3.18.6 \(\int \frac {1}{(1-2 x)^3 (3+5 x)^3} \, dx\) [1706]

Optimal. Leaf size=65 \[ \frac {2}{1331 (1-2 x)^2}+\frac {60}{14641 (1-2 x)}-\frac {25}{2662 (3+5 x)^2}-\frac {150}{14641 (3+5 x)}-\frac {600 \log (1-2 x)}{161051}+\frac {600 \log (3+5 x)}{161051} \]

[Out]

2/1331/(1-2*x)^2+60/14641/(1-2*x)-25/2662/(3+5*x)^2-150/14641/(3+5*x)-600/161051*ln(1-2*x)+600/161051*ln(3+5*x
)

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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {46} \begin {gather*} \frac {60}{14641 (1-2 x)}-\frac {150}{14641 (5 x+3)}+\frac {2}{1331 (1-2 x)^2}-\frac {25}{2662 (5 x+3)^2}-\frac {600 \log (1-2 x)}{161051}+\frac {600 \log (5 x+3)}{161051} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

2/(1331*(1 - 2*x)^2) + 60/(14641*(1 - 2*x)) - 25/(2662*(3 + 5*x)^2) - 150/(14641*(3 + 5*x)) - (600*Log[1 - 2*x
])/161051 + (600*Log[3 + 5*x])/161051

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {8}{1331 (-1+2 x)^3}+\frac {120}{14641 (-1+2 x)^2}-\frac {1200}{161051 (-1+2 x)}+\frac {125}{1331 (3+5 x)^3}+\frac {750}{14641 (3+5 x)^2}+\frac {3000}{161051 (3+5 x)}\right ) \, dx\\ &=\frac {2}{1331 (1-2 x)^2}+\frac {60}{14641 (1-2 x)}-\frac {25}{2662 (3+5 x)^2}-\frac {150}{14641 (3+5 x)}-\frac {600 \log (1-2 x)}{161051}+\frac {600 \log (3+5 x)}{161051}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 48, normalized size = 0.74 \begin {gather*} \frac {-\frac {11 \left (-301-5960 x+1800 x^2+12000 x^3\right )}{\left (-3+x+10 x^2\right )^2}-1200 \log (1-2 x)+1200 \log (3+5 x)}{322102} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-301 - 5960*x + 1800*x^2 + 12000*x^3))/(-3 + x + 10*x^2)^2 - 1200*Log[1 - 2*x] + 1200*Log[3 + 5*x])/322
102

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Maple [A]
time = 0.09, size = 54, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {6000}{14641} x^{3}-\frac {900}{14641} x^{2}+\frac {2980}{14641} x +\frac {301}{29282}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {600 \ln \left (-1+2 x \right )}{161051}+\frac {600 \ln \left (3+5 x \right )}{161051}\)	\(49\)
norman	\(\frac {\frac {149000}{43923} x^{4}+\frac {11800}{43923} x^{3}-\frac {90610}{43923} x^{2}+\frac {9241}{29282}}{\left (-1+2 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {600 \ln \left (-1+2 x \right )}{161051}+\frac {600 \ln \left (3+5 x \right )}{161051}\)	\(50\)
default	\(\frac {2}{1331 \left (-1+2 x \right )^{2}}-\frac {60}{14641 \left (-1+2 x \right )}-\frac {600 \ln \left (-1+2 x \right )}{161051}-\frac {25}{2662 \left (3+5 x \right )^{2}}-\frac {150}{14641 \left (3+5 x \right )}+\frac {600 \ln \left (3+5 x \right )}{161051}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^3/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

2/1331/(-1+2*x)^2-60/14641/(-1+2*x)-600/161051*ln(-1+2*x)-25/2662/(3+5*x)^2-150/14641/(3+5*x)+600/161051*ln(3+
5*x)

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Maxima [A]
time = 0.41, size = 56, normalized size = 0.86 \begin {gather*} -\frac {12000 \, x^{3} + 1800 \, x^{2} - 5960 \, x - 301}{29282 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {600}{161051} \, \log \left (5 \, x + 3\right ) - \frac {600}{161051} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(12000*x^3 + 1800*x^2 - 5960*x - 301)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 600/161051*log(5*x + 3)
- 600/161051*log(2*x - 1)

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Fricas [A]
time = 0.78, size = 95, normalized size = 1.46 \begin {gather*} -\frac {132000 \, x^{3} + 19800 \, x^{2} - 1200 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1200 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 65560 \, x - 3311}{322102 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/322102*(132000*x^3 + 19800*x^2 - 1200*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 1200*(100*x^4 +
20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) - 65560*x - 3311)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [A]
time = 0.08, size = 54, normalized size = 0.83 \begin {gather*} - \frac {12000 x^{3} + 1800 x^{2} - 5960 x - 301}{2928200 x^{4} + 585640 x^{3} - 1727638 x^{2} - 175692 x + 263538} - \frac {600 \log {\left (x - \frac {1}{2} \right )}}{161051} + \frac {600 \log {\left (x + \frac {3}{5} \right )}}{161051} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12000*x**3 + 1800*x**2 - 5960*x - 301)/(2928200*x**4 + 585640*x**3 - 1727638*x**2 - 175692*x + 263538) - 600
*log(x - 1/2)/161051 + 600*log(x + 3/5)/161051

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Giac [A]
time = 0.62, size = 46, normalized size = 0.71 \begin {gather*} -\frac {12000 \, x^{3} + 1800 \, x^{2} - 5960 \, x - 301}{29282 \, {\left (10 \, x^{2} + x - 3\right )}^{2}} + \frac {600}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {600}{161051} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(12000*x^3 + 1800*x^2 - 5960*x - 301)/(10*x^2 + x - 3)^2 + 600/161051*log(abs(5*x + 3)) - 600/161051*
log(abs(2*x - 1))

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Mupad [B]
time = 1.06, size = 45, normalized size = 0.69 \begin {gather*} \frac {1200\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{161051}+\frac {-\frac {60\,x^3}{14641}-\frac {9\,x^2}{14641}+\frac {149\,x}{73205}+\frac {301}{2928200}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((2*x - 1)^3*(5*x + 3)^3),x)

[Out]

(1200*atanh((20*x)/11 + 1/11))/161051 + ((149*x)/73205 - (9*x^2)/14641 - (60*x^3)/14641 + 301/2928200)/(x^3/5
- (59*x^2)/100 - (3*x)/50 + x^4 + 9/100)

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