3.1231 \(\int \frac{1-2 x}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{21}{3 x+2}+\frac{68}{5 x+3}-\frac{11}{2 (5 x+3)^2}-309 \log (3 x+2)+309 \log (5 x+3) \]

[Out]

21/(2 + 3*x) - 11/(2*(3 + 5*x)^2) + 68/(3 + 5*x) - 309*Log[2 + 3*x] + 309*Log[3 + 5*x]

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Rubi [A]  time = 0.0210872, antiderivative size = 46, 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} \[ \frac{21}{3 x+2}+\frac{68}{5 x+3}-\frac{11}{2 (5 x+3)^2}-309 \log (3 x+2)+309 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2 + 3*x) - 11/(2*(3 + 5*x)^2) + 68/(3 + 5*x) - 309*Log[2 + 3*x] + 309*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)^2 (3+5 x)^3} \, dx &=\int \left (-\frac{63}{(2+3 x)^2}-\frac{927}{2+3 x}+\frac{55}{(3+5 x)^3}-\frac{340}{(3+5 x)^2}+\frac{1545}{3+5 x}\right ) \, dx\\ &=\frac{21}{2+3 x}-\frac{11}{2 (3+5 x)^2}+\frac{68}{3+5 x}-309 \log (2+3 x)+309 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.019711, size = 48, normalized size = 1.04 \[ \frac{21}{3 x+2}+\frac{68}{5 x+3}-\frac{11}{2 (5 x+3)^2}-309 \log (3 x+2)+309 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2 + 3*x) - 11/(2*(3 + 5*x)^2) + 68/(3 + 5*x) - 309*Log[2 + 3*x] + 309*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.008, size = 45, normalized size = 1. \begin{align*} 21\, \left ( 2+3\,x \right ) ^{-1}-{\frac{11}{2\, \left ( 3+5\,x \right ) ^{2}}}+68\, \left ( 3+5\,x \right ) ^{-1}-309\,\ln \left ( 2+3\,x \right ) +309\,\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)^2/(3+5*x)^3,x)

[Out]

21/(2+3*x)-11/2/(3+5*x)^2+68/(3+5*x)-309*ln(2+3*x)+309*ln(3+5*x)

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Maxima [A]  time = 1.08227, size = 62, normalized size = 1.35 \begin{align*} \frac{3090 \, x^{2} + 3811 \, x + 1172}{2 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + 309 \, \log \left (5 \, x + 3\right ) - 309 \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/2*(3090*x^2 + 3811*x + 1172)/(75*x^3 + 140*x^2 + 87*x + 18) + 309*log(5*x + 3) - 309*log(3*x + 2)

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Fricas [A]  time = 1.51904, size = 220, normalized size = 4.78 \begin{align*} \frac{3090 \, x^{2} + 618 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 618 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) + 3811 \, x + 1172}{2 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3090*x^2 + 618*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 618*(75*x^3 + 140*x^2 + 87*x + 18)*log(3*x +
2) + 3811*x + 1172)/(75*x^3 + 140*x^2 + 87*x + 18)

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Sympy [A]  time = 0.148221, size = 41, normalized size = 0.89 \begin{align*} \frac{3090 x^{2} + 3811 x + 1172}{150 x^{3} + 280 x^{2} + 174 x + 36} + 309 \log{\left (x + \frac{3}{5} \right )} - 309 \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)**2/(3+5*x)**3,x)

[Out]

(3090*x**2 + 3811*x + 1172)/(150*x**3 + 280*x**2 + 174*x + 36) + 309*log(x + 3/5) - 309*log(x + 2/3)

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Giac [A]  time = 2.71856, size = 66, normalized size = 1.43 \begin{align*} \frac{21}{3 \, x + 2} - \frac{15 \,{\left (\frac{202}{3 \, x + 2} - 845\right )}}{2 \,{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + 309 \, \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+3*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

21/(3*x + 2) - 15/2*(202/(3*x + 2) - 845)/(1/(3*x + 2) - 5)^2 + 309*log(abs(-1/(3*x + 2) + 5))