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

Optimal. Leaf size=48 \[ -\frac {81 x}{40}-\frac {33271}{1936 (1-2 x)}+\frac {2401}{352 (1-2 x)^2}-\frac {153811 \log (1-2 x)}{21296}+\frac {\log (5 x+3)}{33275} \]

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Rubi [A]  time = 0.02, antiderivative size = 48, 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 {81 x}{40}-\frac {33271}{1936 (1-2 x)}+\frac {2401}{352 (1-2 x)^2}-\frac {153811 \log (1-2 x)}{21296}+\frac {\log (5 x+3)}{33275} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2401/(352*(1 - 2*x)^2) - 33271/(1936*(1 - 2*x)) - (81*x)/40 - (153811*Log[1 - 2*x])/21296 + Log[3 + 5*x]/33275

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)^4}{(1-2 x)^3 (3+5 x)} \, dx &=\int \left (-\frac {81}{40}-\frac {2401}{88 (-1+2 x)^3}-\frac {33271}{968 (-1+2 x)^2}-\frac {153811}{10648 (-1+2 x)}+\frac {1}{6655 (3+5 x)}\right ) \, dx\\ &=\frac {2401}{352 (1-2 x)^2}-\frac {33271}{1936 (1-2 x)}-\frac {81 x}{40}-\frac {153811 \log (1-2 x)}{21296}+\frac {\log (3+5 x)}{33275}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 46, normalized size = 0.96 \begin {gather*} \frac {-431244 (5 x+3)+\frac {18299050}{2 x-1}+\frac {7263025}{(1-2 x)^2}-7690550 \log (5-10 x)+32 \log (5 x+3)}{1064800} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7263025/(1 - 2*x)^2 + 18299050/(-1 + 2*x) - 431244*(3 + 5*x) - 7690550*Log[5 - 10*x] + 32*Log[3 + 5*x])/10648
00

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.46, size = 65, normalized size = 1.35 \begin {gather*} -\frac {8624880 \, x^{3} - 8624880 \, x^{2} - 32 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 7690550 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 34441880 \, x + 11036025}{1064800 \, {\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)^4/(1-2*x)^3/(3+5*x),x, algorithm="fricas")

[Out]

-1/1064800*(8624880*x^3 - 8624880*x^2 - 32*(4*x^2 - 4*x + 1)*log(5*x + 3) + 7690550*(4*x^2 - 4*x + 1)*log(2*x
- 1) - 34441880*x + 11036025)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.16, size = 36, normalized size = 0.75 \begin {gather*} -\frac {81}{40} \, x + \frac {343 \, {\left (388 \, x - 117\right )}}{3872 \, {\left (2 \, x - 1\right )}^{2}} + \frac {1}{33275} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {153811}{21296} \, \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)^4/(1-2*x)^3/(3+5*x),x, algorithm="giac")

[Out]

-81/40*x + 343/3872*(388*x - 117)/(2*x - 1)^2 + 1/33275*log(abs(5*x + 3)) - 153811/21296*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 39, normalized size = 0.81 \begin {gather*} -\frac {81 x}{40}-\frac {153811 \ln \left (2 x -1\right )}{21296}+\frac {\ln \left (5 x +3\right )}{33275}+\frac {2401}{352 \left (2 x -1\right )^{2}}+\frac {33271}{1936 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/40*x+1/33275*ln(5*x+3)+2401/352/(2*x-1)^2+33271/1936/(2*x-1)-153811/21296*ln(2*x-1)

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maxima [A]  time = 0.55, size = 39, normalized size = 0.81 \begin {gather*} -\frac {81}{40} \, x + \frac {343 \, {\left (388 \, x - 117\right )}}{3872 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1}{33275} \, \log \left (5 \, x + 3\right ) - \frac {153811}{21296} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/40*x + 343/3872*(388*x - 117)/(4*x^2 - 4*x + 1) + 1/33275*log(5*x + 3) - 153811/21296*log(2*x - 1)

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mupad [B]  time = 1.07, size = 32, normalized size = 0.67 \begin {gather*} \frac {\ln \left (x+\frac {3}{5}\right )}{33275}-\frac {153811\,\ln \left (x-\frac {1}{2}\right )}{21296}-\frac {81\,x}{40}+\frac {\frac {33271\,x}{3872}-\frac {40131}{15488}}{x^2-x+\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + 3/5)/33275 - (153811*log(x - 1/2))/21296 - (81*x)/40 + ((33271*x)/3872 - 40131/15488)/(x^2 - x + 1/4)

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sympy [A]  time = 0.18, size = 37, normalized size = 0.77 \begin {gather*} - \frac {81 x}{40} - \frac {40131 - 133084 x}{15488 x^{2} - 15488 x + 3872} - \frac {153811 \log {\left (x - \frac {1}{2} \right )}}{21296} + \frac {\log {\left (x + \frac {3}{5} \right )}}{33275} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x/40 - (40131 - 133084*x)/(15488*x**2 - 15488*x + 3872) - 153811*log(x - 1/2)/21296 + log(x + 3/5)/33275

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