3.1431 \(\int \frac{(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx\)

Optimal. Leaf size=86 \[ \frac{953535}{3 x+2}+\frac{617100}{5 x+3}+\frac{64317}{(3 x+2)^2}-\frac{33275}{2 (5 x+3)^2}+\frac{5236}{(3 x+2)^3}+\frac{1617}{4 (3 x+2)^4}+\frac{343}{15 (3 x+2)^5}-6618975 \log (3 x+2)+6618975 \log (5 x+3) \]

[Out]

343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/
(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6618975*Log[2 + 3*x] + 6618975*Log[3 + 5*x]

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Rubi [A]  time = 0.0480218, antiderivative size = 86, 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} \[ \frac{953535}{3 x+2}+\frac{617100}{5 x+3}+\frac{64317}{(3 x+2)^2}-\frac{33275}{2 (5 x+3)^2}+\frac{5236}{(3 x+2)^3}+\frac{1617}{4 (3 x+2)^4}+\frac{343}{15 (3 x+2)^5}-6618975 \log (3 x+2)+6618975 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/
(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6618975*Log[2 + 3*x] + 6618975*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)^3}{(2+3 x)^6 (3+5 x)^3} \, dx &=\int \left (-\frac{343}{(2+3 x)^6}-\frac{4851}{(2+3 x)^5}-\frac{47124}{(2+3 x)^4}-\frac{385902}{(2+3 x)^3}-\frac{2860605}{(2+3 x)^2}-\frac{19856925}{2+3 x}+\frac{166375}{(3+5 x)^3}-\frac{3085500}{(3+5 x)^2}+\frac{33094875}{3+5 x}\right ) \, dx\\ &=\frac{343}{15 (2+3 x)^5}+\frac{1617}{4 (2+3 x)^4}+\frac{5236}{(2+3 x)^3}+\frac{64317}{(2+3 x)^2}+\frac{953535}{2+3 x}-\frac{33275}{2 (3+5 x)^2}+\frac{617100}{3+5 x}-6618975 \log (2+3 x)+6618975 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0648542, size = 88, normalized size = 1.02 \[ \frac{953535}{3 x+2}+\frac{617100}{5 x+3}+\frac{64317}{(3 x+2)^2}-\frac{33275}{2 (5 x+3)^2}+\frac{5236}{(3 x+2)^3}+\frac{1617}{4 (3 x+2)^4}+\frac{343}{15 (3 x+2)^5}-6618975 \log (5 (3 x+2))+6618975 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/
(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6618975*Log[5*(2 + 3*x)] + 6618975*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 81, normalized size = 0.9 \begin{align*}{\frac{343}{15\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{1617}{4\, \left ( 2+3\,x \right ) ^{4}}}+5236\, \left ( 2+3\,x \right ) ^{-3}+64317\, \left ( 2+3\,x \right ) ^{-2}+953535\, \left ( 2+3\,x \right ) ^{-1}-{\frac{33275}{2\, \left ( 3+5\,x \right ) ^{2}}}+617100\, \left ( 3+5\,x \right ) ^{-1}-6618975\,\ln \left ( 2+3\,x \right ) +6618975\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/15/(2+3*x)^5+1617/4/(2+3*x)^4+5236/(2+3*x)^3+64317/(2+3*x)^2+953535/(2+3*x)-33275/2/(3+5*x)^2+617100/(3+5*
x)-6618975*ln(2+3*x)+6618975*ln(3+5*x)

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Maxima [A]  time = 1.02518, size = 116, normalized size = 1.35 \begin{align*} \frac{160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \,{\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 6618975 \, \log \left (5 \, x + 3\right ) - 6618975 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 11148662
9505*x + 12050702538)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) +
6618975*log(5*x + 3) - 6618975*log(3*x + 2)

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Fricas [A]  time = 1.35695, size = 595, normalized size = 6.92 \begin{align*} \frac{160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 397138500 \,{\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 ) - 397138500 \,{\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 ) + 111486629505 \, x + 12050702538}{60 \,{\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)^3/(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 39713850
0*(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) - 3971385
00*(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) + 111486
629505*x + 12050702538)/(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 = 0.214785, size = 82, normalized size = 0.95 \begin{align*} \frac{160841092500 x^{6} + 627280260750 x^{5} + 1018898535600 x^{4} + 882286862985 x^{3} + 429553050280 x^{2} + 111486629505 x + 12050702538}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 6618975 \log{\left (x + \frac{3}{5} \right )} - 6618975 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(160841092500*x**6 + 627280260750*x**5 + 1018898535600*x**4 + 882286862985*x**3 + 429553050280*x**2 + 11148662
9505*x + 12050702538)/(364500*x**7 + 1652400*x**6 + 3209220*x**5 + 3461400*x**4 + 2239200*x**3 + 868800*x**2 +
187200*x + 17280) + 6618975*log(x + 3/5) - 6618975*log(x + 2/3)

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Giac [A]  time = 2.50877, size = 88, normalized size = 1.02 \begin{align*} \frac{160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{5}} + 6618975 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 6618975 \, \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)^3/(2+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 11148662
9505*x + 12050702538)/((5*x + 3)^2*(3*x + 2)^5) + 6618975*log(abs(5*x + 3)) - 6618975*log(abs(3*x + 2))