3.1513 \(\int \frac{1}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{34371}{343 (3 x+2)}-\frac{625}{11 (5 x+3)}-\frac{324}{49 (3 x+2)^2}-\frac{3}{7 (3 x+2)^3}-\frac{32 \log (1-2 x)}{290521}+\frac{1612242 \log (3 x+2)}{2401}-\frac{81250}{121} \log (5 x+3) \]

[Out]

-3/(7*(2 + 3*x)^3) - 324/(49*(2 + 3*x)^2) - 34371/(343*(2 + 3*x)) - 625/(11*(3 + 5*x)) - (32*Log[1 - 2*x])/290
521 + (1612242*Log[2 + 3*x])/2401 - (81250*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0348619, antiderivative size = 75, 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} \[ -\frac{34371}{343 (3 x+2)}-\frac{625}{11 (5 x+3)}-\frac{324}{49 (3 x+2)^2}-\frac{3}{7 (3 x+2)^3}-\frac{32 \log (1-2 x)}{290521}+\frac{1612242 \log (3 x+2)}{2401}-\frac{81250}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-3/(7*(2 + 3*x)^3) - 324/(49*(2 + 3*x)^2) - 34371/(343*(2 + 3*x)) - 625/(11*(3 + 5*x)) - (32*Log[1 - 2*x])/290
521 + (1612242*Log[2 + 3*x])/2401 - (81250*Log[3 + 5*x])/121

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}{(1-2 x) (2+3 x)^4 (3+5 x)^2} \, dx &=\int \left (-\frac{64}{290521 (-1+2 x)}+\frac{27}{7 (2+3 x)^4}+\frac{1944}{49 (2+3 x)^3}+\frac{103113}{343 (2+3 x)^2}+\frac{4836726}{2401 (2+3 x)}+\frac{3125}{11 (3+5 x)^2}-\frac{406250}{121 (3+5 x)}\right ) \, dx\\ &=-\frac{3}{7 (2+3 x)^3}-\frac{324}{49 (2+3 x)^2}-\frac{34371}{343 (2+3 x)}-\frac{625}{11 (3+5 x)}-\frac{32 \log (1-2 x)}{290521}+\frac{1612242 \log (2+3 x)}{2401}-\frac{81250}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0765995, size = 62, normalized size = 0.83 \[ \frac{2 \left (-\frac{77 \left (22801770 x^3+44843517 x^2+29372133 x+6406511\right )}{2 (3 x+2)^3 (5 x+3)}-16 \log (1-2 x)[phone] \log (6 x+4)-97540625 \log (10 x+6)\right )}{290521} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)*(2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

(2*((-77*(6406511 + 29372133*x + 44843517*x^2 + 22801770*x^3))/(2*(2 + 3*x)^3*(3 + 5*x)) - 16*Log[1 - 2*x] + 9
7540641*Log[4 + 6*x] - 97540625*Log[6 + 10*x]))/290521

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Maple [A]  time = 0.01, size = 62, normalized size = 0.8 \begin{align*} -{\frac{32\,\ln \left ( 2\,x-1 \right ) }{290521}}-{\frac{3}{7\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{324}{49\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{34371}{686+1029\,x}}+{\frac{1612242\,\ln \left ( 2+3\,x \right ) }{2401}}-{\frac{625}{33+55\,x}}-{\frac{81250\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x)

[Out]

-32/290521*ln(2*x-1)-3/7/(2+3*x)^3-324/49/(2+3*x)^2-34371/343/(2+3*x)+1612242/2401*ln(2+3*x)-625/11/(3+5*x)-81
250/121*ln(3+5*x)

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Maxima [A]  time = 1.06427, size = 86, normalized size = 1.15 \begin{align*} -\frac{22801770 \, x^{3} + 44843517 \, x^{2} + 29372133 \, x + 6406511}{3773 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} - \frac{81250}{121} \, \log \left (5 \, x + 3\right ) + \frac{1612242}{2401} \, \log \left (3 \, x + 2\right ) - \frac{32}{290521} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/3773*(22801770*x^3 + 44843517*x^2 + 29372133*x + 6406511)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24) - 8125
0/121*log(5*x + 3) + 1612242/2401*log(3*x + 2) - 32/290521*log(2*x - 1)

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Fricas [B]  time = 1.53302, size = 421, normalized size = 5.61 \begin{align*} -\frac{1755736290 \, x^{3} + 3452950809 \, x^{2} + 195081250 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 195081282 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 32 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (2 \, x - 1\right ) + 2261654241 \, x + 493301347}{290521 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/290521*(1755736290*x^3 + 3452950809*x^2 + 195081250*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(5*x + 3)
- 195081282*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(3*x + 2) + 32*(135*x^4 + 351*x^3 + 342*x^2 + 148*x
+ 24)*log(2*x - 1) + 2261654241*x + 493301347)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

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Sympy [A]  time = 0.220775, size = 65, normalized size = 0.87 \begin{align*} - \frac{22801770 x^{3} + 44843517 x^{2} + 29372133 x + 6406511}{509355 x^{4} + 1324323 x^{3} + 1290366 x^{2} + 558404 x + 90552} - \frac{32 \log{\left (x - \frac{1}{2} \right )}}{290521} - \frac{81250 \log{\left (x + \frac{3}{5} \right )}}{121} + \frac{1612242 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)**4/(3+5*x)**2,x)

[Out]

-(22801770*x**3 + 44843517*x**2 + 29372133*x + 6406511)/(509355*x**4 + 1324323*x**3 + 1290366*x**2 + 558404*x
+ 90552) - 32*log(x - 1/2)/290521 - 81250*log(x + 3/5)/121 + 1612242*log(x + 2/3)/2401

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Giac [A]  time = 2.28156, size = 99, normalized size = 1.32 \begin{align*} -\frac{625}{11 \,{\left (5 \, x + 3\right )}} + \frac{135 \,{\left (\frac{37929}{5 \, x + 3} + \frac{7564}{{\left (5 \, x + 3\right )}^{2}} + 49386\right )}}{343 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{3}} + \frac{1612242}{2401} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{32}{290521} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-625/11/(5*x + 3) + 135/343*(37929/(5*x + 3) + 7564/(5*x + 3)^2 + 49386)/(1/(5*x + 3) + 3)^3 + 1612242/2401*lo
g(abs(-1/(5*x + 3) - 3)) - 32/290521*log(abs(-11/(5*x + 3) + 2))