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

Optimal. Leaf size=54 \[ \frac {37}{1331 (1-2 x)}-\frac {5}{1331 (5 x+3)}+\frac {7}{242 (1-2 x)^2}-\frac {195 \log (1-2 x)}{14641}+\frac {195 \log (5 x+3)}{14641} \]

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Rubi [A]  time = 0.02, antiderivative size = 54, 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 {37}{1331 (1-2 x)}-\frac {5}{1331 (5 x+3)}+\frac {7}{242 (1-2 x)^2}-\frac {195 \log (1-2 x)}{14641}+\frac {195 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(242*(1 - 2*x)^2) + 37/(1331*(1 - 2*x)) - 5/(1331*(3 + 5*x)) - (195*Log[1 - 2*x])/14641 + (195*Log[3 + 5*x])
/14641

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 {2+3 x}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac {14}{121 (-1+2 x)^3}+\frac {74}{1331 (-1+2 x)^2}-\frac {390}{14641 (-1+2 x)}+\frac {25}{1331 (3+5 x)^2}+\frac {975}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {7}{242 (1-2 x)^2}+\frac {37}{1331 (1-2 x)}-\frac {5}{1331 (3+5 x)}-\frac {195 \log (1-2 x)}{14641}+\frac {195 \log (3+5 x)}{14641}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 62, normalized size = 1.15 \begin {gather*} \frac {10}{1331 (5 (1-2 x)-11)}+\frac {37}{1331 (1-2 x)}+\frac {7}{242 (1-2 x)^2}+\frac {195 \log (11-5 (1-2 x))}{14641}-\frac {195 \log (1-2 x)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

10/(1331*(-11 + 5*(1 - 2*x))) + 7/(242*(1 - 2*x)^2) + 37/(1331*(1 - 2*x)) + (195*Log[11 - 5*(1 - 2*x)])/14641
- (195*Log[1 - 2*x])/14641

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 75, normalized size = 1.39 \begin {gather*} -\frac {8580 \, x^{2} - 390 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 390 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 3861 \, x - 4873}{29282 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(8580*x^2 - 390*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) + 390*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1)
- 3861*x - 4873)/(20*x^3 - 8*x^2 - 7*x + 3)

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giac [A]  time = 1.21, size = 51, normalized size = 0.94 \begin {gather*} -\frac {5}{1331 \, {\left (5 \, x + 3\right )}} + \frac {10 \, {\left (\frac {792}{5 \, x + 3} - 109\right )}}{14641 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}^{2}} - \frac {195}{14641} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/1331/(5*x + 3) + 10/14641*(792/(5*x + 3) - 109)/(11/(5*x + 3) - 2)^2 - 195/14641*log(abs(-11/(5*x + 3) + 2)
)

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \begin {gather*} -\frac {195 \ln \left (2 x -1\right )}{14641}+\frac {195 \ln \left (5 x +3\right )}{14641}-\frac {5}{1331 \left (5 x +3\right )}+\frac {7}{242 \left (2 x -1\right )^{2}}-\frac {37}{1331 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/1331/(5*x+3)+195/14641*ln(5*x+3)+7/242/(2*x-1)^2-37/1331/(2*x-1)-195/14641*ln(2*x-1)

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maxima [A]  time = 0.49, size = 46, normalized size = 0.85 \begin {gather*} -\frac {780 \, x^{2} - 351 \, x - 443}{2662 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac {195}{14641} \, \log \left (5 \, x + 3\right ) - \frac {195}{14641} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2662*(780*x^2 - 351*x - 443)/(20*x^3 - 8*x^2 - 7*x + 3) + 195/14641*log(5*x + 3) - 195/14641*log(2*x - 1)

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mupad [B]  time = 0.04, size = 38, normalized size = 0.70 \begin {gather*} \frac {390\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{14641}-\frac {-\frac {39\,x^2}{2662}+\frac {351\,x}{53240}+\frac {443}{53240}}{-x^3+\frac {2\,x^2}{5}+\frac {7\,x}{20}-\frac {3}{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(390*atanh((20*x)/11 + 1/11))/14641 - ((351*x)/53240 - (39*x^2)/2662 + 443/53240)/((7*x)/20 + (2*x^2)/5 - x^3
- 3/20)

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sympy [A]  time = 0.16, size = 44, normalized size = 0.81 \begin {gather*} - \frac {780 x^{2} - 351 x - 443}{53240 x^{3} - 21296 x^{2} - 18634 x + 7986} - \frac {195 \log {\left (x - \frac {1}{2} \right )}}{14641} + \frac {195 \log {\left (x + \frac {3}{5} \right )}}{14641} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(780*x**2 - 351*x - 443)/(53240*x**3 - 21296*x**2 - 18634*x + 7986) - 195*log(x - 1/2)/14641 + 195*log(x + 3/
5)/14641

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