3.16.18 \(\int \frac {(2+3 x)^6}{(1-2 x) (3+5 x)^3} \, dx\) [1518]

Optimal. Leaf size=62 \[ -\frac {216999 x}{25000}-\frac {19683 x^2}{5000}-\frac {243 x^3}{250}-\frac {1}{343750 (3+5 x)^2}-\frac {8}{75625 (3+5 x)}-\frac {117649 \log (1-2 x)}{21296}+\frac {3347 \log (3+5 x)}{4159375} \]

[Out]

-216999/25000*x-19683/5000*x^2-243/250*x^3-1/343750/(3+5*x)^2-8/75625/(3+5*x)-117649/21296*ln(1-2*x)+3347/4159
375*ln(3+5*x)

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Rubi [A]
time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {90} \begin {gather*} -\frac {243 x^3}{250}-\frac {19683 x^2}{5000}-\frac {216999 x}{25000}-\frac {8}{75625 (5 x+3)}-\frac {1}{343750 (5 x+3)^2}-\frac {117649 \log (1-2 x)}{21296}+\frac {3347 \log (5 x+3)}{4159375} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-216999*x)/25000 - (19683*x^2)/5000 - (243*x^3)/250 - 1/(343750*(3 + 5*x)^2) - 8/(75625*(3 + 5*x)) - (117649*
Log[1 - 2*x])/21296 + (3347*Log[3 + 5*x])/4159375

Rule 90

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 {(2+3 x)^6}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac {216999}{25000}-\frac {19683 x}{2500}-\frac {729 x^2}{250}-\frac {117649}{10648 (-1+2 x)}+\frac {1}{34375 (3+5 x)^3}+\frac {8}{15125 (3+5 x)^2}+\frac {3347}{831875 (3+5 x)}\right ) \, dx\\ &=-\frac {216999 x}{25000}-\frac {19683 x^2}{5000}-\frac {243 x^3}{250}-\frac {1}{343750 (3+5 x)^2}-\frac {8}{75625 (3+5 x)}-\frac {117649 \log (1-2 x)}{21296}+\frac {3347 \log (3+5 x)}{4159375}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.90 \begin {gather*} \frac {11 \left (329460615-525137580 x-238164300 x^2-58806000 x^3-\frac {176}{(3+5 x)^2}-\frac {6400}{3+5 x}\right )-3676531250 \log (1-2 x)+535520 \log (6+10 x)}{665500000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(329460615 - 525137580*x - 238164300*x^2 - 58806000*x^3 - 176/(3 + 5*x)^2 - 6400/(3 + 5*x)) - 3676531250*L
og[1 - 2*x] + 535520*Log[6 + 10*x])/665500000

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Maple [A]
time = 0.11, size = 49, normalized size = 0.79

method	result	size
risch	\(-\frac {243 x^{3}}{250}-\frac {19683 x^{2}}{5000}-\frac {216999 x}{25000}+\frac {-\frac {8 x}{15125}-\frac {1211}{3781250}}{\left (3+5 x \right )^{2}}-\frac {117649 \ln \left (-1+2 x \right )}{21296}+\frac {3347 \ln \left (3+5 x \right )}{4159375}\)	\(45\)
default	\(-\frac {243 x^{3}}{250}-\frac {19683 x^{2}}{5000}-\frac {216999 x}{25000}-\frac {117649 \ln \left (-1+2 x \right )}{21296}-\frac {1}{343750 \left (3+5 x \right )^{2}}-\frac {8}{75625 \left (3+5 x \right )}+\frac {3347 \ln \left (3+5 x \right )}{4159375}\)	\(49\)
norman	\(\frac {-\frac {141786169}{1815000} x -\frac {322155941}{1089000} x^{2}-\frac {68769}{200} x^{3}-\frac {5103}{40} x^{4}-\frac {243}{10} x^{5}}{\left (3+5 x \right )^{2}}-\frac {117649 \ln \left (-1+2 x \right )}{21296}+\frac {3347 \ln \left (3+5 x \right )}{4159375}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^3-19683/5000*x^2-216999/25000*x-117649/21296*ln(-1+2*x)-1/343750/(3+5*x)^2-8/75625/(3+5*x)+3347/415
9375*ln(3+5*x)

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Maxima [A]
time = 0.29, size = 49, normalized size = 0.79 \begin {gather*} -\frac {243}{250} \, x^{3} - \frac {19683}{5000} \, x^{2} - \frac {216999}{25000} \, x - \frac {2000 \, x + 1211}{3781250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3347}{4159375} \, \log \left (5 \, x + 3\right ) - \frac {117649}{21296} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^3 - 19683/5000*x^2 - 216999/25000*x - 1/3781250*(2000*x + 1211)/(25*x^2 + 30*x + 9) + 3347/4159375*
log(5*x + 3) - 117649/21296*log(2*x - 1)

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Fricas [A]
time = 0.93, size = 75, normalized size = 1.21 \begin {gather*} -\frac {8085825000 \, x^{5} + 42450581250 \, x^{4} + 114414423750 \, x^{3} + 98436833550 \, x^{2} - 267760 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1838265625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 25994486210 \, x + 106568}{332750000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/332750000*(8085825000*x^5 + 42450581250*x^4 + 114414423750*x^3 + 98436833550*x^2 - 267760*(25*x^2 + 30*x +
9)*log(5*x + 3) + 1838265625*(25*x^2 + 30*x + 9)*log(2*x - 1) + 25994486210*x + 106568)/(25*x^2 + 30*x + 9)

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Sympy [A]
time = 0.08, size = 53, normalized size = 0.85 \begin {gather*} - \frac {243 x^{3}}{250} - \frac {19683 x^{2}}{5000} - \frac {216999 x}{25000} - \frac {2000 x + 1211}{94531250 x^{2} + 113437500 x + 34031250} - \frac {117649 \log {\left (x - \frac {1}{2} \right )}}{21296} + \frac {3347 \log {\left (x + \frac {3}{5} \right )}}{4159375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**3/250 - 19683*x**2/5000 - 216999*x/25000 - (2000*x + 1211)/(94531250*x**2 + 113437500*x + 34031250) -
117649*log(x - 1/2)/21296 + 3347*log(x + 3/5)/4159375

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Giac [A]
time = 1.68, size = 46, normalized size = 0.74 \begin {gather*} -\frac {243}{250} \, x^{3} - \frac {19683}{5000} \, x^{2} - \frac {216999}{25000} \, x - \frac {2000 \, x + 1211}{3781250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {3347}{4159375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {117649}{21296} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^3 - 19683/5000*x^2 - 216999/25000*x - 1/3781250*(2000*x + 1211)/(5*x + 3)^2 + 3347/4159375*log(abs(
5*x + 3)) - 117649/21296*log(abs(2*x - 1))

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Mupad [B]
time = 0.04, size = 43, normalized size = 0.69 \begin {gather*} \frac {3347\,\ln \left (x+\frac {3}{5}\right )}{4159375}-\frac {117649\,\ln \left (x-\frac {1}{2}\right )}{21296}-\frac {216999\,x}{25000}-\frac {\frac {8\,x}{378125}+\frac {1211}{94531250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {19683\,x^2}{5000}-\frac {243\,x^3}{250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3347*log(x + 3/5))/4159375 - (117649*log(x - 1/2))/21296 - (216999*x)/25000 - ((8*x)/378125 + 1211/94531250)/
((6*x)/5 + x^2 + 9/25) - (19683*x^2)/5000 - (243*x^3)/250

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