3.1336 \(\int \frac{(1-2 x)^2}{(2+3 x)^7 (3+5 x)^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{2958125}{3 x+2}+\frac{1615625}{5 x+3}+\frac{424975}{2 (3 x+2)^2}-\frac{75625}{2 (5 x+3)^2}+\frac{57110}{3 (3 x+2)^3}+\frac{3467}{2 (3 x+2)^4}+\frac{707}{5 (3 x+2)^5}+\frac{49}{6 (3 x+2)^6}-19637500 \log (3 x+2)[phone] \log (5 x+3) \]

[Out]

49/(6*(2 + 3*x)^6) + 707/(5*(2 + 3*x)^5) + 3467/(2*(2 + 3*x)^4) + 57110/(3*(2 + 3*x)^3) + 424975/(2*(2 + 3*x)^
2) + 2958125/(2 + 3*x) - 75625/(2*(3 + 5*x)^2) + 1615625/(3 + 5*x) - 19637500*Log[2 + 3*x] + 19637500*Log[3 +
5*x]

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Rubi [A]  time = 0.0539351, antiderivative size = 101, 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} \[ \frac{2958125}{3 x+2}+\frac{1615625}{5 x+3}+\frac{424975}{2 (3 x+2)^2}-\frac{75625}{2 (5 x+3)^2}+\frac{57110}{3 (3 x+2)^3}+\frac{3467}{2 (3 x+2)^4}+\frac{707}{5 (3 x+2)^5}+\frac{49}{6 (3 x+2)^6}-19637500 \log (3 x+2)[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(6*(2 + 3*x)^6) + 707/(5*(2 + 3*x)^5) + 3467/(2*(2 + 3*x)^4) + 57110/(3*(2 + 3*x)^3) + 424975/(2*(2 + 3*x)^
2) + 2958125/(2 + 3*x) - 75625/(2*(3 + 5*x)^2) + 1615625/(3 + 5*x) - 19637500*Log[2 + 3*x] + 19637500*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)^7 (3+5 x)^3} \, dx &=\int \left (-\frac{147}{(2+3 x)^7}-\frac{2121}{(2+3 x)^6}-\frac{20802}{(2+3 x)^5}-\frac{171330}{(2+3 x)^4}-\frac{1274925}{(2+3 x)^3}-\frac{8874375}{(2+3 x)^2}-\frac{58912500}{2+3 x}+\frac{378125}{(3+5 x)^3}-\frac{8078125}{(3+5 x)^2}+\frac{98187500}{3+5 x}\right ) \, dx\\ &=\frac{49}{6 (2+3 x)^6}+\frac{707}{5 (2+3 x)^5}+\frac{3467}{2 (2+3 x)^4}+\frac{57110}{3 (2+3 x)^3}+\frac{424975}{2 (2+3 x)^2}+\frac{2958125}{2+3 x}-\frac{75625}{2 (3+5 x)^2}+\frac{1615625}{3+5 x}-19637500 \log (2+3 x)[phone] \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0677463, size = 103, normalized size = 1.02 \[ \frac{2958125}{3 x+2}+\frac{1615625}{5 x+3}+\frac{424975}{2 (3 x+2)^2}-\frac{75625}{2 (5 x+3)^2}+\frac{57110}{3 (3 x+2)^3}+\frac{3467}{2 (3 x+2)^4}+\frac{707}{5 (3 x+2)^5}+\frac{49}{6 (3 x+2)^6}-19637500 \log (5 (3 x+2))[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(6*(2 + 3*x)^6) + 707/(5*(2 + 3*x)^5) + 3467/(2*(2 + 3*x)^4) + 57110/(3*(2 + 3*x)^3) + 424975/(2*(2 + 3*x)^
2) + 2958125/(2 + 3*x) - 75625/(2*(3 + 5*x)^2) + 1615625/(3 + 5*x) - 19637500*Log[5*(2 + 3*x)] + 19637500*Log[
3 + 5*x]

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*}{\frac{49}{6\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{707}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{3467}{2\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{57110}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{424975}{2\, \left ( 2+3\,x \right ) ^{2}}}+2958125\, \left ( 2+3\,x \right ) ^{-1}-{\frac{75625}{2\, \left ( 3+5\,x \right ) ^{2}}}+1615625\, \left ( 3+5\,x \right ) ^{-1}-19637500\,\ln \left ( 2+3\,x \right ) [phone]\,\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)^7/(3+5*x)^3,x)

[Out]

49/6/(2+3*x)^6+707/5/(2+3*x)^5+3467/2/(2+3*x)^4+57110/3/(2+3*x)^3+424975/2/(2+3*x)^2+2958125/(2+3*x)-75625/2/(
3+5*x)^2+1615625/(3+5*x)-19637500*ln(2+3*x)[phone]*ln(3+5*x)

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Maxima [A]  time = 1.00653, size = 130, normalized size = 1.29 \begin{align*} \frac{238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 128130976648 \, x + 11917538647}{10 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} + 19637500 \, \log \left (5 \, x + 3\right ) - 19637500 \, \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)^7/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 2316445391250*x^4 + 1509746867100*x^3 + 59018
8362770*x^2 + 128130976648*x + 11917538647)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 11
8080*x^3 + 38320*x^2 + 7104*x + 576) + 19637500*log(5*x + 3) - 19637500*log(3*x + 2)

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Fricas [A]  time = 1.40381, size = 694, normalized size = 6.87 \begin{align*} \frac{238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 196375000 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (5 \, x + 3\right ) - 196375000 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (3 \, x + 2\right ) + 128130976648 \, x + 11917538647}{10 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 2316445391250*x^4 + 1509746867100*x^3 + 59018
8362770*x^2 + 196375000*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2
+ 7104*x + 576)*log(5*x + 3) - 196375000*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 1180
80*x^3 + 38320*x^2 + 7104*x + 576)*log(3*x + 2) + 128130976648*x + 11917538647)/(18225*x^8 + 94770*x^7 + 21554
1*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)

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Sympy [A]  time = 0.230287, size = 92, normalized size = 0.91 \begin{align*} \frac{238595625000 x^{7} + 1089586687500 x^{6} + 2131807725000 x^{5} + 2316445391250 x^{4} + 1509746867100 x^{3} + 590188362770 x^{2} + 128130976648 x + 11917538647}{182250 x^{8} + 947700 x^{7} + 2155410 x^{6} + 2800440 x^{5} + 2273400 x^{4} + 1180800 x^{3} + 383200 x^{2} + 71040 x + 5760} + 19637500 \log{\left (x + \frac{3}{5} \right )} - 19637500 \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)**7/(3+5*x)**3,x)

[Out]

(238595625000*x**7 + 1089586687500*x**6 + 2131807725000*x**5 + 2316445391250*x**4 + 1509746867100*x**3 + 59018
8362770*x**2 + 128130976648*x + 11917538647)/(182250*x**8 + 947700*x**7 + 2155410*x**6 + 2800440*x**5 + 227340
0*x**4 + 1180800*x**3 + 383200*x**2 + 71040*x + 5760) + 19637500*log(x + 3/5) - 19637500*log(x + 2/3)

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Giac [A]  time = 3.27379, size = 95, normalized size = 0.94 \begin{align*} \frac{238595625000 \, x^{7} + 1089586687500 \, x^{6} + 2131807725000 \, x^{5} + 2316445391250 \, x^{4} + 1509746867100 \, x^{3} + 590188362770 \, x^{2} + 128130976648 \, x + 11917538647}{10 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{6}} + 19637500 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 19637500 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(238595625000*x^7 + 1089586687500*x^6 + 2131807725000*x^5 + 2316445391250*x^4 + 1509746867100*x^3 + 59018
8362770*x^2 + 128130976648*x + 11917538647)/((5*x + 3)^2*(3*x + 2)^6) + 19637500*log(abs(5*x + 3)) - 19637500*
log(abs(3*x + 2))