3.1519 \(\int \frac {(2+3 x)^5}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac {243 x^2}{500}-\frac {4941 x}{2500}-\frac {167}{378125 (5 x+3)}-\frac {1}{68750 (5 x+3)^2}-\frac {16807 \log (1-2 x)}{10648}+\frac {11224 \log (5 x+3)}{4159375} \]

[Out]

-4941/2500*x-243/500*x^2-1/68750/(3+5*x)^2-167/378125/(3+5*x)-16807/10648*ln(1-2*x)+11224/4159375*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 55, 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} \[ -\frac {243 x^2}{500}-\frac {4941 x}{2500}-\frac {167}{378125 (5 x+3)}-\frac {1}{68750 (5 x+3)^2}-\frac {16807 \log (1-2 x)}{10648}+\frac {11224 \log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4941*x)/2500 - (243*x^2)/500 - 1/(68750*(3 + 5*x)^2) - 167/(378125*(3 + 5*x)) - (16807*Log[1 - 2*x])/10648 +
(11224*Log[3 + 5*x])/4159375

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 {(2+3 x)^5}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac {4941}{2500}-\frac {243 x}{250}-\frac {16807}{5324 (-1+2 x)}+\frac {1}{6875 (3+5 x)^3}+\frac {167}{75625 (3+5 x)^2}+\frac {11224}{831875 (3+5 x)}\right ) \, dx\\ &=-\frac {4941 x}{2500}-\frac {243 x^2}{500}-\frac {1}{68750 (3+5 x)^2}-\frac {167}{378125 (3+5 x)}-\frac {16807 \log (1-2 x)}{10648}+\frac {11224 \log (3+5 x)}{4159375}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 0.91 \[ \frac {-\frac {11 \left (73507500 x^4+387139500 x^3+217337175 x^2-93782210 x-60415061\right )}{(5 x+3)^2}-105043750 \log (1-2 x)+179584 \log (10 x+6)}{66550000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-60415061 - 93782210*x + 217337175*x^2 + 387139500*x^3 + 73507500*x^4))/(3 + 5*x)^2 - 105043750*Log[1 -
2*x] + 179584*Log[6 + 10*x])/66550000

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fricas [A]  time = 0.89, size = 70, normalized size = 1.27 \[ -\frac {404291250 \, x^{4} + 2129267250 \, x^{3} + 2118486150 \, x^{2} - 89792 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 52521875 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 591955870 \, x + 44572}{33275000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/33275000*(404291250*x^4 + 2129267250*x^3 + 2118486150*x^2 - 89792*(25*x^2 + 30*x + 9)*log(5*x + 3) + 525218
75*(25*x^2 + 30*x + 9)*log(2*x - 1) + 591955870*x + 44572)/(25*x^2 + 30*x + 9)

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giac [A]  time = 0.99, size = 41, normalized size = 0.75 \[ -\frac {243}{500} \, x^{2} - \frac {4941}{2500} \, x - \frac {1670 \, x + 1013}{756250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {11224}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {16807}{10648} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/500*x^2 - 4941/2500*x - 1/756250*(1670*x + 1013)/(5*x + 3)^2 + 11224/4159375*log(abs(5*x + 3)) - 16807/10
648*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 44, normalized size = 0.80 \[ -\frac {243 x^{2}}{500}-\frac {4941 x}{2500}-\frac {16807 \ln \left (2 x -1\right )}{10648}+\frac {11224 \ln \left (5 x +3\right )}{4159375}-\frac {1}{68750 \left (5 x +3\right )^{2}}-\frac {167}{378125 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/500*x^2-4941/2500*x-1/68750/(5*x+3)^2-167/378125/(5*x+3)+11224/4159375*ln(5*x+3)-16807/10648*ln(2*x-1)

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maxima [A]  time = 0.64, size = 44, normalized size = 0.80 \[ -\frac {243}{500} \, x^{2} - \frac {4941}{2500} \, x - \frac {1670 \, x + 1013}{756250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {11224}{4159375} \, \log \left (5 \, x + 3\right ) - \frac {16807}{10648} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/500*x^2 - 4941/2500*x - 1/756250*(1670*x + 1013)/(25*x^2 + 30*x + 9) + 11224/4159375*log(5*x + 3) - 16807
/10648*log(2*x - 1)

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mupad [B]  time = 0.04, size = 38, normalized size = 0.69 \[ \frac {11224\,\ln \left (x+\frac {3}{5}\right )}{4159375}-\frac {16807\,\ln \left (x-\frac {1}{2}\right )}{10648}-\frac {4941\,x}{2500}-\frac {\frac {167\,x}{1890625}+\frac {1013}{18906250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {243\,x^2}{500} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11224*log(x + 3/5))/4159375 - (16807*log(x - 1/2))/10648 - (4941*x)/2500 - ((167*x)/1890625 + 1013/18906250)/
((6*x)/5 + x^2 + 9/25) - (243*x^2)/500

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sympy [A]  time = 0.18, size = 46, normalized size = 0.84 \[ - \frac {243 x^{2}}{500} - \frac {4941 x}{2500} - \frac {1670 x + 1013}{18906250 x^{2} + 22687500 x + 6806250} - \frac {16807 \log {\left (x - \frac {1}{2} \right )}}{10648} + \frac {11224 \log {\left (x + \frac {3}{5} \right )}}{4159375} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**2/500 - 4941*x/2500 - (1670*x + 1013)/(18906250*x**2 + 22687500*x + 6806250) - 16807*log(x - 1/2)/1064
8 + 11224*log(x + 3/5)/4159375

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