (3 x+2)^2}-\frac{154}{3 (3 x+2)^3}-\frac{49}{12 (3 x+2)^4}+46475 \log (3 x+2)-46475 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(12*(2 + 3*x)^4) - 154/(3*(2 + 3*x)^3) - 1133/(2*(2 + 3*x)^2) - 7480/(2 + 3*x) - 3025/(3 + 5*x) + 46475*Lo
g[2 + 3*x] - 46475*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)^2}{(2+3 x)^5 (3+5 x)^2} \, dx &=\int \left (\frac{49}{(2+3 x)^5}+\frac{462}{(2+3 x)^4}+\frac{3399}{(2+3 x)^3}+\frac{22440}{(2+3 x)^2}+\frac{139425}{2+3 x}+\frac{15125}{(3+5 x)^2}-\frac{232375}{3+5 x}\right ) \, dx\\ &=-\frac{49}{12 (2+3 x)^4}-\frac{154}{3 (2+3 x)^3}-\frac{1133}{2 (2+3 x)^2}-\frac{7480}{2+3 x}-\frac{3025}{3+5 x}+46475 \log (2+3 x)-46475 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0768305, size = 57, normalized size = 0.84 \[ -\frac{5019300 x^4+13217490 x^3+13046462 x^2+5720639 x+940153}{4 (3 x+2)^4 (5 x+3)}+46475 \log (5 (3 x+2))-46475 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(940153 + 5720639*x + 13046462*x^2 + 13217490*x^3 + 5019300*x^4)/(4*(2 + 3*x)^4*(3 + 5*x)) + 46475*Log[5*(2 +
3*x)] - 46475*Log[3 + 5*x]

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \begin{align*} -{\frac{49}{12\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{154}{3\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{1133}{2\, \left ( 2+3\,x \right ) ^{2}}}-7480\, \left ( 2+3\,x \right ) ^{-1}-3025\, \left ( 3+5\,x \right ) ^{-1}+46475\,\ln \left ( 2+3\,x \right ) -46475\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/12/(2+3*x)^4-154/3/(2+3*x)^3-1133/2/(2+3*x)^2-7480/(2+3*x)-3025/(3+5*x)+46475*ln(2+3*x)-46475*ln(3+5*x)

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Maxima [A]  time = 1.16256, size = 89, normalized size = 1.31 \begin{align*} -\frac{5019300 \, x^{4} + 13217490 \, x^{3} + 13046462 \, x^{2} + 5720639 \, x + 940153}{4 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} - 46475 \, \log \left (5 \, x + 3\right ) + 46475 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(5019300*x^4 + 13217490*x^3 + 13046462*x^2 + 5720639*x + 940153)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^
2 + 368*x + 48) - 46475*log(5*x + 3) + 46475*log(3*x + 2)

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Fricas [A]  time = 1.18086, size = 382, normalized size = 5.62 \begin{align*} -\frac{5019300 \, x^{4} + 13217490 \, x^{3} + 13046462 \, x^{2} + 185900 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (5 \, x + 3\right ) - 185900 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )} \log \left (3 \, x + 2\right ) + 5720639 \, x + 940153}{4 \,{\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)^2/(2+3*x)^5/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/4*(5019300*x^4 + 13217490*x^3 + 13046462*x^2 + 185900*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 4
8)*log(5*x + 3) - 185900*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*log(3*x + 2) + 5720639*x + 94
0153)/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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Sympy [A]  time = 0.18085, size = 61, normalized size = 0.9 \begin{align*} - \frac{5019300 x^{4} + 13217490 x^{3} + 13046462 x^{2} + 5720639 x + 940153}{1620 x^{5} + 5292 x^{4} + 6912 x^{3} + 4512 x^{2} + 1472 x + 192} - 46475 \log{\left (x + \frac{3}{5} \right )} + 46475 \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/(2+3*x)**5/(3+5*x)**2,x)

[Out]

-(5019300*x**4 + 13217490*x**3 + 13046462*x**2 + 5720639*x + 940153)/(1620*x**5 + 5292*x**4 + 6912*x**3 + 4512
*x**2 + 1472*x + 192) - 46475*log(x + 3/5) + 46475*log(x + 2/3)

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Giac [A]  time = 2.82564, size = 90, normalized size = 1.32 \begin{align*} -\frac{3025}{5 \, x + 3} + \frac{25 \,{\left (\frac{884412}{5 \, x + 3} + \frac{341028}{{\left (5 \, x + 3\right )}^{2}} + \frac{45688}{{\left (5 \, x + 3\right )}^{3}} + 784485\right )}}{4 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{4}} + 46475 \, \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/(2+3*x)^5/(3+5*x)^2,x, algorithm="giac")

[Out]

-3025/(5*x + 3) + 25/4*(884412/(5*x + 3) + 341028/(5*x + 3)^2 + 45688/(5*x + 3)^3 + 784485)/(1/(5*x + 3) + 3)^
4 + 46475*log(abs(-1/(5*x + 3) - 3))