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

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

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Rubi [A]  time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \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) \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 45, normalized size = 0.76 \begin {gather*} \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)) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.01, size = 95, normalized size = 1.61 \begin {gather*} \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 {gather*}

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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giac [A]  time = 1.17, size = 52, normalized size = 0.88 \begin {gather*} \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 {gather*}

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))

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maple [A]  time = 0.01, size = 54, normalized size = 0.92 \begin {gather*} -1375 \ln \left (3 x +2\right )+1375 \ln \left (5 x +3\right )+\frac {7}{12 \left (3 x +2\right )^{4}}+\frac {11}{3 \left (3 x +2\right )^{3}}+\frac {55}{2 \left (3 x +2\right )^{2}}+\frac {275}{3 x +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)/(3*x+2)^5/(5*x+3),x)

[Out]

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

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maxima [A]  time = 0.49, size = 56, normalized size = 0.95 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.05, size = 45, normalized size = 0.76 \begin {gather*} \frac {\frac {275\,x^3}{3}+\frac {3355\,x^2}{18}+\frac {10241\,x}{81}+\frac {27815}{972}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}}-2750\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)/((3*x + 2)^5*(5*x + 3)),x)

[Out]

((10241*x)/81 + (3355*x^2)/18 + (275*x^3)/3 + 27815/972)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81) - 2
750*atanh(30*x + 19)

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sympy [A]  time = 0.17, size = 53, normalized size = 0.90 \begin {gather*} - \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 {gather*}

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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