3.1502 \(\int \frac{(2+3 x)^7}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=65 \[ -\frac{2187 x^5}{250}-\frac{86751 x^4}{2000}-\frac{495477 x^3}{5000}-\frac{14750667 x^2}{100000}-\frac{19846971 x}{100000}-\frac{1}{859375 (5 x+3)}-\frac{823543 \log (1-2 x)}{7744}+\frac{233 \log (5 x+3)}{9453125} \]

[Out]

(-19846971*x)/100000 - (14750667*x^2)/100000 - (495477*x^3)/5000 - (86751*x^4)/2000 - (2187*x^5)/250 - 1/(8593
75*(3 + 5*x)) - (823543*Log[1 - 2*x])/7744 + (233*Log[3 + 5*x])/9453125

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Rubi [A]  time = 0.0325331, antiderivative size = 65, 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} \[ -\frac{2187 x^5}{250}-\frac{86751 x^4}{2000}-\frac{495477 x^3}{5000}-\frac{14750667 x^2}{100000}-\frac{19846971 x}{100000}-\frac{1}{859375 (5 x+3)}-\frac{823543 \log (1-2 x)}{7744}+\frac{233 \log (5 x+3)}{9453125} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^7/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

(-19846971*x)/100000 - (14750667*x^2)/100000 - (495477*x^3)/5000 - (86751*x^4)/2000 - (2187*x^5)/250 - 1/(8593
75*(3 + 5*x)) - (823543*Log[1 - 2*x])/7744 + (233*Log[3 + 5*x])/9453125

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)^7}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac{19846971}{100000}-\frac{14750667 x}{50000}-\frac{1486431 x^2}{5000}-\frac{86751 x^3}{500}-\frac{2187 x^4}{50}-\frac{823543}{3872 (-1+2 x)}+\frac{1}{171875 (3+5 x)^2}+\frac{233}{1890625 (3+5 x)}\right ) \, dx\\ &=-\frac{19846971 x}{100000}-\frac{14750667 x^2}{100000}-\frac{495477 x^3}{5000}-\frac{86751 x^4}{2000}-\frac{2187 x^5}{250}-\frac{1}{859375 (3+5 x)}-\frac{823543 \log (1-2 x)}{7744}+\frac{233 \log (3+5 x)}{9453125}\\ \end{align*}

Mathematica [A]  time = 0.0305617, size = 60, normalized size = 0.92 \[ \frac{-\frac{11 \left (9622800000 x^6+53486730000 x^5+137632770000 x^4+227660301000 x^3+315671083200 x^2-35641061775 x-99978641969\right )}{5 x+3}-257357187500 \log (1-2 x)+59648 \log (10 x+6)}{2420000000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^7/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

((-11*(-99978641969 - 35641061775*x + 315671083200*x^2 + 227660301000*x^3 + 137632770000*x^4 + 53486730000*x^5
+ 9622800000*x^6))/(3 + 5*x) - 257357187500*Log[1 - 2*x] + 59648*Log[6 + 10*x])/2420000000

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Maple [A]  time = 0.007, size = 50, normalized size = 0.8 \begin{align*} -{\frac{2187\,{x}^{5}}{250}}-{\frac{86751\,{x}^{4}}{2000}}-{\frac{495477\,{x}^{3}}{5000}}-{\frac{14750667\,{x}^{2}}{100000}}-{\frac{19846971\,x}{100000}}-{\frac{823543\,\ln \left ( 2\,x-1 \right ) }{7744}}-{\frac{1}{2578125+4296875\,x}}+{\frac{233\,\ln \left ( 3+5\,x \right ) }{9453125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^7/(1-2*x)/(3+5*x)^2,x)

[Out]

-2187/250*x^5-86751/2000*x^4-495477/5000*x^3-14750667/100000*x^2-19846971/100000*x-823543/7744*ln(2*x-1)-1/859
375/(3+5*x)+233/9453125*ln(3+5*x)

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Maxima [A]  time = 1.35074, size = 66, normalized size = 1.02 \begin{align*} -\frac{2187}{250} \, x^{5} - \frac{86751}{2000} \, x^{4} - \frac{495477}{5000} \, x^{3} - \frac{14750667}{100000} \, x^{2} - \frac{19846971}{100000} \, x - \frac{1}{859375 \,{\left (5 \, x + 3\right )}} + \frac{233}{9453125} \, \log \left (5 \, x + 3\right ) - \frac{823543}{7744} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-2187/250*x^5 - 86751/2000*x^4 - 495477/5000*x^3 - 14750667/100000*x^2 - 19846971/100000*x - 1/859375/(5*x + 3
) + 233/9453125*log(5*x + 3) - 823543/7744*log(2*x - 1)

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Fricas [A]  time = 1.35881, size = 282, normalized size = 4.34 \begin{align*} -\frac{26462700000 \, x^{6} + 147088507500 \, x^{5} + 378490117500 \, x^{4} + 626065827750 \, x^{3} + 868095478800 \, x^{2} - 14912 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 64339296875 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 360222523650 \, x + 704}{605000000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/605000000*(26462700000*x^6 + 147088507500*x^5 + 378490117500*x^4 + 626065827750*x^3 + 868095478800*x^2 - 14
912*(5*x + 3)*log(5*x + 3) + 64339296875*(5*x + 3)*log(2*x - 1) + 360222523650*x + 704)/(5*x + 3)

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Sympy [A]  time = 0.155497, size = 58, normalized size = 0.89 \begin{align*} - \frac{2187 x^{5}}{250} - \frac{86751 x^{4}}{2000} - \frac{495477 x^{3}}{5000} - \frac{14750667 x^{2}}{100000} - \frac{19846971 x}{100000} - \frac{823543 \log{\left (x - \frac{1}{2} \right )}}{7744} + \frac{233 \log{\left (x + \frac{3}{5} \right )}}{9453125} - \frac{1}{4296875 x + 2578125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**7/(1-2*x)/(3+5*x)**2,x)

[Out]

-2187*x**5/250 - 86751*x**4/2000 - 495477*x**3/5000 - 14750667*x**2/100000 - 19846971*x/100000 - 823543*log(x
- 1/2)/7744 + 233*log(x + 3/5)/9453125 - 1/(4296875*x + 2578125)

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Giac [A]  time = 2.79493, size = 122, normalized size = 1.88 \begin{align*} -\frac{27}{12500000} \,{\left (5 \, x + 3\right )}^{5}{\left (\frac{12690}{5 \, x + 3} + \frac{98100}{{\left (5 \, x + 3\right )}^{2}} + \frac{813525}{{\left (5 \, x + 3\right )}^{3}} + \frac{8951575}{{\left (5 \, x + 3\right )}^{4}} + 1296\right )} - \frac{1}{859375 \,{\left (5 \, x + 3\right )}} + \frac{531729603}{5000000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{823543}{7744} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)/(3+5*x)^2,x, algorithm="giac")

[Out]

-27/12500000*(5*x + 3)^5*(12690/(5*x + 3) + 98100/(5*x + 3)^2 + 813525/(5*x + 3)^3 + 8951575/(5*x + 3)^4 + 129
6) - 1/859375/(5*x + 3) + 531729603/5000000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 823543/7744*log(abs(-11/(5*x +
3) + 2))