3.1233 \(\int \frac{1-2 x}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=66 \[ \frac{3060}{3 x+2}+\frac{3350}{5 x+3}+\frac{309}{2 (3 x+2)^2}-\frac{275}{2 (5 x+3)^2}+\frac{7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (5 x+3) \]

[Out]

7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*
x] + 25350*Log[3 + 5*x]

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Rubi [A]  time = 0.0310509, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{3060}{3 x+2}+\frac{3350}{5 x+3}+\frac{309}{2 (3 x+2)^2}-\frac{275}{2 (5 x+3)^2}+\frac{7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*
x] + 25350*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)^4 (3+5 x)^3} \, dx &=\int \left (-\frac{63}{(2+3 x)^4}-\frac{927}{(2+3 x)^3}-\frac{9180}{(2+3 x)^2}-\frac{76050}{2+3 x}+\frac{1375}{(3+5 x)^3}-\frac{16750}{(3+5 x)^2}+\frac{126750}{3+5 x}\right ) \, dx\\ &=\frac{7}{(2+3 x)^3}+\frac{309}{2 (2+3 x)^2}+\frac{3060}{2+3 x}-\frac{275}{2 (3+5 x)^2}+\frac{3350}{3+5 x}-25350 \log (2+3 x)+25350 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0229768, size = 68, normalized size = 1.03 \[ \frac{3060}{3 x+2}+\frac{3350}{5 x+3}+\frac{309}{2 (3 x+2)^2}-\frac{275}{2 (5 x+3)^2}+\frac{7}{(3 x+2)^3}-25350 \log (3 x+2)+25350 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

7/(2 + 3*x)^3 + 309/(2*(2 + 3*x)^2) + 3060/(2 + 3*x) - 275/(2*(3 + 5*x)^2) + 3350/(3 + 5*x) - 25350*Log[2 + 3*
x] + 25350*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.009, size = 63, normalized size = 1. \begin{align*} 7\, \left ( 2+3\,x \right ) ^{-3}+{\frac{309}{2\, \left ( 2+3\,x \right ) ^{2}}}+3060\, \left ( 2+3\,x \right ) ^{-1}-{\frac{275}{2\, \left ( 3+5\,x \right ) ^{2}}}+3350\, \left ( 3+5\,x \right ) ^{-1}-25350\,\ln \left ( 2+3\,x \right ) +25350\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)/(2+3*x)^4/(3+5*x)^3,x)

[Out]

7/(2+3*x)^3+309/2/(2+3*x)^2+3060/(2+3*x)-275/2/(3+5*x)^2+3350/(3+5*x)-25350*ln(2+3*x)+25350*ln(3+5*x)

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Maxima [A]  time = 1.02904, size = 89, normalized size = 1.35 \begin{align*} \frac{2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 25350 \, \log \left (5 \, x + 3\right ) - 25350 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 +
564*x + 72) + 25350*log(5*x + 3) - 25350*log(3*x + 2)

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Fricas [A]  time = 1.49649, size = 375, normalized size = 5.68 \begin{align*} \frac{2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 50700 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 50700 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 2404363 \, x + 384608}{2 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 50700*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*l
og(5*x + 3) - 50700*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 2404363*x + 384608)
/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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Sympy [A]  time = 0.179372, size = 61, normalized size = 0.92 \begin{align*} \frac{2281500 x^{4} + 5855850 x^{3} + 5631080 x^{2} + 2404363 x + 384608}{1350 x^{5} + 4320 x^{4} + 5526 x^{3} + 3532 x^{2} + 1128 x + 144} + 25350 \log{\left (x + \frac{3}{5} \right )} - 25350 \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+3*x)**4/(3+5*x)**3,x)

[Out]

(2281500*x**4 + 5855850*x**3 + 5631080*x**2 + 2404363*x + 384608)/(1350*x**5 + 4320*x**4 + 5526*x**3 + 3532*x*
*2 + 1128*x + 144) + 25350*log(x + 3/5) - 25350*log(x + 2/3)

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Giac [A]  time = 1.57149, size = 74, normalized size = 1.12 \begin{align*} \frac{2281500 \, x^{4} + 5855850 \, x^{3} + 5631080 \, x^{2} + 2404363 \, x + 384608}{2 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{3}} + 25350 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 25350 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

1/2*(2281500*x^4 + 5855850*x^3 + 5631080*x^2 + 2404363*x + 384608)/((5*x + 3)^2*(3*x + 2)^3) + 25350*log(abs(5
*x + 3)) - 25350*log(abs(3*x + 2))