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

Optimal. Leaf size=48 \[ -\frac {81 x}{250}-\frac {134}{75625 (5 x+3)}-\frac {1}{13750 (5 x+3)^2}-\frac {2401 \log (1-2 x)}{5324}+\frac {6802 \log (5 x+3)}{831875} \]

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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}{250}-\frac {134}{75625 (5 x+3)}-\frac {1}{13750 (5 x+3)^2}-\frac {2401 \log (1-2 x)}{5324}+\frac {6802 \log (5 x+3)}{831875} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-81*x)/250 - 1/(13750*(3 + 5*x)^2) - 134/(75625*(3 + 5*x)) - (2401*Log[1 - 2*x])/5324 + (6802*Log[3 + 5*x])/8
31875

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+5 x)^3} \, dx &=\int \left (-\frac {81}{250}-\frac {2401}{2662 (-1+2 x)}+\frac {1}{1375 (3+5 x)^3}+\frac {134}{15125 (3+5 x)^2}+\frac {6802}{166375 (3+5 x)}\right ) \, dx\\ &=-\frac {81 x}{250}-\frac {1}{13750 (3+5 x)^2}-\frac {134}{75625 (3+5 x)}-\frac {2401 \log (1-2 x)}{5324}+\frac {6802 \log (3+5 x)}{831875}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 45, normalized size = 0.94 \begin {gather*} \frac {-\frac {55 \left (490050 x^3+343035 x^2-117076 x-87883\right )}{(5 x+3)^2}-1500625 \log (1-2 x)+27208 \log (10 x+6)}{3327500} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(-87883 - 117076*x + 343035*x^2 + 490050*x^3))/(3 + 5*x)^2 - 1500625*Log[1 - 2*x] + 27208*Log[6 + 10*x])
/3327500

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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+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.02, size = 65, normalized size = 1.35 \begin {gather*} -\frac {26952750 \, x^{3} + 32343300 \, x^{2} - 27208 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1500625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 9732470 \, x + 17930}{3327500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3327500*(26952750*x^3 + 32343300*x^2 - 27208*(25*x^2 + 30*x + 9)*log(5*x + 3) + 1500625*(25*x^2 + 30*x + 9)
*log(2*x - 1) + 9732470*x + 17930)/(25*x^2 + 30*x + 9)

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giac [A]  time = 0.96, size = 36, normalized size = 0.75 \begin {gather*} -\frac {81}{250} \, x - \frac {268 \, x + 163}{30250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {6802}{831875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {2401}{5324} \, \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+5*x)^3,x, algorithm="giac")

[Out]

-81/250*x - 1/30250*(268*x + 163)/(5*x + 3)^2 + 6802/831875*log(abs(5*x + 3)) - 2401/5324*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}{250}-\frac {2401 \ln \left (2 x -1\right )}{5324}+\frac {6802 \ln \left (5 x +3\right )}{831875}-\frac {1}{13750 \left (5 x +3\right )^{2}}-\frac {134}{75625 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/250*x-1/13750/(5*x+3)^2-134/75625/(5*x+3)+6802/831875*ln(5*x+3)-2401/5324*ln(2*x-1)

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maxima [A]  time = 0.46, size = 39, normalized size = 0.81 \begin {gather*} -\frac {81}{250} \, x - \frac {268 \, x + 163}{30250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {6802}{831875} \, \log \left (5 \, x + 3\right ) - \frac {2401}{5324} \, \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+5*x)^3,x, algorithm="maxima")

[Out]

-81/250*x - 1/30250*(268*x + 163)/(25*x^2 + 30*x + 9) + 6802/831875*log(5*x + 3) - 2401/5324*log(2*x - 1)

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mupad [B]  time = 0.04, size = 33, normalized size = 0.69 \begin {gather*} \frac {6802\,\ln \left (x+\frac {3}{5}\right )}{831875}-\frac {2401\,\ln \left (x-\frac {1}{2}\right )}{5324}-\frac {81\,x}{250}-\frac {\frac {134\,x}{378125}+\frac {163}{756250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6802*log(x + 3/5))/831875 - (2401*log(x - 1/2))/5324 - (81*x)/250 - ((134*x)/378125 + 163/756250)/((6*x)/5 +
x^2 + 9/25)

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sympy [A]  time = 0.18, size = 39, normalized size = 0.81 \begin {gather*} - \frac {81 x}{250} - \frac {268 x + 163}{756250 x^{2} + 907500 x + 272250} - \frac {2401 \log {\left (x - \frac {1}{2} \right )}}{5324} + \frac {6802 \log {\left (x + \frac {3}{5} \right )}}{831875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x/250 - (268*x + 163)/(756250*x**2 + 907500*x + 272250) - 2401*log(x - 1/2)/5324 + 6802*log(x + 3/5)/83187
5

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