3.1232 \(\int \frac{1-2 x}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=57 \[ \frac{309}{3 x+2}+\frac{505}{5 x+3}+\frac{21}{2 (3 x+2)^2}-\frac{55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (5 x+3) \]

[Out]

21/(2*(2 + 3*x)^2) + 309/(2 + 3*x) - 55/(2*(3 + 5*x)^2) + 505/(3 + 5*x) - 3060*Log[2 + 3*x] + 3060*Log[3 + 5*x
]

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Rubi [A]  time = 0.0260223, antiderivative size = 57, 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} \[ \frac{309}{3 x+2}+\frac{505}{5 x+3}+\frac{21}{2 (3 x+2)^2}-\frac{55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2*(2 + 3*x)^2) + 309/(2 + 3*x) - 55/(2*(3 + 5*x)^2) + 505/(3 + 5*x) - 3060*Log[2 + 3*x] + 3060*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)^3} \, dx &=\int \left (-\frac{63}{(2+3 x)^3}-\frac{927}{(2+3 x)^2}-\frac{9180}{2+3 x}+\frac{275}{(3+5 x)^3}-\frac{2525}{(3+5 x)^2}+\frac{15300}{3+5 x}\right ) \, dx\\ &=\frac{21}{2 (2+3 x)^2}+\frac{309}{2+3 x}-\frac{55}{2 (3+5 x)^2}+\frac{505}{3+5 x}-3060 \log (2+3 x)+3060 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0310341, size = 59, normalized size = 1.04 \[ \frac{309}{3 x+2}+\frac{505}{5 x+3}+\frac{21}{2 (3 x+2)^2}-\frac{55}{2 (5 x+3)^2}-3060 \log (3 x+2)+3060 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(2*(2 + 3*x)^2) + 309/(2 + 3*x) - 55/(2*(3 + 5*x)^2) + 505/(3 + 5*x) - 3060*Log[2 + 3*x] + 3060*Log[-3*(3 +
5*x)]

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Maple [A]  time = 0.008, size = 54, normalized size = 1. \begin{align*}{\frac{21}{2\, \left ( 2+3\,x \right ) ^{2}}}+309\, \left ( 2+3\,x \right ) ^{-1}-{\frac{55}{2\, \left ( 3+5\,x \right ) ^{2}}}+505\, \left ( 3+5\,x \right ) ^{-1}-3060\,\ln \left ( 2+3\,x \right ) +3060\,\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)^3,x)

[Out]

21/2/(2+3*x)^2+309/(2+3*x)-55/2/(3+5*x)^2+505/(3+5*x)-3060*ln(2+3*x)+3060*ln(3+5*x)

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Maxima [A]  time = 2.17863, size = 76, normalized size = 1.33 \begin{align*} \frac{91800 \, x^{3} + 174420 \, x^{2} + 110296 \, x + 23213}{2 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} + 3060 \, \log \left (5 \, x + 3\right ) - 3060 \, \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)^3,x, algorithm="maxima")

[Out]

1/2*(91800*x^3 + 174420*x^2 + 110296*x + 23213)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36) + 3060*log(5*x + 3)
- 3060*log(3*x + 2)

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Fricas [A]  time = 1.56989, size = 294, normalized size = 5.16 \begin{align*} \frac{91800 \, x^{3} + 174420 \, x^{2} + 6120 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 6120 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 110296 \, x + 23213}{2 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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)^3,x, algorithm="fricas")

[Out]

1/2*(91800*x^3 + 174420*x^2 + 6120*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(5*x + 3) - 6120*(225*x^4 + 5
70*x^3 + 541*x^2 + 228*x + 36)*log(3*x + 2) + 110296*x + 23213)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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Sympy [A]  time = 0.156531, size = 51, normalized size = 0.89 \begin{align*} \frac{91800 x^{3} + 174420 x^{2} + 110296 x + 23213}{450 x^{4} + 1140 x^{3} + 1082 x^{2} + 456 x + 72} + 3060 \log{\left (x + \frac{3}{5} \right )} - 3060 \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)**3,x)

[Out]

(91800*x**3 + 174420*x**2 + 110296*x + 23213)/(450*x**4 + 1140*x**3 + 1082*x**2 + 456*x + 72) + 3060*log(x + 3
/5) - 3060*log(x + 2/3)

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Giac [A]  time = 2.04619, size = 65, normalized size = 1.14 \begin{align*} \frac{91800 \, x^{3} + 174420 \, x^{2} + 110296 \, x + 23213}{2 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}^{2}} + 3060 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 3060 \, \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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

1/2*(91800*x^3 + 174420*x^2 + 110296*x + 23213)/(15*x^2 + 19*x + 6)^2 + 3060*log(abs(5*x + 3)) - 3060*log(abs(
3*x + 2))