3.1217 \(\int \frac{1-2 x}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{68}{3 x+2}-\frac{55}{5 x+3}-\frac{7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (5 x+3) \]

[Out]

-7/(2*(2 + 3*x)^2) - 68/(2 + 3*x) - 55/(3 + 5*x) + 505*Log[2 + 3*x] - 505*Log[3 + 5*x]

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Rubi [A]  time = 0.0207971, 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} \[ -\frac{68}{3 x+2}-\frac{55}{5 x+3}-\frac{7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(2*(2 + 3*x)^2) - 68/(2 + 3*x) - 55/(3 + 5*x) + 505*Log[2 + 3*x] - 505*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)^3 (3+5 x)^2} \, dx &=\int \left (\frac{21}{(2+3 x)^3}+\frac{204}{(2+3 x)^2}+\frac{1515}{2+3 x}+\frac{275}{(3+5 x)^2}-\frac{2525}{3+5 x}\right ) \, dx\\ &=-\frac{7}{2 (2+3 x)^2}-\frac{68}{2+3 x}-\frac{55}{3+5 x}+505 \log (2+3 x)-505 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0177877, size = 48, normalized size = 1.04 \[ -\frac{68}{3 x+2}-\frac{55}{5 x+3}-\frac{7}{2 (3 x+2)^2}+505 \log (3 x+2)-505 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7/(2*(2 + 3*x)^2) - 68/(2 + 3*x) - 55/(3 + 5*x) + 505*Log[2 + 3*x] - 505*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.008, size = 45, normalized size = 1. \begin{align*} -{\frac{7}{2\, \left ( 2+3\,x \right ) ^{2}}}-68\, \left ( 2+3\,x \right ) ^{-1}-55\, \left ( 3+5\,x \right ) ^{-1}+505\,\ln \left ( 2+3\,x \right ) -505\,\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)^3/(3+5*x)^2,x)

[Out]

-7/2/(2+3*x)^2-68/(2+3*x)-55/(3+5*x)+505*ln(2+3*x)-505*ln(3+5*x)

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Maxima [A]  time = 1.52552, size = 62, normalized size = 1.35 \begin{align*} -\frac{3030 \, x^{2} + 3939 \, x + 1277}{2 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - 505 \, \log \left (5 \, x + 3\right ) + 505 \, \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)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/2*(3030*x^2 + 3939*x + 1277)/(45*x^3 + 87*x^2 + 56*x + 12) - 505*log(5*x + 3) + 505*log(3*x + 2)

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Fricas [A]  time = 1.64916, size = 220, normalized size = 4.78 \begin{align*} -\frac{3030 \, x^{2} + 1010 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 1010 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 3939 \, x + 1277}{2 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3030*x^2 + 1010*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 1010*(45*x^3 + 87*x^2 + 56*x + 12)*log(3*x
+ 2) + 3939*x + 1277)/(45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [A]  time = 0.144746, size = 41, normalized size = 0.89 \begin{align*} - \frac{3030 x^{2} + 3939 x + 1277}{90 x^{3} + 174 x^{2} + 112 x + 24} - 505 \log{\left (x + \frac{3}{5} \right )} + 505 \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)**3/(3+5*x)**2,x)

[Out]

-(3030*x**2 + 3939*x + 1277)/(90*x**3 + 174*x**2 + 112*x + 24) - 505*log(x + 3/5) + 505*log(x + 2/3)

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Giac [A]  time = 2.344, size = 66, normalized size = 1.43 \begin{align*} -\frac{55}{5 \, x + 3} + \frac{15 \,{\left (\frac{206}{5 \, x + 3} + 513\right )}}{2 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{2}} + 505 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-55/(5*x + 3) + 15/2*(206/(5*x + 3) + 513)/(1/(5*x + 3) + 3)^2 + 505*log(abs(-1/(5*x + 3) - 3))