3.1203 \(\int \frac{1-2 x}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{55}{3 x+2}+\frac{11}{2 (3 x+2)^2}+\frac{7}{9 (3 x+2)^3}-275 \log (3 x+2)+275 \log (5 x+3) \]

[Out]

7/(9*(2 + 3*x)^3) + 11/(2*(2 + 3*x)^2) + 55/(2 + 3*x) - 275*Log[2 + 3*x] + 275*Log[3 + 5*x]

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Rubi [A]  time = 0.0191768, antiderivative size = 48, 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} \[ \frac{55}{3 x+2}+\frac{11}{2 (3 x+2)^2}+\frac{7}{9 (3 x+2)^3}-275 \log (3 x+2)+275 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(9*(2 + 3*x)^3) + 11/(2*(2 + 3*x)^2) + 55/(2 + 3*x) - 275*Log[2 + 3*x] + 275*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)} \, dx &=\int \left (-\frac{7}{(2+3 x)^4}-\frac{33}{(2+3 x)^3}-\frac{165}{(2+3 x)^2}-\frac{825}{2+3 x}+\frac{1375}{3+5 x}\right ) \, dx\\ &=\frac{7}{9 (2+3 x)^3}+\frac{11}{2 (2+3 x)^2}+\frac{55}{2+3 x}-275 \log (2+3 x)+275 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0242245, size = 40, normalized size = 0.83 \[ \frac{8910 x^2+12177 x+4172}{18 (3 x+2)^3}-275 \log (3 x+2)+275 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4172 + 12177*x + 8910*x^2)/(18*(2 + 3*x)^3) - 275*Log[2 + 3*x] + 275*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.006, size = 45, normalized size = 0.9 \begin{align*}{\frac{7}{9\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{11}{2\, \left ( 2+3\,x \right ) ^{2}}}+55\, \left ( 2+3\,x \right ) ^{-1}-275\,\ln \left ( 2+3\,x \right ) +275\,\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),x)

[Out]

7/9/(2+3*x)^3+11/2/(2+3*x)^2+55/(2+3*x)-275*ln(2+3*x)+275*ln(3+5*x)

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Maxima [A]  time = 1.16814, size = 62, normalized size = 1.29 \begin{align*} \frac{8910 \, x^{2} + 12177 \, x + 4172}{18 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + 275 \, \log \left (5 \, x + 3\right ) - 275 \, \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),x, algorithm="maxima")

[Out]

1/18*(8910*x^2 + 12177*x + 4172)/(27*x^3 + 54*x^2 + 36*x + 8) + 275*log(5*x + 3) - 275*log(3*x + 2)

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Fricas [A]  time = 1.71802, size = 217, normalized size = 4.52 \begin{align*} \frac{8910 \, x^{2} + 4950 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 4950 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 12177 \, x + 4172}{18 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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),x, algorithm="fricas")

[Out]

1/18*(8910*x^2 + 4950*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 4950*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x +
2) + 12177*x + 4172)/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 0.143739, size = 41, normalized size = 0.85 \begin{align*} \frac{8910 x^{2} + 12177 x + 4172}{486 x^{3} + 972 x^{2} + 648 x + 144} + 275 \log{\left (x + \frac{3}{5} \right )} - 275 \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),x)

[Out]

(8910*x**2 + 12177*x + 4172)/(486*x**3 + 972*x**2 + 648*x + 144) + 275*log(x + 3/5) - 275*log(x + 2/3)

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Giac [A]  time = 2.25271, size = 51, normalized size = 1.06 \begin{align*} \frac{8910 \, x^{2} + 12177 \, x + 4172}{18 \,{\left (3 \, x + 2\right )}^{3}} + 275 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 275 \, \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),x, algorithm="giac")

[Out]

1/18*(8910*x^2 + 12177*x + 4172)/(3*x + 2)^3 + 275*log(abs(5*x + 3)) - 275*log(abs(3*x + 2))