\(\int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx\) [1335]

Optimal result

Integrand size = 22, antiderivative size = 88 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {49}{5 (2+3 x)^5}+\frac {707}{4 (2+3 x)^4}+\frac {6934}{3 (2+3 x)^3}+\frac {28555}{(2+3 x)^2}+\frac {424975}{2+3 x}-\frac {15125}{2 (3+5 x)^2}+\frac {277750}{3+5 x}-2958125 \log (2+3 x)+2958125 \log (3+5 x) \]

[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*ln(2+3*x)+2958125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {424975}{3 x+2}+\frac {277750}{5 x+3}+\frac {28555}{(3 x+2)^2}-\frac {15125}{2 (5 x+3)^2}+\frac {6934}{3 (3 x+2)^3}+\frac {707}{4 (3 x+2)^4}+\frac {49}{5 (3 x+2)^5}-2958125 \log (3 x+2)+2958125 \log (5 x+3) \]

[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 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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {49}{5 (2+3 x)^5}+\frac {707}{4 (2+3 x)^4}+\frac {6934}{3 (2+3 x)^3}+\frac {28555}{(2+3 x)^2}+\frac {424975}{2+3 x}-\frac {15125}{2 (3+5 x)^2}+\frac {277750}{3+5 x}-2958125 \log (5 (2+3 x))+2958125 \log (3+5 x) \]

[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]

Maple [A] (verified)

Time = 2.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72

method	result	size
norman	\(\frac {1198040625 x^{6}[phone] x^{4}+\frac {3321676301}{4} x +\frac {9344716875}{2} x^{5}+\frac {9598703854}{3} x^{2}+\frac {26287200325}{4} x^{3}+\frac {897608377}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {1198040625 x^{6}[phone] x^{4}+\frac {3321676301}{4} x +\frac {9344716875}{2} x^{5}+\frac {9598703854}{3} x^{2}+\frac {26287200325}{4} x^{3}+\frac {897608377}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\)	\(64\)
default	\(\frac {49}{5 \left (2+3 x \right )^{5}}+\frac {707}{4 \left (2+3 x \right )^{4}}+\frac {6934}{3 \left (2+3 x \right )^{3}}+\frac {28555}{\left (2+3 x \right )^{2}}+\frac {424975}{2+3 x}-\frac {15125}{2 \left (3+5 x \right )^{2}}+\frac {277750}{3+5 x}-2958125 \ln \left (2+3 x \right )+2958125 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(-\frac {408931199520 x -123360912000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-26580528000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-51755355000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-2453587200000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-317944008000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-455677147800000 \ln \left (x +\frac {3}{5}\right ) x^{5}-491484186000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-234624276000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\)	\(185\)

[In]

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

[Out]

(1198040625*x^6+7589365500*x^4+3321676301/4*x+9344716875/2*x^5+9598703854/3*x^2+26287200325/4*x^3+897608377/10
)/(2+3*x)^5/(3+5*x)^2-2958125*ln(2+3*x)+2958125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.76 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 177487500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 x^{6} + 280341506250 x^{5} + 455361930000 x^{4} + 394308004875 x^{3} + 191974077080 x^{2} + 49825144515 x + 5385650262}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 2958125 \log {\left (x + \frac {3}{5} \right )} - 2958125 \log {\left (x + \frac {2}{3} \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 2958125 \, \log \left (5 \, x + 3\right ) - 2958125 \, \log \left (3 \, x + 2\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {71882437500 \, x^{6} + 280341506250 \, x^{5} + 455361930000 \, x^{4} + 394308004875 \, x^{3} + 191974077080 \, x^{2} + 49825144515 \, x + 5385650262}{60 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 2958125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 2958125 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 1.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.85 \[ \int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {\frac {591625\,x^6}{3}+\frac {1538225\,x^5}{2}+\frac {101191540\,x^4}{81}+\frac {1051488013\,x^3}{972}+\frac {9598703854\,x^2}{18225}+\frac {3321676301\,x}{24300}+\frac {897608377}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-5916250\,\mathrm {atanh}\left (30\,x+19\right ) \]

[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)