3.1235 \(\int \frac{1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{189375}{3 x+2}+\frac{125000}{5 x+3}+\frac{12675}{(3 x+2)^2}-\frac{6875}{2 (5 x+3)^2}+\frac{1020}{(3 x+2)^3}+\frac{309}{4 (3 x+2)^4}+\frac{21}{5 (3 x+2)^5}-1321875 \log (3 x+2)+1321875 \log (5 x+3) \]

[Out]

21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3*x)^2 + 189375/(2 + 3*x) - 6875/(2*(
3 + 5*x)^2) + 125000/(3 + 5*x) - 1321875*Log[2 + 3*x] + 1321875*Log[3 + 5*x]

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Rubi [A]  time = 0.0460292, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{189375}{3 x+2}+\frac{125000}{5 x+3}+\frac{12675}{(3 x+2)^2}-\frac{6875}{2 (5 x+3)^2}+\frac{1020}{(3 x+2)^3}+\frac{309}{4 (3 x+2)^4}+\frac{21}{5 (3 x+2)^5}-1321875 \log (3 x+2)+1321875 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3*x)^2 + 189375/(2 + 3*x) - 6875/(2*(
3 + 5*x)^2) + 125000/(3 + 5*x) - 1321875*Log[2 + 3*x] + 1321875*Log[3 + 5*x]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx &=\int \left (-\frac{63}{(2+3 x)^6}-\frac{927}{(2+3 x)^5}-\frac{9180}{(2+3 x)^4}-\frac{76050}{(2+3 x)^3}-\frac{568125}{(2+3 x)^2}-\frac{3965625}{2+3 x}+\frac{34375}{(3+5 x)^3}-\frac{625000}{(3+5 x)^2}+\frac{6609375}{3+5 x}\right ) \, dx\\ &=\frac{21}{5 (2+3 x)^5}+\frac{309}{4 (2+3 x)^4}+\frac{1020}{(2+3 x)^3}+\frac{12675}{(2+3 x)^2}+\frac{189375}{2+3 x}-\frac{6875}{2 (3+5 x)^2}+\frac{125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0305123, size = 88, normalized size = 1.02 \[ \frac{189375}{3 x+2}+\frac{125000}{5 x+3}+\frac{12675}{(3 x+2)^2}-\frac{6875}{2 (5 x+3)^2}+\frac{1020}{(3 x+2)^3}+\frac{309}{4 (3 x+2)^4}+\frac{21}{5 (3 x+2)^5}-1321875 \log (3 x+2)+1321875 \log (-3 (5 x+3)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3*x)^2 + 189375/(2 + 3*x) - 6875/(2*(
3 + 5*x)^2) + 125000/(3 + 5*x) - 1321875*Log[2 + 3*x] + 1321875*Log[-3*(3 + 5*x)]

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Maple [A]  time = 0.008, size = 81, normalized size = 0.9 \begin{align*}{\frac{21}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{309}{4\, \left ( 2+3\,x \right ) ^{4}}}+1020\, \left ( 2+3\,x \right ) ^{-3}+12675\, \left ( 2+3\,x \right ) ^{-2}+189375\, \left ( 2+3\,x \right ) ^{-1}-{\frac{6875}{2\, \left ( 3+5\,x \right ) ^{2}}}+125000\, \left ( 3+5\,x \right ) ^{-1}-1321875\,\ln \left ( 2+3\,x \right ) +1321875\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/5/(2+3*x)^5+309/4/(2+3*x)^4+1020/(2+3*x)^3+12675/(2+3*x)^2+189375/(2+3*x)-6875/2/(3+5*x)^2+125000/(3+5*x)-1
321875*ln(2+3*x)+1321875*ln(3+5*x)

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Maxima [A]  time = 0.991766, size = 116, normalized size = 1.35 \begin{align*} \frac{10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 1321875 \, \log \left (5 \, x + 3\right ) - 1321875 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 7421662135*x +
802214966)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 1321875*lo
g(5*x + 3) - 1321875*log(3*x + 2)

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Fricas [A]  time = 1.48755, size = 579, normalized size = 6.73 \begin{align*} \frac{10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 26437500 \,{\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 ) - 26437500 \,{\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 ) + 7421662135 \, x + 802214966}{20 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 26437500*(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) - 26437500*(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) + 7421662135*x +
802214966)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)

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Sympy [A]  time = 0.222474, size = 82, normalized size = 0.95 \begin{align*} \frac{10707187500 x^{6} + 41758031250 x^{5} + 67828050000 x^{4} + 58733814375 x^{3} + 28595335800 x^{2} + 7421662135 x + 802214966}{121500 x^{7} + 550800 x^{6} + 1069740 x^{5} + 1153800 x^{4} + 746400 x^{3} + 289600 x^{2} + 62400 x + 5760} + 1321875 \log{\left (x + \frac{3}{5} \right )} - 1321875 \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+3*x)**6/(3+5*x)**3,x)

[Out]

(10707187500*x**6 + 41758031250*x**5 + 67828050000*x**4 + 58733814375*x**3 + 28595335800*x**2 + 7421662135*x +
802214966)/(121500*x**7 + 550800*x**6 + 1069740*x**5 + 1153800*x**4 + 746400*x**3 + 289600*x**2 + 62400*x + 5
760) + 1321875*log(x + 3/5) - 1321875*log(x + 2/3)

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Giac [A]  time = 2.38476, size = 88, normalized size = 1.02 \begin{align*} \frac{10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{5}} + 1321875 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 1321875 \, \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+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 7421662135*x +
802214966)/((5*x + 3)^2*(3*x + 2)^5) + 1321875*log(abs(5*x + 3)) - 1321875*log(abs(3*x + 2))