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*Log[2 + 3*x] + 424975*Log[3 + 5*x]

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Rubi [A]  time = 0.0379856, antiderivative size = 77, 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} \[ \frac{57110}{3 x+2}+\frac{46475}{5 x+3}+\frac{3467}{(3 x+2)^2}-\frac{3025}{2 (5 x+3)^2}+\frac{707}{3 (3 x+2)^3}+\frac{49}{4 (3 x+2)^4}-424975 \log (3 x+2)+424975 \log (5 x+3) \]

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*}

Mathematica [A]  time = 0.0534782, 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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Maple [A]  time = 0.009, size = 72, normalized size = 0.9 \begin{align*}{\frac{49}{4\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{707}{3\, \left ( 2+3\,x \right ) ^{3}}}+3467\, \left ( 2+3\,x \right ) ^{-2}+57110\, \left ( 2+3\,x \right ) ^{-1}-{\frac{3025}{2\, \left ( 3+5\,x \right ) ^{2}}}+46475\, \left ( 3+5\,x \right ) ^{-1}-424975\,\ln \left ( 2+3\,x \right ) +424975\,\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)^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*ln(2+3*x)+42
4975*ln(3+5*x)

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Maxima [A]  time = 1.09161, size = 103, normalized size = 1.34 \begin{align*} \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 ) \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)^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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Fricas [A]  time = 1.29718, size = 486, normalized size = 6.31 \begin{align*} \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 )}} \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)^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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Sympy [A]  time = 0.19635, size = 71, normalized size = 0.92 \begin{align*} \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 )} \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)**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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Giac [A]  time = 2.24529, size = 103, normalized size = 1.34 \begin{align*} \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 ) \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)^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))