3.1320 \(\int \frac{(1-2 x)^2}{(2+3 x)^8 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{1615625}{3 x+2}-\frac{378125}{5 x+3}-\frac{138875}{(3 x+2)^2}-\frac{46475}{3 (3 x+2)^3}-\frac{1870}{(3 x+2)^4}-\frac{1133}{5 (3 x+2)^5}-\frac{77}{3 (3 x+2)^6}-\frac{7}{3 (3 x+2)^7}+9212500 \log (3 x+2)-9212500 \log (5 x+3) \]

[Out]

-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 13
8875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[2 + 3*x] - 9212500*Log[3 + 5*x]

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Rubi [A]  time = 0.0476409, antiderivative size = 97, 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} \[ -\frac{1615625}{3 x+2}-\frac{378125}{5 x+3}-\frac{138875}{(3 x+2)^2}-\frac{46475}{3 (3 x+2)^3}-\frac{1870}{(3 x+2)^4}-\frac{1133}{5 (3 x+2)^5}-\frac{77}{3 (3 x+2)^6}-\frac{7}{3 (3 x+2)^7}+9212500 \log (3 x+2)-9212500 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 13
8875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[2 + 3*x] - 9212500*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)^8 (3+5 x)^2} \, dx &=\int \left (\frac{49}{(2+3 x)^8}+\frac{462}{(2+3 x)^7}+\frac{3399}{(2+3 x)^6}+\frac{22440}{(2+3 x)^5}+\frac{139425}{(2+3 x)^4}+\frac{833250}{(2+3 x)^3}+\frac{4846875}{(2+3 x)^2}+\frac{27637500}{2+3 x}+\frac{1890625}{(3+5 x)^2}-\frac{46062500}{3+5 x}\right ) \, dx\\ &=-\frac{7}{3 (2+3 x)^7}-\frac{77}{3 (2+3 x)^6}-\frac{1133}{5 (2+3 x)^5}-\frac{1870}{(2+3 x)^4}-\frac{46475}{3 (2+3 x)^3}-\frac{138875}{(2+3 x)^2}-\frac{1615625}{2+3 x}-\frac{378125}{3+5 x}+9212500 \log (2+3 x)-9212500 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0682139, size = 99, normalized size = 1.02 \[ -\frac{1615625}{3 x+2}-\frac{378125}{5 x+3}-\frac{138875}{(3 x+2)^2}-\frac{46475}{3 (3 x+2)^3}-\frac{1870}{(3 x+2)^4}-\frac{1133}{5 (3 x+2)^5}-\frac{77}{3 (3 x+2)^6}-\frac{7}{3 (3 x+2)^7}+9212500 \log (5 (3 x+2))-9212500 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-7/(3*(2 + 3*x)^7) - 77/(3*(2 + 3*x)^6) - 1133/(5*(2 + 3*x)^5) - 1870/(2 + 3*x)^4 - 46475/(3*(2 + 3*x)^3) - 13
8875/(2 + 3*x)^2 - 1615625/(2 + 3*x) - 378125/(3 + 5*x) + 9212500*Log[5*(2 + 3*x)] - 9212500*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 90, normalized size = 0.9 \begin{align*} -{\frac{7}{3\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{77}{3\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{1133}{5\, \left ( 2+3\,x \right ) ^{5}}}-1870\, \left ( 2+3\,x \right ) ^{-4}-{\frac{46475}{3\, \left ( 2+3\,x \right ) ^{3}}}-138875\, \left ( 2+3\,x \right ) ^{-2}-1615625\, \left ( 2+3\,x \right ) ^{-1}-378125\, \left ( 3+5\,x \right ) ^{-1}+9212500\,\ln \left ( 2+3\,x \right ) -9212500\,\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)^8/(3+5*x)^2,x)

[Out]

-7/3/(2+3*x)^7-77/3/(2+3*x)^6-1133/5/(2+3*x)^5-1870/(2+3*x)^4-46475/3/(2+3*x)^3-138875/(2+3*x)^2-1615625/(2+3*
x)-378125/(3+5*x)+9212500*ln(2+3*x)-9212500*ln(3+5*x)

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Maxima [A]  time = 1.14374, size = 130, normalized size = 1.34 \begin{align*} -\frac{100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 59178013234 \, x + 5590850403}{15 \,{\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} - 9212500 \, \log \left (5 \, x + 3\right ) + 9212500 \, \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)^8/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900125*x^4 + 676227617505*x^3 + 2684085
63588*x^2 + 59178013234*x + 5590850403)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*
x^3 + 24864*x^2 + 4672*x + 384) - 9212500*log(5*x + 3) + 9212500*log(3*x + 2)

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Fricas [A]  time = 1.33269, size = 684, normalized size = 7.05 \begin{align*} -\frac{100738687500 \, x^{7} + 466755918750 \, x^{6} + 926721303750 \, x^{5} + 1022059900125 \, x^{4} + 676227617505 \, x^{3} + 268408563588 \, x^{2} + 138187500 \,{\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (5 \, x + 3\right ) - 138187500 \,{\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (3 \, x + 2\right ) + 59178013234 \, x + 5590850403}{15 \,{\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/15*(100738687500*x^7 + 466755918750*x^6 + 926721303750*x^5 + 1022059900125*x^4 + 676227617505*x^3 + 2684085
63588*x^2 + 138187500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 +
4672*x + 384)*log(5*x + 3) - 138187500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x
^3 + 24864*x^2 + 4672*x + 384)*log(3*x + 2) + 59178013234*x + 5590850403)/(10935*x^8 + 57591*x^7 + 132678*x^6
+ 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)

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Sympy [A]  time = 0.228968, size = 92, normalized size = 0.95 \begin{align*} - \frac{100738687500 x^{7} + 466755918750 x^{6} + 926721303750 x^{5} + 1022059900125 x^{4} + 676227617505 x^{3} + 268408563588 x^{2} + 59178013234 x + 5590850403}{164025 x^{8} + 863865 x^{7} + 1990170 x^{6} + 2619540 x^{5} + 2154600 x^{4} + 1134000 x^{3} + 372960 x^{2} + 70080 x + 5760} - 9212500 \log{\left (x + \frac{3}{5} \right )} + 9212500 \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)**8/(3+5*x)**2,x)

[Out]

-(100738687500*x**7 + 466755918750*x**6 + 926721303750*x**5 + 1022059900125*x**4 + 676227617505*x**3 + 2684085
63588*x**2 + 59178013234*x + 5590850403)/(164025*x**8 + 863865*x**7 + 1990170*x**6 + 2619540*x**5 + 2154600*x*
*4 + 1134000*x**3 + 372960*x**2 + 70080*x + 5760) - 9212500*log(x + 3/5) + 9212500*log(x + 2/3)

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Giac [A]  time = 2.54105, size = 127, normalized size = 1.31 \begin{align*} -\frac{378125}{5 \, x + 3} + \frac{625 \,{\left (\frac{120779019}{5 \, x + 3} + \frac{110006829}{{\left (5 \, x + 3\right )}^{2}} + \frac{54129465}{{\left (5 \, x + 3\right )}^{3}} + \frac{15246900}{{\left (5 \, x + 3\right )}^{4}} + \frac{2349450}{{\left (5 \, x + 3\right )}^{5}} + \frac{157100}{{\left (5 \, x + 3\right )}^{6}} + 55800576\right )}}{{\left (\frac{1}{5 \, x + 3} + 3\right )}^{7}} + 9212500 \, \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)^8/(3+5*x)^2,x, algorithm="giac")

[Out]

-378125/(5*x + 3) + 625*(120779019/(5*x + 3) + 110006829/(5*x + 3)^2 + 54129465/(5*x + 3)^3 + 15246900/(5*x +
3)^4 + 2349450/(5*x + 3)^5 + 157100/(5*x + 3)^6 + 55800576)/(1/(5*x + 3) + 3)^7 + 9212500*log(abs(-1/(5*x + 3)
- 3))