3.16.5 \(\int \frac {(2+3 x)^4}{(1-2 x) (3+5 x)^2} \, dx\) [1505]

Optimal. Leaf size=44 \[ -\frac {1593 x}{500}-\frac {81 x^2}{100}-\frac {1}{6875 (3+5 x)}-\frac {2401}{968} \log (1-2 x)+\frac {134 \log (3+5 x)}{75625} \]

[Out]

-1593/500*x-81/100*x^2-1/6875/(3+5*x)-2401/968*ln(1-2*x)+134/75625*ln(3+5*x)

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Rubi [A]
time = 0.01, antiderivative size = 44, 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 = {90} \begin {gather*} -\frac {81 x^2}{100}-\frac {1593 x}{500}-\frac {1}{6875 (5 x+3)}-\frac {2401}{968} \log (1-2 x)+\frac {134 \log (5 x+3)}{75625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1593*x)/500 - (81*x^2)/100 - 1/(6875*(3 + 5*x)) - (2401*Log[1 - 2*x])/968 + (134*Log[3 + 5*x])/75625

Rule 90

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)^2} \, dx &=\int \left (-\frac {1593}{500}-\frac {81 x}{50}-\frac {2401}{484 (-1+2 x)}+\frac {1}{1375 (3+5 x)^2}+\frac {134}{15125 (3+5 x)}\right ) \, dx\\ &=-\frac {1593 x}{500}-\frac {81 x^2}{100}-\frac {1}{6875 (3+5 x)}-\frac {2401}{968} \log (1-2 x)+\frac {134 \log (3+5 x)}{75625}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 52, normalized size = 1.18 \begin {gather*} \frac {999}{500} (1-2 x)-\frac {81}{400} (1-2 x)^2-\frac {1}{6875 (3+5 x)}-\frac {2401}{968} \log (1-2 x)+\frac {134 \log (6+10 x)}{75625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(999*(1 - 2*x))/500 - (81*(1 - 2*x)^2)/400 - 1/(6875*(3 + 5*x)) - (2401*Log[1 - 2*x])/968 + (134*Log[6 + 10*x]
)/75625

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Maple [A]
time = 0.13, size = 35, normalized size = 0.80

method	result	size
risch	\(-\frac {81 x^{2}}{100}-\frac {1593 x}{500}-\frac {1}{34375 \left (x +\frac {3}{5}\right )}-\frac {2401 \ln \left (-1+2 x \right )}{968}+\frac {134 \ln \left (3+5 x \right )}{75625}\)	\(33\)
default	\(-\frac {81 x^{2}}{100}-\frac {1593 x}{500}-\frac {2401 \ln \left (-1+2 x \right )}{968}-\frac {1}{6875 \left (3+5 x \right )}+\frac {134 \ln \left (3+5 x \right )}{75625}\)	\(35\)
norman	\(\frac {-\frac {157703}{16500} x -\frac {459}{25} x^{2}-\frac {81}{20} x^{3}}{3+5 x}-\frac {2401 \ln \left (-1+2 x \right )}{968}+\frac {134 \ln \left (3+5 x \right )}{75625}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/100*x^2-1593/500*x-2401/968*ln(-1+2*x)-1/6875/(3+5*x)+134/75625*ln(3+5*x)

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Maxima [A]
time = 0.34, size = 34, normalized size = 0.77 \begin {gather*} -\frac {81}{100} \, x^{2} - \frac {1593}{500} \, x - \frac {1}{6875 \, {\left (5 \, x + 3\right )}} + \frac {134}{75625} \, \log \left (5 \, x + 3\right ) - \frac {2401}{968} \, \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)^2,x, algorithm="maxima")

[Out]

-81/100*x^2 - 1593/500*x - 1/6875/(5*x + 3) + 134/75625*log(5*x + 3) - 2401/968*log(2*x - 1)

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Fricas [A]
time = 0.99, size = 50, normalized size = 1.14 \begin {gather*} -\frac {2450250 \, x^{3} + 11107800 \, x^{2} - 1072 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1500625 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 5782590 \, x + 88}{605000 \, {\left (5 \, x + 3\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)^2,x, algorithm="fricas")

[Out]

-1/605000*(2450250*x^3 + 11107800*x^2 - 1072*(5*x + 3)*log(5*x + 3) + 1500625*(5*x + 3)*log(2*x - 1) + 5782590
*x + 88)/(5*x + 3)

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Sympy [A]
time = 0.06, size = 37, normalized size = 0.84 \begin {gather*} - \frac {81 x^{2}}{100} - \frac {1593 x}{500} - \frac {2401 \log {\left (x - \frac {1}{2} \right )}}{968} + \frac {134 \log {\left (x + \frac {3}{5} \right )}}{75625} - \frac {1}{34375 x + 20625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x**2/100 - 1593*x/500 - 2401*log(x - 1/2)/968 + 134*log(x + 3/5)/75625 - 1/(34375*x + 20625)

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Giac [A]
time = 1.57, size = 63, normalized size = 1.43 \begin {gather*} -\frac {27}{2500} \, {\left (5 \, x + 3\right )}^{2} {\left (\frac {41}{5 \, x + 3} + 3\right )} - \frac {1}{6875 \, {\left (5 \, x + 3\right )}} + \frac {12393}{5000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {2401}{968} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \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)^2,x, algorithm="giac")

[Out]

-27/2500*(5*x + 3)^2*(41/(5*x + 3) + 3) - 1/6875/(5*x + 3) + 12393/5000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 24
01/968*log(abs(-11/(5*x + 3) + 2))

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Mupad [B]
time = 0.04, size = 30, normalized size = 0.68 \begin {gather*} \frac {134\,\ln \left (x+\frac {3}{5}\right )}{75625}-\frac {2401\,\ln \left (x-\frac {1}{2}\right )}{968}-\frac {1593\,x}{500}-\frac {1}{34375\,\left (x+\frac {3}{5}\right )}-\frac {81\,x^2}{100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(134*log(x + 3/5))/75625 - (2401*log(x - 1/2))/968 - (1593*x)/500 - 1/(34375*(x + 3/5)) - (81*x^2)/100

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