3.1417 \(\int \frac{(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{3584625}{3 x+2}-\frac{831875}{5 x+3}-\frac{308550}{(3 x+2)^2}-\frac{34485}{(3 x+2)^3}-\frac{8349}{2 (3 x+2)^4}-\frac{2541}{5 (3 x+2)^5}-\frac{1568}{27 (3 x+2)^6}-\frac{49}{9 (3 x+2)^7}[phone] \log (3 x+2)-20418750 \log (5 x+3) \]

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[2 + 3*x] - 20418750*Log[3 + 5*x]

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Rubi [A]  time = 0.0503811, 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} \[ -\frac{3584625}{3 x+2}-\frac{831875}{5 x+3}-\frac{308550}{(3 x+2)^2}-\frac{34485}{(3 x+2)^3}-\frac{8349}{2 (3 x+2)^4}-\frac{2541}{5 (3 x+2)^5}-\frac{1568}{27 (3 x+2)^6}-\frac{49}{9 (3 x+2)^7}[phone] \log (3 x+2)-20418750 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[2 + 3*x] - 20418750*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)^8 (3+5 x)^2} \, dx &=\int \left (\frac{343}{3 (2+3 x)^8}+\frac{3136}{3 (2+3 x)^7}+\frac{7623}{(2+3 x)^6}+\frac{50094}{(2+3 x)^5}+\frac{310365}{(2+3 x)^4}+\frac{1851300}{(2+3 x)^3}+\frac{10753875}{(2+3 x)^2}+\frac{61256250}{2+3 x}+\frac{4159375}{(3+5 x)^2}-\frac{102093750}{3+5 x}\right ) \, dx\\ &=-\frac{49}{9 (2+3 x)^7}-\frac{1568}{27 (2+3 x)^6}-\frac{2541}{5 (2+3 x)^5}-\frac{8349}{2 (2+3 x)^4}-\frac{34485}{(2+3 x)^3}-\frac{308550}{(2+3 x)^2}-\frac{3584625}{2+3 x}-\frac{831875}{3+5 x}[phone] \log (2+3 x)-20418750 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0943523, size = 99, normalized size = 1.02 \[ -\frac{3584625}{3 x+2}-\frac{831875}{5 x+3}-\frac{308550}{(3 x+2)^2}-\frac{34485}{(3 x+2)^3}-\frac{8349}{2 (3 x+2)^4}-\frac{2541}{5 (3 x+2)^5}-\frac{1568}{27 (3 x+2)^6}-\frac{49}{9 (3 x+2)^7}[phone] \log (5 (3 x+2))-20418750 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[5*(2 + 3*x)] - 20418750*Log[3 + 5*x
]

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Maple [A]  time = 0.008, size = 90, normalized size = 0.9 \begin{align*} -{\frac{49}{9\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{1568}{27\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{2541}{5\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{8349}{2\, \left ( 2+3\,x \right ) ^{4}}}-34485\, \left ( 2+3\,x \right ) ^{-3}-308550\, \left ( 2+3\,x \right ) ^{-2}-3584625\, \left ( 2+3\,x \right ) ^{-1}-831875\, \left ( 3+5\,x \right ) ^{-1}[phone]\,\ln \left ( 2+3\,x \right ) -20418750\,\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)^8/(3+5*x)^2,x)

[Out]

-49/9/(2+3*x)^7-1568/27/(2+3*x)^6-2541/5/(2+3*x)^5-8349/2/(2+3*x)^4-34485/(2+3*x)^3-308550/(2+3*x)^2-3584625/(
2+3*x)-831875/(3+5*x)[phone]*ln(2+3*x)-20418750*ln(3+5*x)

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Maxima [A]  time = 1.02364, size = 130, normalized size = 1.34 \begin{align*} -\frac{4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 2360937751874 \, x + 223049897418}{270 \,{\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 )}} - 20418750 \, \log \left (5 \, x + 3\right ) + 20418750 \, \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)^8/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 2360937751874*x + 223049897418)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 1436
40*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384) - 20418750*log(5*x + 3) + 20418750*log(3*x + 2)

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Fricas [A]  time = 1.32422, size = 707, normalized size = 7.29 \begin{align*} -\frac{4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 5513062500 \,{\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 ) - 5513062500 \,{\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 ) + 2360937751874 \, x + 223049897418}{270 \,{\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)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 5513062500*(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) - 5513062500*(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) + 2360937751874*x + 223049897418)/(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 = 0.236657, size = 92, normalized size = 0.95 \begin{align*} - \frac{4019022562500 x^{7} + 18621471206250 x^{6} + 36972030521250 x^{5} + 40775613627375 x^{4} + 26978454053595 x^{3} + 10708299857748 x^{2} + 2360937751874 x + 223049897418}{2952450 x^{8} + 15549570 x^{7} + 35823060 x^{6} + 47151720 x^{5} + 38782800 x^{4} + 20412000 x^{3} + 6713280 x^{2} + 1261440 x + 103680} - 20418750 \log{\left (x + \frac{3}{5} \right )} + 20418750 \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)**8/(3+5*x)**2,x)

[Out]

-(4019022562500*x**7 + 18621471206250*x**6 + 36972030521250*x**5 + 40775613627375*x**4 + 26978454053595*x**3 +
10708299857748*x**2 + 2360937751874*x + 223049897418)/(2952450*x**8 + 15549570*x**7 + 35823060*x**6 + 4715172
0*x**5 + 38782800*x**4 + 20412000*x**3 + 6713280*x**2 + 1261440*x + 103680) - 20418750*log(x + 3/5) + 20418750
*log(x + 2/3)

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Giac [A]  time = 3.58906, size = 127, normalized size = 1.31 \begin{align*} -\frac{831875}{5 \, x + 3} + \frac{625 \,{\left (\frac{537521373}{5 \, x + 3} + \frac{489712095}{{\left (5 \, x + 3\right )}^{2}} + \frac{241051911}{{\left (5 \, x + 3\right )}^{3}} + \frac{67932770}{{\left (5 \, x + 3\right )}^{4}} + \frac{10476370}{{\left (5 \, x + 3\right )}^{5}} + \frac{701580}{{\left (5 \, x + 3\right )}^{6}} + 248285331\right )}}{2 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{7}} + 20418750 \, \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)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="giac")

[Out]

-831875/(5*x + 3) + 625/2*(537521373/(5*x + 3) + 489712095/(5*x + 3)^2 + 241051911/(5*x + 3)^3 + 67932770/(5*x
+ 3)^4 + 10476370/(5*x + 3)^5 + 701580/(5*x + 3)^6 + 248285331)/(1/(5*x + 3) + 3)^7 + 20418750*log(abs(-1/(5*
x + 3) - 3))