3.1230 \(\int \frac{1-2 x}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=37 \[ \frac{7}{5 x+3}-\frac{11}{10 (5 x+3)^2}-21 \log (3 x+2)+21 \log (5 x+3) \]

[Out]

-11/(10*(3 + 5*x)^2) + 7/(3 + 5*x) - 21*Log[2 + 3*x] + 21*Log[3 + 5*x]

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Rubi [A]  time = 0.0168432, antiderivative size = 37, 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} \[ \frac{7}{5 x+3}-\frac{11}{10 (5 x+3)^2}-21 \log (3 x+2)+21 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-11/(10*(3 + 5*x)^2) + 7/(3 + 5*x) - 21*Log[2 + 3*x] + 21*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+5 x)^3} \, dx &=\int \left (-\frac{63}{2+3 x}+\frac{11}{(3+5 x)^3}-\frac{35}{(3+5 x)^2}+\frac{105}{3+5 x}\right ) \, dx\\ &=-\frac{11}{10 (3+5 x)^2}+\frac{7}{3+5 x}-21 \log (2+3 x)+21 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0131317, size = 48, normalized size = 1.3 \[ \frac{350 x-210 (5 x+3)^2 \log (5 (3 x+2))+210 (5 x+3)^2 \log (5 x+3)+199}{10 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(199 + 350*x - 210*(3 + 5*x)^2*Log[5*(2 + 3*x)] + 210*(3 + 5*x)^2*Log[3 + 5*x])/(10*(3 + 5*x)^2)

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Maple [A]  time = 0.007, size = 36, normalized size = 1. \begin{align*} -{\frac{11}{10\, \left ( 3+5\,x \right ) ^{2}}}+7\, \left ( 3+5\,x \right ) ^{-1}-21\,\ln \left ( 2+3\,x \right ) +21\,\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+5*x)^3,x)

[Out]

-11/10/(3+5*x)^2+7/(3+5*x)-21*ln(2+3*x)+21*ln(3+5*x)

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Maxima [A]  time = 1.11458, size = 49, normalized size = 1.32 \begin{align*} \frac{350 \, x + 199}{10 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + 21 \, \log \left (5 \, x + 3\right ) - 21 \, \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+5*x)^3,x, algorithm="maxima")

[Out]

1/10*(350*x + 199)/(25*x^2 + 30*x + 9) + 21*log(5*x + 3) - 21*log(3*x + 2)

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Fricas [A]  time = 1.50256, size = 159, normalized size = 4.3 \begin{align*} \frac{210 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 210 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (3 \, x + 2\right ) + 350 \, x + 199}{10 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(210*(25*x^2 + 30*x + 9)*log(5*x + 3) - 210*(25*x^2 + 30*x + 9)*log(3*x + 2) + 350*x + 199)/(25*x^2 + 30*
x + 9)

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Sympy [A]  time = 0.13038, size = 31, normalized size = 0.84 \begin{align*} \frac{350 x + 199}{250 x^{2} + 300 x + 90} + 21 \log{\left (x + \frac{3}{5} \right )} - 21 \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+5*x)**3,x)

[Out]

(350*x + 199)/(250*x**2 + 300*x + 90) + 21*log(x + 3/5) - 21*log(x + 2/3)

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Giac [A]  time = 2.2246, size = 45, normalized size = 1.22 \begin{align*} \frac{350 \, x + 199}{10 \,{\left (5 \, x + 3\right )}^{2}} + 21 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 21 \, \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+5*x)^3,x, algorithm="giac")

[Out]

1/10*(350*x + 199)/(5*x + 3)^2 + 21*log(abs(5*x + 3)) - 21*log(abs(3*x + 2))