3.1334 \(\int \frac {(1-2 x)^2}{(2+3 x)^5 (3+5 x)^3} \, dx\)

Optimal. Leaf size=77 \[ \frac {57110}{3 x+2}+\frac {46475}{5 x+3}+\frac {3467}{(3 x+2)^2}-\frac {3025}{2 (5 x+3)^2}+\frac {707}{3 (3 x+2)^3}+\frac {49}{4 (3 x+2)^4}-424975 \log (3 x+2)+424975 \log (5 x+3) \]

[Out]

49/4/(2+3*x)^4+707/3/(2+3*x)^3+3467/(2+3*x)^2+57110/(2+3*x)-3025/2/(3+5*x)^2+46475/(3+5*x)-424975*ln(2+3*x)+42
4975*ln(3+5*x)

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Rubi [A]  time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, 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} \[ \frac {57110}{3 x+2}+\frac {46475}{5 x+3}+\frac {3467}{(3 x+2)^2}-\frac {3025}{2 (5 x+3)^2}+\frac {707}{3 (3 x+2)^3}+\frac {49}{4 (3 x+2)^4}-424975 \log (3 x+2)+424975 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(4*(2 + 3*x)^4) + 707/(3*(2 + 3*x)^3) + 3467/(2 + 3*x)^2 + 57110/(2 + 3*x) - 3025/(2*(3 + 5*x)^2) + 46475/(
3 + 5*x) - 424975*Log[2 + 3*x] + 424975*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)^3} \, dx &=\int \left (-\frac {147}{(2+3 x)^5}-\frac {2121}{(2+3 x)^4}-\frac {20802}{(2+3 x)^3}-\frac {171330}{(2+3 x)^2}-\frac {1274925}{2+3 x}+\frac {15125}{(3+5 x)^3}-\frac {232375}{(3+5 x)^2}+\frac {2124875}{3+5 x}\right ) \, dx\\ &=\frac {49}{4 (2+3 x)^4}+\frac {707}{3 (2+3 x)^3}+\frac {3467}{(2+3 x)^2}+\frac {57110}{2+3 x}-\frac {3025}{2 (3+5 x)^2}+\frac {46475}{3+5 x}-424975 \log (2+3 x)+424975 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 79, normalized size = 1.03 \[ \frac {57110}{3 x+2}+\frac {46475}{5 x+3}+\frac {3467}{(3 x+2)^2}-\frac {3025}{2 (5 x+3)^2}+\frac {707}{3 (3 x+2)^3}+\frac {49}{4 (3 x+2)^4}-424975 \log (5 (3 x+2))+424975 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(4*(2 + 3*x)^4) + 707/(3*(2 + 3*x)^3) + 3467/(2 + 3*x)^2 + 57110/(2 + 3*x) - 3025/(2*(3 + 5*x)^2) + 46475/(
3 + 5*x) - 424975*Log[5*(2 + 3*x)] + 424975*Log[3 + 5*x]

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fricas [A]  time = 0.84, size = 135, normalized size = 1.75 \[ \frac {688459500 \, x^{5} + 2226019050 \, x^{4} + 2877250740 \, x^{3} + 1858347679 \, x^{2} + 5099700 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 5099700 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 599747838 \, x + 77372211}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(688459500*x^5 + 2226019050*x^4 + 2877250740*x^3 + 1858347679*x^2 + 5099700*(2025*x^6 + 7830*x^5 + 12609*
x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 5099700*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3
+ 5224*x^2 + 1344*x + 144)*log(3*x + 2) + 599747838*x + 77372211)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^
3 + 5224*x^2 + 1344*x + 144)

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giac [A]  time = 0.90, size = 76, normalized size = 0.99 \[ \frac {57110}{3 \, x + 2} - \frac {4125 \, {\left (\frac {404}{3 \, x + 2} - 1855\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {3467}{{\left (3 \, x + 2\right )}^{2}} + \frac {707}{3 \, {\left (3 \, x + 2\right )}^{3}} + \frac {49}{4 \, {\left (3 \, x + 2\right )}^{4}} + 424975 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

57110/(3*x + 2) - 4125/2*(404/(3*x + 2) - 1855)/(1/(3*x + 2) - 5)^2 + 3467/(3*x + 2)^2 + 707/3/(3*x + 2)^3 + 4
9/4/(3*x + 2)^4 + 424975*log(abs(-1/(3*x + 2) + 5))

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maple [A]  time = 0.01, size = 72, normalized size = 0.94 \[ -424975 \ln \left (3 x +2\right )+424975 \ln \left (5 x +3\right )+\frac {49}{4 \left (3 x +2\right )^{4}}+\frac {707}{3 \left (3 x +2\right )^{3}}+\frac {3467}{\left (3 x +2\right )^{2}}+\frac {57110}{3 x +2}-\frac {3025}{2 \left (5 x +3\right )^{2}}+\frac {46475}{5 x +3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/4/(3*x+2)^4+707/3/(3*x+2)^3+3467/(3*x+2)^2+57110/(3*x+2)-3025/2/(5*x+3)^2+46475/(5*x+3)-424975*ln(3*x+2)+42
4975*ln(5*x+3)

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maxima [A]  time = 0.56, size = 76, normalized size = 0.99 \[ \frac {688459500 \, x^{5} + 2226019050 \, x^{4} + 2877250740 \, x^{3} + 1858347679 \, x^{2} + 599747838 \, x + 77372211}{12 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 424975 \, \log \left (5 \, x + 3\right ) - 424975 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(688459500*x^5 + 2226019050*x^4 + 2877250740*x^3 + 1858347679*x^2 + 599747838*x + 77372211)/(2025*x^6 + 7
830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 424975*log(5*x + 3) - 424975*log(3*x + 2)

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mupad [B]  time = 1.12, size = 65, normalized size = 0.84 \[ \frac {\frac {84995\,x^5}{3}+\frac {1648903\,x^4}{18}+\frac {47954179\,x^3}{405}+\frac {1858347679\,x^2}{24300}+\frac {99957973\,x}{4050}+\frac {25790737}{8100}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}}-849950\,\mathrm {atanh}\left (30\,x+19\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((99957973*x)/4050 + (1858347679*x^2)/24300 + (47954179*x^3)/405 + (1648903*x^4)/18 + (84995*x^5)/3 + 25790737
/8100)/((448*x)/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225) - 849950*a
tanh(30*x + 19)

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sympy [A]  time = 0.20, size = 71, normalized size = 0.92 \[ \frac {688459500 x^{5} + 2226019050 x^{4} + 2877250740 x^{3} + 1858347679 x^{2} + 599747838 x + 77372211}{24300 x^{6} + 93960 x^{5} + 151308 x^{4} + 129888 x^{3} + 62688 x^{2} + 16128 x + 1728} + 424975 \log {\left (x + \frac {3}{5} \right )} - 424975 \log {\left (x + \frac {2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(688459500*x**5 + 2226019050*x**4 + 2877250740*x**3 + 1858347679*x**2 + 599747838*x + 77372211)/(24300*x**6 +
93960*x**5 + 151308*x**4 + 129888*x**3 + 62688*x**2 + 16128*x + 1728) + 424975*log(x + 3/5) - 424975*log(x + 2
/3)

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