3.1318 \(\int \frac{(1-2 x)^2}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac{46475}{3 x+2}-\frac{15125}{5 x+3}-\frac{3740}{(3 x+2)^2}-\frac{1133}{3 (3 x+2)^3}-\frac{77}{2 (3 x+2)^4}-\frac{49}{15 (3 x+2)^5}+277750 \log (3 x+2)-277750 \log (5 x+3) \]

[Out]

-49/(15*(2 + 3*x)^5) - 77/(2*(2 + 3*x)^4) - 1133/(3*(2 + 3*x)^3) - 3740/(2 + 3*x)^2 - 46475/(2 + 3*x) - 15125/
(3 + 5*x) + 277750*Log[2 + 3*x] - 277750*Log[3 + 5*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0356603, antiderivative size = 77, 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} \[ -\frac{46475}{3 x+2}-\frac{15125}{5 x+3}-\frac{3740}{(3 x+2)^2}-\frac{1133}{3 (3 x+2)^3}-\frac{77}{2 (3 x+2)^4}-\frac{49}{15 (3 x+2)^5}+277750 \log (3 x+2)-277750 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(15*(2 + 3*x)^5) - 77/(2*(2 + 3*x)^4) - 1133/(3*(2 + 3*x)^3) - 3740/(2 + 3*x)^2 - 46475/(2 + 3*x) - 15125/
(3 + 5*x) + 277750*Log[2 + 3*x] - 277750*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)^2} \, dx &=\int \left (\frac{49}{(2+3 x)^6}+\frac{462}{(2+3 x)^5}+\frac{3399}{(2+3 x)^4}+\frac{22440}{(2+3 x)^3}+\frac{139425}{(2+3 x)^2}+\frac{833250}{2+3 x}+\frac{75625}{(3+5 x)^2}-\frac{1388750}{3+5 x}\right ) \, dx\\ &=-\frac{49}{15 (2+3 x)^5}-\frac{77}{2 (2+3 x)^4}-\frac{1133}{3 (2+3 x)^3}-\frac{3740}{(2+3 x)^2}-\frac{46475}{2+3 x}-\frac{15125}{3+5 x}+277750 \log (2+3 x)-277750 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0611766, size = 62, normalized size = 0.81 \[ -\frac{674932500 x^5+2227277250 x^4+2939206050 x^3+1938789435 x^2+639246515 x+84279984}{30 (3 x+2)^5 (5 x+3)}+277750 \log (5 (3 x+2))-277750 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(84279984 + 639246515*x + 1938789435*x^2 + 2939206050*x^3 + 2227277250*x^4 + 674932500*x^5)/(30*(2 + 3*x)^5*(
3 + 5*x)) + 277750*Log[5*(2 + 3*x)] - 277750*Log[3 + 5*x]

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 72, normalized size = 0.9 \begin{align*} -{\frac{49}{15\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{77}{2\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{1133}{3\, \left ( 2+3\,x \right ) ^{3}}}-3740\, \left ( 2+3\,x \right ) ^{-2}-46475\, \left ( 2+3\,x \right ) ^{-1}-15125\, \left ( 3+5\,x \right ) ^{-1}+277750\,\ln \left ( 2+3\,x \right ) -277750\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/15/(2+3*x)^5-77/2/(2+3*x)^4-1133/3/(2+3*x)^3-3740/(2+3*x)^2-46475/(2+3*x)-15125/(3+5*x)+277750*ln(2+3*x)-2
77750*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.28562, size = 103, normalized size = 1.34 \begin{align*} -\frac{674932500 \, x^{5} + 2227277250 \, x^{4} + 2939206050 \, x^{3} + 1938789435 \, x^{2} + 639246515 \, x + 84279984}{30 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - 277750 \, \log \left (5 \, x + 3\right ) + 277750 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(674932500*x^5 + 2227277250*x^4 + 2939206050*x^3 + 1938789435*x^2 + 639246515*x + 84279984)/(1215*x^6 +
4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 277750*log(5*x + 3) + 277750*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.42596, size = 471, normalized size = 6.12 \begin{align*} -\frac{674932500 \, x^{5} + 2227277250 \, x^{4} + 2939206050 \, x^{3} + 1938789435 \, x^{2} + 8332500 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 8332500 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 639246515 \, x + 84279984}{30 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(674932500*x^5 + 2227277250*x^4 + 2939206050*x^3 + 1938789435*x^2 + 8332500*(1215*x^6 + 4779*x^5 + 7830*
x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 8332500*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 33
60*x^2 + 880*x + 96)*log(3*x + 2) + 639246515*x + 84279984)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*
x^2 + 880*x + 96)

________________________________________________________________________________________

Sympy [A]  time = 0.195798, size = 71, normalized size = 0.92 \begin{align*} - \frac{674932500 x^{5} + 2227277250 x^{4} + 2939206050 x^{3} + 1938789435 x^{2} + 639246515 x + 84279984}{36450 x^{6} + 143370 x^{5} + 234900 x^{4} + 205200 x^{3} + 100800 x^{2} + 26400 x + 2880} - 277750 \log{\left (x + \frac{3}{5} \right )} + 277750 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(674932500*x**5 + 2227277250*x**4 + 2939206050*x**3 + 1938789435*x**2 + 639246515*x + 84279984)/(36450*x**6 +
143370*x**5 + 234900*x**4 + 205200*x**3 + 100800*x**2 + 26400*x + 2880) - 277750*log(x + 3/5) + 277750*log(x
+ 2/3)

________________________________________________________________________________________

Giac [A]  time = 1.84867, size = 103, normalized size = 1.34 \begin{align*} -\frac{15125}{5 \, x + 3} + \frac{125 \,{\left (\frac{2338497}{5 \, x + 3} + \frac{1317834}{{\left (5 \, x + 3\right )}^{2}} + \frac{338628}{{\left (5 \, x + 3\right )}^{3}} + \frac{33998}{{\left (5 \, x + 3\right )}^{4}} + 1583793\right )}}{2 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{5}} + 277750 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15125/(5*x + 3) + 125/2*(2338497/(5*x + 3) + 1317834/(5*x + 3)^2 + 338628/(5*x + 3)^3 + 33998/(5*x + 3)^4 + 1
583793)/(1/(5*x + 3) + 3)^5 + 277750*log(abs(-1/(5*x + 3) - 3))