3.17.79 \(\int \frac {1}{(1-2 x)^3 (3+5 x)} \, dx\) [1679]

Optimal. Leaf size=43 \[ \frac {1}{22 (1-2 x)^2}+\frac {5}{121 (1-2 x)}-\frac {25 \log (1-2 x)}{1331}+\frac {25 \log (3+5 x)}{1331} \]

[Out]

1/22/(1-2*x)^2+5/121/(1-2*x)-25/1331*ln(1-2*x)+25/1331*ln(3+5*x)

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Rubi [A]
time = 0.01, antiderivative size = 43, 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 {5}{121 (1-2 x)}+\frac {1}{22 (1-2 x)^2}-\frac {25 \log (1-2 x)}{1331}+\frac {25 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^3*(3 + 5*x)),x]

[Out]

1/(22*(1 - 2*x)^2) + 5/(121*(1 - 2*x)) - (25*Log[1 - 2*x])/1331 + (25*Log[3 + 5*x])/1331

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)} \, dx &=\int \left (-\frac {2}{11 (-1+2 x)^3}+\frac {10}{121 (-1+2 x)^2}-\frac {50}{1331 (-1+2 x)}+\frac {125}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {1}{22 (1-2 x)^2}+\frac {5}{121 (1-2 x)}-\frac {25 \log (1-2 x)}{1331}+\frac {25 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 46, normalized size = 1.07 \begin {gather*} \frac {231-220 x-50 (1-2 x)^2 \log (1-2 x)+50 (1-2 x)^2 \log (6+10 x)}{2662 (1-2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(231 - 220*x - 50*(1 - 2*x)^2*Log[1 - 2*x] + 50*(1 - 2*x)^2*Log[6 + 10*x])/(2662*(1 - 2*x)^2)

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Maple [A]
time = 0.10, size = 36, normalized size = 0.84

method	result	size
risch	\(\frac {-\frac {10 x}{121}+\frac {21}{242}}{\left (-1+2 x \right )^{2}}-\frac {25 \ln \left (-1+2 x \right )}{1331}+\frac {25 \ln \left (3+5 x \right )}{1331}\)	\(32\)
norman	\(\frac {\frac {32}{121} x -\frac {42}{121} x^{2}}{\left (-1+2 x \right )^{2}}-\frac {25 \ln \left (-1+2 x \right )}{1331}+\frac {25 \ln \left (3+5 x \right )}{1331}\)	\(35\)
default	\(\frac {1}{22 \left (-1+2 x \right )^{2}}-\frac {5}{121 \left (-1+2 x \right )}-\frac {25 \ln \left (-1+2 x \right )}{1331}+\frac {25 \ln \left (3+5 x \right )}{1331}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22/(-1+2*x)^2-5/121/(-1+2*x)-25/1331*ln(-1+2*x)+25/1331*ln(3+5*x)

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Maxima [A]
time = 0.38, size = 36, normalized size = 0.84 \begin {gather*} -\frac {20 \, x - 21}{242 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {25}{1331} \, \log \left (5 \, x + 3\right ) - \frac {25}{1331} \, \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),x, algorithm="maxima")

[Out]

-1/242*(20*x - 21)/(4*x^2 - 4*x + 1) + 25/1331*log(5*x + 3) - 25/1331*log(2*x - 1)

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Fricas [A]
time = 0.41, size = 55, normalized size = 1.28 \begin {gather*} \frac {50 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 50 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 220 \, x + 231}{2662 \, {\left (4 \, x^{2} - 4 \, 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),x, algorithm="fricas")

[Out]

1/2662*(50*(4*x^2 - 4*x + 1)*log(5*x + 3) - 50*(4*x^2 - 4*x + 1)*log(2*x - 1) - 220*x + 231)/(4*x^2 - 4*x + 1)

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Sympy [A]
time = 0.06, size = 34, normalized size = 0.79 \begin {gather*} - \frac {20 x - 21}{968 x^{2} - 968 x + 242} - \frac {25 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {25 \log {\left (x + \frac {3}{5} \right )}}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*x - 21)/(968*x**2 - 968*x + 242) - 25*log(x - 1/2)/1331 + 25*log(x + 3/5)/1331

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Giac [A]
time = 1.54, size = 33, normalized size = 0.77 \begin {gather*} -\frac {20 \, x - 21}{242 \, {\left (2 \, x - 1\right )}^{2}} + \frac {25}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {25}{1331} \, \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),x, algorithm="giac")

[Out]

-1/242*(20*x - 21)/(2*x - 1)^2 + 25/1331*log(abs(5*x + 3)) - 25/1331*log(abs(2*x - 1))

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Mupad [B]
time = 0.04, size = 26, normalized size = 0.60 \begin {gather*} \frac {50\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {5\,x}{242}-\frac {21}{968}}{x^2-x+\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((2*x - 1)^3*(5*x + 3)),x)

[Out]

(50*atanh((20*x)/11 + 1/11))/1331 - ((5*x)/242 - 21/968)/(x^2 - x + 1/4)

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