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

Optimal. Leaf size=75 \[ \frac {25350}{3 x+2}+\frac {20875}{5 x+3}+\frac {1530}{(3 x+2)^2}-\frac {1375}{2 (5 x+3)^2}+\frac {103}{(3 x+2)^3}+\frac {21}{4 (3 x+2)^4}-189375 \log (3 x+2)+189375 \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {25350}{3 x+2}+\frac {20875}{5 x+3}+\frac {1530}{(3 x+2)^2}-\frac {1375}{2 (5 x+3)^2}+\frac {103}{(3 x+2)^3}+\frac {21}{4 (3 x+2)^4}-189375 \log (3 x+2)+189375 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) - 1375/(2*(3 + 5*x)^2) + 20875/(3 +
5*x) - 189375*Log[2 + 3*x] + 189375*Log[3 + 5*x]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx &=\int \left (-\frac {63}{(2+3 x)^5}-\frac {927}{(2+3 x)^4}-\frac {9180}{(2+3 x)^3}-\frac {76050}{(2+3 x)^2}-\frac {568125}{2+3 x}+\frac {6875}{(3+5 x)^3}-\frac {104375}{(3+5 x)^2}+\frac {946875}{3+5 x}\right ) \, dx\\ &=\frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 77, normalized size = 1.03 \begin {gather*} \frac {25350}{3 x+2}+\frac {20875}{5 x+3}+\frac {1530}{(3 x+2)^2}-\frac {1375}{2 (5 x+3)^2}+\frac {103}{(3 x+2)^3}+\frac {21}{4 (3 x+2)^4}-189375 \log (3 x+2)+189375 \log (-3 (5 x+3)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) - 1375/(2*(3 + 5*x)^2) + 20875/(3 +
5*x) - 189375*Log[2 + 3*x] + 189375*Log[-3*(3 + 5*x)]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.22, size = 135, normalized size = 1.80 \begin {gather*} \frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 757500 \, {\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 ) - 757500 \, {\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 ) + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 757500*(2025*x^6 + 7830*x^5 + 12609*x^4 +
10824*x^3 + 5224*x^2 + 1344*x + 144)*log(5*x + 3) - 757500*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 522
4*x^2 + 1344*x + 144)*log(3*x + 2) + 89085434*x + 11492725)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 522
4*x^2 + 1344*x + 144)

________________________________________________________________________________________

giac [A]  time = 1.13, size = 76, normalized size = 1.01 \begin {gather*} \frac {25350}{3 \, x + 2} - \frac {9375 \, {\left (\frac {80}{3 \, x + 2} - 367\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {1530}{{\left (3 \, x + 2\right )}^{2}} + \frac {103}{{\left (3 \, x + 2\right )}^{3}} + \frac {21}{4 \, {\left (3 \, x + 2\right )}^{4}} + 189375 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25350/(3*x + 2) - 9375/2*(80/(3*x + 2) - 367)/(1/(3*x + 2) - 5)^2 + 1530/(3*x + 2)^2 + 103/(3*x + 2)^3 + 21/4/
(3*x + 2)^4 + 189375*log(abs(-1/(3*x + 2) + 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 72, normalized size = 0.96 \begin {gather*} -189375 \ln \left (3 x +2\right )+189375 \ln \left (5 x +3\right )+\frac {21}{4 \left (3 x +2\right )^{4}}+\frac {103}{\left (3 x +2\right )^{3}}+\frac {1530}{\left (3 x +2\right )^{2}}+\frac {25350}{3 x +2}-\frac {1375}{2 \left (5 x +3\right )^{2}}+\frac {20875}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/4/(3*x+2)^4+103/(3*x+2)^3+1530/(3*x+2)^2+25350/(3*x+2)-1375/2/(5*x+3)^2+20875/(5*x+3)-189375*ln(3*x+2)+1893
75*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 76, normalized size = 1.01 \begin {gather*} \frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 189375 \, \log \left (5 \, x + 3\right ) - 189375 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 89085434*x + 11492725)/(2025*x^6 + 7830*x
^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144) + 189375*log(5*x + 3) - 189375*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 65, normalized size = 0.87 \begin {gather*} \frac {12625\,x^5+\frac {244925\,x^4}{6}+\frac {1424605\,x^3}{27}+\frac {11041421\,x^2}{324}+\frac {44542717\,x}{4050}+\frac {459709}{324}}{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}}-378750\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((44542717*x)/4050 + (11041421*x^2)/324 + (1424605*x^3)/27 + (244925*x^4)/6 + 12625*x^5 + 459709/324)/((448*x)
/675 + (5224*x^2)/2025 + (3608*x^3)/675 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225) - 378750*atanh(30*x + 19)

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 73, normalized size = 0.97 \begin {gather*} - \frac {- 102262500 x^{5} - 330648750 x^{4} - 427381500 x^{3} - 276035525 x^{2} - 89085434 x - 11492725}{8100 x^{6} + 31320 x^{5} + 50436 x^{4} + 43296 x^{3} + 20896 x^{2} + 5376 x + 576} + 189375 \log {\left (x + \frac {3}{5} \right )} - 189375 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-102262500*x**5 - 330648750*x**4 - 427381500*x**3 - 276035525*x**2 - 89085434*x - 11492725)/(8100*x**6 + 313
20*x**5 + 50436*x**4 + 43296*x**3 + 20896*x**2 + 5376*x + 576) + 189375*log(x + 3/5) - 189375*log(x + 2/3)

________________________________________________________________________________________