3.1204 \(\int \frac{1-2 x}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=59 \[ \frac{275}{3 x+2}+\frac{55}{2 (3 x+2)^2}+\frac{11}{3 (3 x+2)^3}+\frac{7}{12 (3 x+2)^4}-1375 \log (3 x+2)+1375 \log (5 x+3) \]

[Out]

7/(12*(2 + 3*x)^4) + 11/(3*(2 + 3*x)^3) + 55/(2*(2 + 3*x)^2) + 275/(2 + 3*x) - 1375*Log[2 + 3*x] + 1375*Log[3
+ 5*x]

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Rubi [A]  time = 0.0218472, antiderivative size = 59, 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} \[ \frac{275}{3 x+2}+\frac{55}{2 (3 x+2)^2}+\frac{11}{3 (3 x+2)^3}+\frac{7}{12 (3 x+2)^4}-1375 \log (3 x+2)+1375 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(12*(2 + 3*x)^4) + 11/(3*(2 + 3*x)^3) + 55/(2*(2 + 3*x)^2) + 275/(2 + 3*x) - 1375*Log[2 + 3*x] + 1375*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)^5 (3+5 x)} \, dx &=\int \left (-\frac{7}{(2+3 x)^5}-\frac{33}{(2+3 x)^4}-\frac{165}{(2+3 x)^3}-\frac{825}{(2+3 x)^2}-\frac{4125}{2+3 x}+\frac{6875}{3+5 x}\right ) \, dx\\ &=\frac{7}{12 (2+3 x)^4}+\frac{11}{3 (2+3 x)^3}+\frac{55}{2 (2+3 x)^2}+\frac{275}{2+3 x}-1375 \log (2+3 x)+1375 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0258437, size = 45, normalized size = 0.76 \[ \frac{89100 x^3+181170 x^2+122892 x+27815}{12 (3 x+2)^4}-1375 \log (3 x+2)+1375 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27815 + 122892*x + 181170*x^2 + 89100*x^3)/(12*(2 + 3*x)^4) - 1375*Log[2 + 3*x] + 1375*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.007, size = 54, normalized size = 0.9 \begin{align*}{\frac{7}{12\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{11}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{55}{2\, \left ( 2+3\,x \right ) ^{2}}}+275\, \left ( 2+3\,x \right ) ^{-1}-1375\,\ln \left ( 2+3\,x \right ) +1375\,\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)^5/(3+5*x),x)

[Out]

7/12/(2+3*x)^4+11/3/(2+3*x)^3+55/2/(2+3*x)^2+275/(2+3*x)-1375*ln(2+3*x)+1375*ln(3+5*x)

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Maxima [A]  time = 1.14705, size = 76, normalized size = 1.29 \begin{align*} \frac{89100 \, x^{3} + 181170 \, x^{2} + 122892 \, x + 27815}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + 1375 \, \log \left (5 \, x + 3\right ) - 1375 \, \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)^5/(3+5*x),x, algorithm="maxima")

[Out]

1/12*(89100*x^3 + 181170*x^2 + 122892*x + 27815)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1375*log(5*x + 3)
- 1375*log(3*x + 2)

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Fricas [A]  time = 1.75282, size = 290, normalized size = 4.92 \begin{align*} \frac{89100 \, x^{3} + 181170 \, x^{2} + 16500 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (5 \, x + 3\right ) - 16500 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 122892 \, x + 27815}{12 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(89100*x^3 + 181170*x^2 + 16500*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(5*x + 3) - 16500*(81*x^4 + 2
16*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 122892*x + 27815)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [A]  time = 0.153872, size = 51, normalized size = 0.86 \begin{align*} \frac{89100 x^{3} + 181170 x^{2} + 122892 x + 27815}{972 x^{4} + 2592 x^{3} + 2592 x^{2} + 1152 x + 192} + 1375 \log{\left (x + \frac{3}{5} \right )} - 1375 \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)**5/(3+5*x),x)

[Out]

(89100*x**3 + 181170*x**2 + 122892*x + 27815)/(972*x**4 + 2592*x**3 + 2592*x**2 + 1152*x + 192) + 1375*log(x +
3/5) - 1375*log(x + 2/3)

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Giac [A]  time = 2.47461, size = 70, normalized size = 1.19 \begin{align*} \frac{275}{3 \, x + 2} + \frac{55}{2 \,{\left (3 \, x + 2\right )}^{2}} + \frac{11}{3 \,{\left (3 \, x + 2\right )}^{3}} + \frac{7}{12 \,{\left (3 \, x + 2\right )}^{4}} + 1375 \, \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)^5/(3+5*x),x, algorithm="giac")

[Out]

275/(3*x + 2) + 55/2/(3*x + 2)^2 + 11/3/(3*x + 2)^3 + 7/12/(3*x + 2)^4 + 1375*log(abs(-1/(3*x + 2) + 5))