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

Optimal. Leaf size=65 \[ -\frac {1331}{2401 (3 x+2)}+\frac {3469}{18522 (3 x+2)^2}-\frac {103}{3969 (3 x+2)^3}+\frac {1}{756 (3 x+2)^4}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (3 x+2)}{16807} \]

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \begin {gather*} -\frac {1331}{2401 (3 x+2)}+\frac {3469}{18522 (3 x+2)^2}-\frac {103}{3969 (3 x+2)^3}+\frac {1}{756 (3 x+2)^4}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (3 x+2)}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(756*(2 + 3*x)^4) - 103/(3969*(2 + 3*x)^3) + 3469/(18522*(2 + 3*x)^2) - 1331/(2401*(2 + 3*x)) - (2662*Log[1
- 2*x])/16807 + (2662*Log[2 + 3*x])/16807

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^5} \, dx &=\int \left (-\frac {5324}{16807 (-1+2 x)}-\frac {1}{63 (2+3 x)^5}+\frac {103}{441 (2+3 x)^4}-\frac {3469}{3087 (2+3 x)^3}+\frac {3993}{2401 (2+3 x)^2}+\frac {7986}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {1}{756 (2+3 x)^4}-\frac {103}{3969 (2+3 x)^3}+\frac {3469}{18522 (2+3 x)^2}-\frac {1331}{2401 (2+3 x)}-\frac {2662 \log (1-2 x)}{16807}+\frac {2662 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 47, normalized size = 0.72 \begin {gather*} \frac {2 \left (-\frac {7 \left (11643588 x^3+21975894 x^2+13836972 x+2906507\right )}{8 (3 x+2)^4}-107811 \log (1-2 x)+107811 \log (6 x+4)\right )}{1361367} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(2906507 + 13836972*x + 21975894*x^2 + 11643588*x^3))/(8*(2 + 3*x)^4) - 107811*Log[1 - 2*x] + 107811*L
og[4 + 6*x]))/1361367

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.75, size = 95, normalized size = 1.46 \begin {gather*} -\frac {81505116 \, x^{3} + 153831258 \, x^{2} - 862488 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 862488 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 96858804 \, x + 20345549}{5445468 \, {\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((3+5*x)^3/(1-2*x)/(2+3*x)^5,x, algorithm="fricas")

[Out]

-1/5445468*(81505116*x^3 + 153831258*x^2 - 862488*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 8624
88*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(2*x - 1) + 96858804*x + 20345549)/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

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giac [A]  time = 1.02, size = 52, normalized size = 0.80 \begin {gather*} -\frac {1331}{2401 \, {\left (3 \, x + 2\right )}} + \frac {3469}{18522 \, {\left (3 \, x + 2\right )}^{2}} - \frac {103}{3969 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{756 \, {\left (3 \, x + 2\right )}^{4}} - \frac {2662}{16807} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/2401/(3*x + 2) + 3469/18522/(3*x + 2)^2 - 103/3969/(3*x + 2)^3 + 1/756/(3*x + 2)^4 - 2662/16807*log(abs(
-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \begin {gather*} -\frac {2662 \ln \left (2 x -1\right )}{16807}+\frac {2662 \ln \left (3 x +2\right )}{16807}+\frac {1}{756 \left (3 x +2\right )^{4}}-\frac {103}{3969 \left (3 x +2\right )^{3}}+\frac {3469}{18522 \left (3 x +2\right )^{2}}-\frac {1331}{2401 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/756/(3*x+2)^4-103/3969/(3*x+2)^3+3469/18522/(3*x+2)^2-1331/2401/(3*x+2)+2662/16807*ln(3*x+2)-2662/16807*ln(2
*x-1)

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maxima [A]  time = 0.54, size = 56, normalized size = 0.86 \begin {gather*} -\frac {11643588 \, x^{3} + 21975894 \, x^{2} + 13836972 \, x + 2906507}{777924 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2662}{16807} \, \log \left (3 \, x + 2\right ) - \frac {2662}{16807} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/777924*(11643588*x^3 + 21975894*x^2 + 13836972*x + 2906507)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 2662
/16807*log(3*x + 2) - 2662/16807*log(2*x - 1)

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mupad [B]  time = 0.05, size = 46, normalized size = 0.71 \begin {gather*} \frac {5324\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {1331\,x^3}{7203}+\frac {406961\,x^2}{1166886}+\frac {1153081\,x}{5250987}+\frac {2906507}{63011844}}{x^4+\frac {8\,x^3}{3}+\frac {8\,x^2}{3}+\frac {32\,x}{27}+\frac {16}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5324*atanh((12*x)/7 + 1/7))/16807 - ((1153081*x)/5250987 + (406961*x^2)/1166886 + (1331*x^3)/7203 + 2906507/6
3011844)/((32*x)/27 + (8*x^2)/3 + (8*x^3)/3 + x^4 + 16/81)

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sympy [A]  time = 0.19, size = 54, normalized size = 0.83 \begin {gather*} - \frac {11643588 x^{3} + 21975894 x^{2} + 13836972 x + 2906507}{63011844 x^{4} + 168031584 x^{3} + 168031584 x^{2} + 74680704 x + 12446784} - \frac {2662 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {2662 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(11643588*x**3 + 21975894*x**2 + 13836972*x + 2906507)/(63011844*x**4 + 168031584*x**3 + 168031584*x**2 + 746
80704*x + 12446784) - 2662*log(x - 1/2)/16807 + 2662*log(x + 2/3)/16807

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