3.12.96 \(\int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx\)

Optimal. Leaf size=88 \[ \frac {424975}{3 x+2}+\frac {277750}{5 x+3}+\frac {28555}{(3 x+2)^2}-\frac {15125}{2 (5 x+3)^2}+\frac {6934}{3 (3 x+2)^3}+\frac {707}{4 (3 x+2)^4}+\frac {49}{5 (3 x+2)^5}-2958125 \log (3 x+2)+2958125 \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 88, 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 {424975}{3 x+2}+\frac {277750}{5 x+3}+\frac {28555}{(3 x+2)^2}-\frac {15125}{2 (5 x+3)^2}+\frac {6934}{3 (3 x+2)^3}+\frac {707}{4 (3 x+2)^4}+\frac {49}{5 (3 x+2)^5}-2958125 \log (3 x+2)+2958125 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^3),x]

[Out]

49/(5*(2 + 3*x)^5) + 707/(4*(2 + 3*x)^4) + 6934/(3*(2 + 3*x)^3) + 28555/(2 + 3*x)^2 + 424975/(2 + 3*x) - 15125
/(2*(3 + 5*x)^2) + 277750/(3 + 5*x) - 2958125*Log[2 + 3*x] + 2958125*Log[3 + 5*x]

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 {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx &=\int \left (-\frac {147}{(2+3 x)^6}-\frac {2121}{(2+3 x)^5}-\frac {20802}{(2+3 x)^4}-\frac {171330}{(2+3 x)^3}-\frac {1274925}{(2+3 x)^2}-\frac {8874375}{2+3 x}+\frac {75625}{(3+5 x)^3}-\frac {1388750}{(3+5 x)^2}+\frac {14790625}{3+5 x}\right ) \, dx\\ &=\frac {49}{5 (2+3 x)^5}+\frac {707}{4 (2+3 x)^4}+\frac {6934}{3 (2+3 x)^3}+\frac {28555}{(2+3 x)^2}+\frac {424975}{2+3 x}-\frac {15125}{2 (3+5 x)^2}+\frac {277750}{3+5 x}-2958125 \log (2+3 x)+2958125 \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 90, normalized size = 1.02 \begin {gather*} \frac {424975}{3 x+2}+\frac {277750}{5 x+3}+\frac {28555}{(3 x+2)^2}-\frac {15125}{2 (5 x+3)^2}+\frac {6934}{3 (3 x+2)^3}+\frac {707}{4 (3 x+2)^4}+\frac {49}{5 (3 x+2)^5}-2958125 \log (5 (3 x+2))+2958125 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^3),x]

[Out]

49/(5*(2 + 3*x)^5) + 707/(4*(2 + 3*x)^4) + 6934/(3*(2 + 3*x)^3) + 28555/(2 + 3*x)^2 + 424975/(2 + 3*x) - 15125
/(2*(3 + 5*x)^2) + 277750/(3 + 5*x) - 2958125*Log[5*(2 + 3*x)] + 2958125*Log[3 + 5*x]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^3),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^3), x]

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 155, normalized size = 1.76 \begin {gather*} \frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875*x^3 + 191974077080*x^2 + 177487500*
(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(5*x + 3) - 177487500
*(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(3*x + 2) + 49825144
515*x + 5385650262)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)

________________________________________________________________________________________

giac [A]  time = 1.03, size = 65, normalized size = 0.74 \begin {gather*} \frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 2958125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 2958125 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875*x^3 + 191974077080*x^2 + 4982514451
5*x + 5385650262)/((5*x + 3)^2*(3*x + 2)^5) + 2958125*log(abs(5*x + 3)) - 2958125*log(abs(3*x + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 81, normalized size = 0.92 \begin {gather*} -2958125 \ln \left (3 x +2\right )+2958125 \ln \left (5 x +3\right )+\frac {49}{5 \left (3 x +2\right )^{5}}+\frac {707}{4 \left (3 x +2\right )^{4}}+\frac {6934}{3 \left (3 x +2\right )^{3}}+\frac {28555}{\left (3 x +2\right )^{2}}+\frac {424975}{3 x +2}-\frac {15125}{2 \left (5 x +3\right )^{2}}+\frac {277750}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2/(3*x+2)^6/(5*x+3)^3,x)

[Out]

49/5/(3*x+2)^5+707/4/(3*x+2)^4+6934/3/(3*x+2)^3+28555/(3*x+2)^2+424975/(3*x+2)-15125/2/(5*x+3)^2+277750/(5*x+3
)-2958125*ln(3*x+2)+2958125*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 86, normalized size = 0.98 \begin {gather*} \frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 2958125 \, \log \left (5 \, x + 3\right ) - 2958125 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^6/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/60*(71882437500*x^6 + 280341506250*x^5 + 455361930000*x^4 + 394308004875*x^3 + 191974077080*x^2 + 4982514451
5*x + 5385650262)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 2958
125*log(5*x + 3) - 2958125*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 75, normalized size = 0.85 \begin {gather*} \frac {\frac {591625\,x^6}{3}+\frac {1538225\,x^5}{2}+\frac {101191540\,x^4}{81}+\frac {1051488013\,x^3}{972}+\frac {9598703854\,x^2}{18225}+\frac {3321676301\,x}{24300}+\frac {897608377}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-5916250\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2/((3*x + 2)^6*(5*x + 3)^3),x)

[Out]

((3321676301*x)/24300 + (9598703854*x^2)/18225 + (1051488013*x^3)/972 + (101191540*x^4)/81 + (1538225*x^5)/2 +
(591625*x^6)/3 + 897608377/60750)/((208*x)/405 + (2896*x^2)/1215 + (2488*x^3)/405 + (1282*x^4)/135 + (1981*x^
5)/225 + (68*x^6)/15 + x^7 + 32/675) - 5916250*atanh(30*x + 19)

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 82, normalized size = 0.93 \begin {gather*} \frac {71882437500 x^{6} + 280341506250 x^{5} + 455361930000 x^{4} + 394308004875 x^{3} + 191974077080 x^{2} + 49825144515 x + 5385650262}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 2958125 \log {\left (x + \frac {3}{5} \right )} - 2958125 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2/(2+3*x)**6/(3+5*x)**3,x)

[Out]

(71882437500*x**6 + 280341506250*x**5 + 455361930000*x**4 + 394308004875*x**3 + 191974077080*x**2 + 4982514451
5*x + 5385650262)/(364500*x**7 + 1652400*x**6 + 3209220*x**5 + 3461400*x**4 + 2239200*x**3 + 868800*x**2 + 187
200*x + 17280) + 2958125*log(x + 3/5) - 2958125*log(x + 2/3)

________________________________________________________________________________________