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

Optimal. Leaf size=55 \[ -\frac {243 x^2}{80}-\frac {10287 x}{400}-\frac {156065}{1936 (1-2 x)}+\frac {16807}{704 (1-2 x)^2}-\frac {543655 \log (1-2 x)}{10648}+\frac {\log (5 x+3)}{166375} \]

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Rubi [A]  time = 0.03, 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} \begin {gather*} -\frac {243 x^2}{80}-\frac {10287 x}{400}-\frac {156065}{1936 (1-2 x)}+\frac {16807}{704 (1-2 x)^2}-\frac {543655 \log (1-2 x)}{10648}+\frac {\log (5 x+3)}{166375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(704*(1 - 2*x)^2) - 156065/(1936*(1 - 2*x)) - (10287*x)/400 - (243*x^2)/80 - (543655*Log[1 - 2*x])/10648
+ Log[3 + 5*x]/166375

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 (3+5 x)} \, dx &=\int \left (-\frac {10287}{400}-\frac {243 x}{40}-\frac {16807}{176 (-1+2 x)^3}-\frac {156065}{968 (-1+2 x)^2}-\frac {543655}{5324 (-1+2 x)}+\frac {1}{33275 (3+5 x)}\right ) \, dx\\ &=\frac {16807}{704 (1-2 x)^2}-\frac {156065}{1936 (1-2 x)}-\frac {10287 x}{400}-\frac {243 x^2}{80}-\frac {543655 \log (1-2 x)}{10648}+\frac {\log (3+5 x)}{166375}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 55, normalized size = 1.00 \begin {gather*} \frac {-1293732 (5 x+3)^2-47005596 (5 x+3)+\frac {858357500}{2 x-1}+\frac {254205875}{(1-2 x)^2}-543655000 \log (5-10 x)+64 \log (5 x+3)}{10648000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(254205875/(1 - 2*x)^2 + 858357500/(-1 + 2*x) - 47005596*(3 + 5*x) - 1293732*(3 + 5*x)^2 - 543655000*Log[5 - 1
0*x] + 64*Log[3 + 5*x])/10648000

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.53, size = 70, normalized size = 1.27 \begin {gather*} -\frac {129373200 \, x^{4} + 965986560 \, x^{3} - 1063016460 \, x^{2} - 64 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 543655000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 1442875060 \, x + 604151625}{10648000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10648000*(129373200*x^4 + 965986560*x^3 - 1063016460*x^2 - 64*(4*x^2 - 4*x + 1)*log(5*x + 3) + 543655000*(4
*x^2 - 4*x + 1)*log(2*x - 1) - 1442875060*x + 604151625)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.20, size = 41, normalized size = 0.75 \begin {gather*} -\frac {243}{80} \, x^{2} - \frac {10287}{400} \, x + \frac {2401 \, {\left (520 \, x - 183\right )}}{7744 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{166375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {543655}{10648} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(2*x - 1)^2 + 1/166375*log(abs(5*x + 3)) - 543655/10648*lo
g(abs(2*x - 1))

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maple [A]  time = 0.01, size = 44, normalized size = 0.80 \begin {gather*} -\frac {243 x^{2}}{80}-\frac {10287 x}{400}-\frac {543655 \ln \left (2 x -1\right )}{10648}+\frac {\ln \left (5 x +3\right )}{166375}+\frac {16807}{704 \left (2 x -1\right )^{2}}+\frac {156065}{1936 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2-10287/400*x+1/166375*ln(5*x+3)+16807/704/(2*x-1)^2+156065/1936/(2*x-1)-543655/10648*ln(2*x-1)

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maxima [A]  time = 0.43, size = 44, normalized size = 0.80 \begin {gather*} -\frac {243}{80} \, x^{2} - \frac {10287}{400} \, x + \frac {2401 \, {\left (520 \, x - 183\right )}}{7744 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{166375} \, \log \left (5 \, x + 3\right ) - \frac {543655}{10648} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/80*x^2 - 10287/400*x + 2401/7744*(520*x - 183)/(4*x^2 - 4*x + 1) + 1/166375*log(5*x + 3) - 543655/10648*l
og(2*x - 1)

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mupad [B]  time = 0.04, size = 37, normalized size = 0.67 \begin {gather*} \frac {\ln \left (x+\frac {3}{5}\right )}{166375}-\frac {543655\,\ln \left (x-\frac {1}{2}\right )}{10648}-\frac {10287\,x}{400}+\frac {\frac {156065\,x}{3872}-\frac {439383}{30976}}{x^2-x+\frac {1}{4}}-\frac {243\,x^2}{80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/5)/166375 - (543655*log(x - 1/2))/10648 - (10287*x)/400 + ((156065*x)/3872 - 439383/30976)/(x^2 - x
+ 1/4) - (243*x^2)/80

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sympy [A]  time = 0.18, size = 44, normalized size = 0.80 \begin {gather*} - \frac {243 x^{2}}{80} - \frac {10287 x}{400} - \frac {439383 - 1248520 x}{30976 x^{2} - 30976 x + 7744} - \frac {543655 \log {\left (x - \frac {1}{2} \right )}}{10648} + \frac {\log {\left (x + \frac {3}{5} \right )}}{166375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**2/80 - 10287*x/400 - (439383 - 1248520*x)/(30976*x**2 - 30976*x + 7744) - 543655*log(x - 1/2)/10648 +
log(x + 3/5)/166375

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