3.17.28 \(\int \frac {1}{(1-2 x)^2 (2+3 x)^3 (3+5 x)^3} \, dx\) [1628]

Optimal. Leaf size=86 \[ \frac {32}{456533 (1-2 x)}+\frac {81}{98 (2+3 x)^2}+\frac {8829}{343 (2+3 x)}-\frac {625}{242 (3+5 x)^2}+\frac {59375}{1331 (3+5 x)}-\frac {6528 \log (1-2 x)}{35153041}-\frac {630342 \log (2+3 x)}{2401}+\frac {3843750 \log (3+5 x)}{14641} \]

[Out]

32/456533/(1-2*x)+81/98/(2+3*x)^2+8829/343/(2+3*x)-625/242/(3+5*x)^2+59375/1331/(3+5*x)-6528/35153041*ln(1-2*x
)-630342/2401*ln(2+3*x)+3843750/14641*ln(3+5*x)

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Rubi [A]
time = 0.03, antiderivative size = 86, 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 {32}{456533 (1-2 x)}+\frac {8829}{343 (3 x+2)}+\frac {59375}{1331 (5 x+3)}+\frac {81}{98 (3 x+2)^2}-\frac {625}{242 (5 x+3)^2}-\frac {6528 \log (1-2 x)}{35153041}-\frac {630342 \log (3 x+2)}{2401}+\frac {3843750 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

32/(456533*(1 - 2*x)) + 81/(98*(2 + 3*x)^2) + 8829/(343*(2 + 3*x)) - 625/(242*(3 + 5*x)^2) + 59375/(1331*(3 +
5*x)) - (6528*Log[1 - 2*x])/35153041 - (630342*Log[2 + 3*x])/2401 + (3843750*Log[3 + 5*x])/14641

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 {1}{(1-2 x)^2 (2+3 x)^3 (3+5 x)^3} \, dx &=\int \left (\frac {64}{456533 (-1+2 x)^2}-\frac {13056}{35153041 (-1+2 x)}-\frac {243}{49 (2+3 x)^3}-\frac {26487}{343 (2+3 x)^2}-\frac {1891026}{2401 (2+3 x)}+\frac {3125}{121 (3+5 x)^3}-\frac {296875}{1331 (3+5 x)^2}+\frac {19218750}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {32}{456533 (1-2 x)}+\frac {81}{98 (2+3 x)^2}+\frac {8829}{343 (2+3 x)}-\frac {625}{242 (3+5 x)^2}+\frac {59375}{1331 (3+5 x)}-\frac {6528 \log (1-2 x)}{35153041}-\frac {630342 \log (2+3 x)}{2401}+\frac {3843750 \log (3+5 x)}{14641}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 79, normalized size = 0.92 \begin {gather*} \frac {2 \left (\frac {77}{4} \left (\frac {64}{1-2 x}+\frac {754677}{(2+3 x)^2}+\frac {23502798}{2+3 x}-\frac {2358125}{(3+5 x)^2}+\frac {40731250}{3+5 x}\right )-3264 \log (1-2 x)-4614418611 \log (4+6 x)[phone] \log (6+10 x)\right )}{35153041} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((77*(64/(1 - 2*x) + 754677/(2 + 3*x)^2 + 23502798/(2 + 3*x) - 2358125/(3 + 5*x)^2 + 40731250/(3 + 5*x)))/4
- 3264*Log[1 - 2*x] - 4614418611*Log[4 + 6*x] + 4614421875*Log[6 + 10*x]))/35153041

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Maple [A]
time = 0.09, size = 71, normalized size = 0.83

method	result	size
risch	\(\frac {\frac {3595608900}{456533} x^{4}+\frac {5033827980}{456533} x^{3}+\frac {904191789}{456533} x^{2}-\frac {1250840457}{456533} x -\frac {909187261}{913066}}{\left (-1+2 x \right ) \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {6528 \ln \left (-1+2 x \right )}{35153041}-\frac {630342 \ln \left (2+3 x \right )}{2401}+\frac {3843750 \ln \left (3+5 x \right )}{14641}\)	\(69\)
default	\(-\frac {32}{456533 \left (-1+2 x \right )}-\frac {6528 \ln \left (-1+2 x \right )}{35153041}+\frac {81}{98 \left (2+3 x \right )^{2}}+\frac {8829}{343 \left (2+3 x \right )}-\frac {630342 \ln \left (2+3 x \right )}{2401}-\frac {625}{242 \left (3+5 x \right )^{2}}+\frac {59375}{1331 \left (3+5 x \right )}+\frac {3843750 \ln \left (3+5 x \right )}{14641}\)	\(71\)
norman	\(\frac {-\frac {191007501005}{10956792} x^{4}-\frac {12883532884}{4108797} x^{3}-\frac {22729681525}{1826132} x^{5}+\frac {4314391651}{2739198} x +\frac {142382725993}{32870376} x^{2}}{\left (-1+2 x \right ) \left (2+3 x \right )^{2} \left (3+5 x \right )^{2}}-\frac {6528 \ln \left (-1+2 x \right )}{35153041}-\frac {630342 \ln \left (2+3 x \right )}{2401}+\frac {3843750 \ln \left (3+5 x \right )}{14641}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/456533/(-1+2*x)-6528/35153041*ln(-1+2*x)+81/98/(2+3*x)^2+8829/343/(2+3*x)-630342/2401*ln(2+3*x)-625/242/(3
+5*x)^2+59375/1331/(3+5*x)+3843750/14641*ln(3+5*x)

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Maxima [A]
time = 0.47, size = 74, normalized size = 0.86 \begin {gather*} \frac {7191217800 \, x^{4} + 10067655960 \, x^{3} + 1808383578 \, x^{2} - 2501680914 \, x - 909187261}{913066 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} + \frac {3843750}{14641} \, \log \left (5 \, x + 3\right ) - \frac {630342}{2401} \, \log \left (3 \, x + 2\right ) - \frac {6528}{35153041} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/913066*(7191217800*x^4 + 10067655960*x^3 + 1808383578*x^2 - 2501680914*x - 909187261)/(450*x^5 + 915*x^4 + 5
12*x^3 - 85*x^2 - 156*x - 36) + 3843750/14641*log(5*x + 3) - 630342/2401*log(3*x + 2) - 6528/35153041*log(2*x
- 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (70) = 140\).
time = 0.39, size = 148, normalized size = 1.72 \begin {gather*} \frac {553723770600 \, x^{4} + 775209508920 \, x^{3} + 139245535506 \, x^{2} + 18457687500 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (5 \, x + 3\right ) - 18457674444 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (3 \, x + 2\right ) - 13056 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (2 \, x - 1\right ) - 192629430378 \, x - 70007419097}{70306082 \, {\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/70306082*(553723770600*x^4 + 775209508920*x^3 + 139245535506*x^2 + 18457687500*(450*x^5 + 915*x^4 + 512*x^3
- 85*x^2 - 156*x - 36)*log(5*x + 3) - 18457674444*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*log(3*x
+ 2) - 13056*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*log(2*x - 1) - 192629430378*x - 70007419097)/
(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)

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Sympy [A]
time = 0.12, size = 75, normalized size = 0.87 \begin {gather*} \frac {7191217800 x^{4} + 10067655960 x^{3} + 1808383578 x^{2} - 2501680914 x - 909187261}{410879700 x^{5} + 835455390 x^{4} + 467489792 x^{3} - 77610610 x^{2} - 142438296 x - 32870376} - \frac {6528 \log {\left (x - \frac {1}{2} \right )}}{35153041} + \frac {3843750 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {630342 \log {\left (x + \frac {2}{3} \right )}}{2401} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7191217800*x**4 + 10067655960*x**3 + 1808383578*x**2 - 2501680914*x - 909187261)/(410879700*x**5 + 835455390*
x**4 + 467489792*x**3 - 77610610*x**2 - 142438296*x - 32870376) - 6528*log(x - 1/2)/35153041 + 3843750*log(x +
3/5)/14641 - 630342*log(x + 2/3)/2401

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Giac [A]
time = 0.57, size = 97, normalized size = 1.13 \begin {gather*} -\frac {32}{456533 \, {\left (2 \, x - 1\right )}} - \frac {4 \, {\left (\frac {207724651275}{2 \, x - 1} + \frac {470659858850}{{\left (2 \, x - 1\right )}^{2}} + \frac {355299675423}{{\left (2 \, x - 1\right )}^{3}} + 30544881750\right )}}{35153041 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2} {\left (\frac {7}{2 \, x - 1} + 3\right )}^{2}} - \frac {630342}{2401} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {3843750}{14641} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/456533/(2*x - 1) - 4/35153041*(207724651275/(2*x - 1) + 470659858850/(2*x - 1)^2 + 355299675423/(2*x - 1)^
3 + 30544881750)/((11/(2*x - 1) + 5)^2*(7/(2*x - 1) + 3)^2) - 630342/2401*log(abs(-7/(2*x - 1) - 3)) + 3843750
/14641*log(abs(-11/(2*x - 1) - 5))

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Mupad [B]
time = 1.07, size = 68, normalized size = 0.79 \begin {gather*} \frac {3843750\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {630342\,\ln \left (x+\frac {2}{3}\right )}{2401}-\frac {6528\,\ln \left (x-\frac {1}{2}\right )}{35153041}-\frac {\frac {7990242\,x^4}{456533}+\frac {55931422\,x^3}{2282665}+\frac {301397263\,x^2}{68479950}-\frac {138982273\,x}{22826650}-\frac {909187261}{410879700}}{-x^5-\frac {61\,x^4}{30}-\frac {256\,x^3}{225}+\frac {17\,x^2}{90}+\frac {26\,x}{75}+\frac {2}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3843750*log(x + 3/5))/14641 - (630342*log(x + 2/3))/2401 - (6528*log(x - 1/2))/35153041 - ((301397263*x^2)/68
479950 - (138982273*x)/22826650 + (55931422*x^3)/2282665 + (7990242*x^4)/456533 - 909187261/410879700)/((26*x)
/75 + (17*x^2)/90 - (256*x^3)/225 - (61*x^4)/30 - x^5 + 2/25)

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