3.14.7 \(\int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx\)

Optimal. Leaf size=54 \[ -\frac {22}{343 (3 x+2)}-\frac {11}{98 (3 x+2)^2}+\frac {1}{63 (3 x+2)^3}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (3 x+2)}{2401} \]

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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 {22}{343 (3 x+2)}-\frac {11}{98 (3 x+2)^2}+\frac {1}{63 (3 x+2)^3}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (3 x+2)}{2401} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

1/(63*(2 + 3*x)^3) - 11/(98*(2 + 3*x)^2) - 22/(343*(2 + 3*x)) - (44*Log[1 - 2*x])/2401 + (44*Log[2 + 3*x])/240
1

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 {3+5 x}{(1-2 x) (2+3 x)^4} \, dx &=\int \left (-\frac {88}{2401 (-1+2 x)}-\frac {1}{7 (2+3 x)^4}+\frac {33}{49 (2+3 x)^3}+\frac {66}{343 (2+3 x)^2}+\frac {132}{2401 (2+3 x)}\right ) \, dx\\ &=\frac {1}{63 (2+3 x)^3}-\frac {11}{98 (2+3 x)^2}-\frac {22}{343 (2+3 x)}-\frac {44 \log (1-2 x)}{2401}+\frac {44 \log (2+3 x)}{2401}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.74 \begin {gather*} \frac {-\frac {7 \left (3564 x^2+6831 x+2872\right )}{(3 x+2)^3}-792 \log (3-6 x)+792 \log (3 x+2)}{43218} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

((-7*(2872 + 6831*x + 3564*x^2))/(2 + 3*x)^3 - 792*Log[3 - 6*x] + 792*Log[2 + 3*x])/43218

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+5 x}{(1-2 x) (2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)*(2 + 3*x)^4), x]

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fricas [A]  time = 1.46, size = 75, normalized size = 1.39 \begin {gather*} -\frac {24948 \, x^{2} - 792 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 792 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x - 1\right ) + 47817 \, x + 20104}{43218 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/43218*(24948*x^2 - 792*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) + 792*(27*x^3 + 54*x^2 + 36*x + 8)*log(2*x
- 1) + 47817*x + 20104)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.92, size = 38, normalized size = 0.70 \begin {gather*} -\frac {3564 \, x^{2} + 6831 \, x + 2872}{6174 \, {\left (3 \, x + 2\right )}^{3}} + \frac {44}{2401} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {44}{2401} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^4,x, algorithm="giac")

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(3*x + 2)^3 + 44/2401*log(abs(3*x + 2)) - 44/2401*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \begin {gather*} -\frac {44 \ln \left (2 x -1\right )}{2401}+\frac {44 \ln \left (3 x +2\right )}{2401}+\frac {1}{63 \left (3 x +2\right )^{3}}-\frac {11}{98 \left (3 x +2\right )^{2}}-\frac {22}{343 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)/(1-2*x)/(3*x+2)^4,x)

[Out]

1/63/(3*x+2)^3-11/98/(3*x+2)^2-22/343/(3*x+2)+44/2401*ln(3*x+2)-44/2401*ln(2*x-1)

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maxima [A]  time = 0.53, size = 46, normalized size = 0.85 \begin {gather*} -\frac {3564 \, x^{2} + 6831 \, x + 2872}{6174 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {44}{2401} \, \log \left (3 \, x + 2\right ) - \frac {44}{2401} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/6174*(3564*x^2 + 6831*x + 2872)/(27*x^3 + 54*x^2 + 36*x + 8) + 44/2401*log(3*x + 2) - 44/2401*log(2*x - 1)

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mupad [B]  time = 0.04, size = 36, normalized size = 0.67 \begin {gather*} \frac {88\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{2401}-\frac {\frac {22\,x^2}{1029}+\frac {253\,x}{6174}+\frac {1436}{83349}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)/((2*x - 1)*(3*x + 2)^4),x)

[Out]

(88*atanh((12*x)/7 + 1/7))/2401 - ((253*x)/6174 + (22*x^2)/1029 + 1436/83349)/((4*x)/3 + 2*x^2 + x^3 + 8/27)

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sympy [A]  time = 0.16, size = 44, normalized size = 0.81 \begin {gather*} - \frac {3564 x^{2} + 6831 x + 2872}{166698 x^{3} + 333396 x^{2} + 222264 x + 49392} - \frac {44 \log {\left (x - \frac {1}{2} \right )}}{2401} + \frac {44 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)/(2+3*x)**4,x)

[Out]

-(3564*x**2 + 6831*x + 2872)/(166698*x**3 + 333396*x**2 + 222264*x + 49392) - 44*log(x - 1/2)/2401 + 44*log(x
+ 2/3)/2401

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