3.1520 \(\int \frac{(2+3 x)^4}{(1-2 x) (3+5 x)^3} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0211931, antiderivative size = 48, normalized size of antiderivative = 1., 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} \[ -\frac{81 x}{250}-\frac{134}{75625 (5 x+3)}-\frac{1}{13750 (5 x+3)^2}-\frac{2401 \log (1-2 x)}{5324}+\frac{6802 \log (5 x+3)}{831875} \]

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*}

Mathematica [A]  time = 0.0249633, size = 45, normalized size = 0.94 \[ \frac{-\frac{55 \left (490050 x^3+343035 x^2-117076 x-87883\right )}{(5 x+3)^2}-1500625 \log (1-2 x)+27208 \log (10 x+6)}{3327500} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 39, normalized size = 0.8 \begin{align*} -{\frac{81\,x}{250}}-{\frac{2401\,\ln \left ( 2\,x-1 \right ) }{5324}}-{\frac{1}{13750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{134}{226875+378125\,x}}+{\frac{6802\,\ln \left ( 3+5\,x \right ) }{831875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0595, size = 53, normalized size = 1.1 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.37052, size = 224, normalized size = 4.67 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.16193, size = 39, normalized size = 0.81 \begin{align*} - \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{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.23831, size = 49, normalized size = 1.02 \begin{align*} -\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{align*}

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))