3.15.48 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx\)

Optimal. Leaf size=109 \[ \frac {21296}{5764801 (1-2 x)}-\frac {17424}{823543 (3 x+2)}-\frac {22506}{823543 (3 x+2)^2}-\frac {4840}{117649 (3 x+2)^3}-\frac {3267}{67228 (3 x+2)^4}+\frac {363}{12005 (3 x+2)^5}-\frac {101}{18522 (3 x+2)^6}+\frac {1}{3087 (3 x+2)^7}-\frac {307824 \log (1-2 x)}{40353607}+\frac {307824 \log (3 x+2)}{40353607} \]

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Rubi [A]  time = 0.05, antiderivative size = 109, 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 = {88} \begin {gather*} \frac {21296}{5764801 (1-2 x)}-\frac {17424}{823543 (3 x+2)}-\frac {22506}{823543 (3 x+2)^2}-\frac {4840}{117649 (3 x+2)^3}-\frac {3267}{67228 (3 x+2)^4}+\frac {363}{12005 (3 x+2)^5}-\frac {101}{18522 (3 x+2)^6}+\frac {1}{3087 (3 x+2)^7}-\frac {307824 \log (1-2 x)}{40353607}+\frac {307824 \log (3 x+2)}{40353607} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^8),x]

[Out]

21296/(5764801*(1 - 2*x)) + 1/(3087*(2 + 3*x)^7) - 101/(18522*(2 + 3*x)^6) + 363/(12005*(2 + 3*x)^5) - 3267/(6
7228*(2 + 3*x)^4) - 4840/(117649*(2 + 3*x)^3) - 22506/(823543*(2 + 3*x)^2) - 17424/(823543*(2 + 3*x)) - (30782
4*Log[1 - 2*x])/40353607 + (307824*Log[2 + 3*x])/40353607

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 {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx &=\int \left (\frac {42592}{5764801 (-1+2 x)^2}-\frac {615648}{40353607 (-1+2 x)}-\frac {1}{147 (2+3 x)^8}+\frac {101}{1029 (2+3 x)^7}-\frac {1089}{2401 (2+3 x)^6}+\frac {9801}{16807 (2+3 x)^5}+\frac {43560}{117649 (2+3 x)^4}+\frac {135036}{823543 (2+3 x)^3}+\frac {52272}{823543 (2+3 x)^2}+\frac {923472}{40353607 (2+3 x)}\right ) \, dx\\ &=\frac {21296}{5764801 (1-2 x)}+\frac {1}{3087 (2+3 x)^7}-\frac {101}{18522 (2+3 x)^6}+\frac {363}{12005 (2+3 x)^5}-\frac {3267}{67228 (2+3 x)^4}-\frac {4840}{117649 (2+3 x)^3}-\frac {22506}{823543 (2+3 x)^2}-\frac {17424}{823543 (2+3 x)}-\frac {307824 \log (1-2 x)}{40353607}+\frac {307824 \log (2+3 x)}{40353607}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 74, normalized size = 0.68 \begin {gather*} \frac {4 \left (-\frac {7 \left (121177995840 x^7+494810149680 x^6+820756518120 x^5+677745912690 x^4+242725322763 x^3-18916696050 x^2-39853850134 x-8381276704\right )}{16 (2 x-1) (3 x+2)^7}-10389060 \log (1-2 x)[phone] \log (6 x+4)\right )}{5447736945} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^8),x]

[Out]

(4*((-7*(-8381276704 - 39853850134*x - 18916696050*x^2 + 242725322763*x^3 + 677745912690*x^4 + 820756518120*x^
5 + 494810149680*x^6 + 121177995840*x^7))/(16*(-1 + 2*x)*(2 + 3*x)^7) - 10389060*Log[1 - 2*x] + 10389060*Log[4
+ 6*x]))/5447736945

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^8),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^8), x]

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fricas [A]  time = 1.52, size = 175, normalized size = 1.61 \begin {gather*} -\frac {848245970880 \, x^{7} + 3463671047760 \, x^{6} + 5745295626840 \, x^{5} + 4744221388830 \, x^{4} + 1699077259341 \, x^{3} - 132416872350 \, x^{2} - 166224960 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (3 \, x + 2\right ) + 166224960 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )} \log \left (2 \, x - 1\right ) - 278976950938 \, x - 58668936928}{21790947780 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/21790947780*(848245970880*x^7 + 3463671047760*x^6 + 5745295626840*x^5 + 4744221388830*x^4 + 1699077259341*x
^3 - 132416872350*x^2 - 166224960*(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x
^2 - 1088*x - 128)*log(3*x + 2) + 166224960*(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*x^
3 - 3360*x^2 - 1088*x - 128)*log(2*x - 1) - 278976950938*x - 58668936928)/(4374*x^8 + 18225*x^7 + 30618*x^6 +
24948*x^5 + 7560*x^4 - 3024*x^3 - 3360*x^2 - 1088*x - 128)

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giac [A]  time = 1.02, size = 96, normalized size = 0.88 \begin {gather*} -\frac {21296}{5764801 \, {\left (2 \, x - 1\right )}} + \frac {4 \, {\left (\frac {108987508287}{2 \, x - 1} + \frac {677288963799}{{\left (2 \, x - 1\right )}^{2}} + \frac {2255033089785}{{\left (2 \, x - 1\right )}^{3}} + \frac {4241269979800}{{\left (2 \, x - 1\right )}^{4}} + \frac {4269658683500}{{\left (2 \, x - 1\right )}^{5}} + \frac {1795850807520}{{\left (2 \, x - 1\right )}^{6}} + 7339564629\right )}}{1412376245 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{7}} + \frac {307824}{40353607} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^8,x, algorithm="giac")

[Out]

-21296/5764801/(2*x - 1) + 4/1412376245*(108987508287/(2*x - 1) + 677288963799/(2*x - 1)^2 + 2255033089785/(2*
x - 1)^3 + 4241269979800/(2*x - 1)^4 + 4269658683500/(2*x - 1)^5 + 1795850807520/(2*x - 1)^6 + 7339564629)/(7/
(2*x - 1) + 3)^7 + 307824/40353607*log(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 90, normalized size = 0.83 \begin {gather*} -\frac {307824 \ln \left (2 x -1\right )}{40353607}+\frac {307824 \ln \left (3 x +2\right )}{40353607}+\frac {1}{3087 \left (3 x +2\right )^{7}}-\frac {101}{18522 \left (3 x +2\right )^{6}}+\frac {363}{12005 \left (3 x +2\right )^{5}}-\frac {3267}{67228 \left (3 x +2\right )^{4}}-\frac {4840}{117649 \left (3 x +2\right )^{3}}-\frac {22506}{823543 \left (3 x +2\right )^{2}}-\frac {17424}{823543 \left (3 x +2\right )}-\frac {21296}{5764801 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)^2/(3*x+2)^8,x)

[Out]

1/3087/(3*x+2)^7-101/18522/(3*x+2)^6+363/12005/(3*x+2)^5-3267/67228/(3*x+2)^4-4840/117649/(3*x+2)^3-22506/8235
43/(3*x+2)^2-17424/823543/(3*x+2)+307824/40353607*ln(3*x+2)-21296/5764801/(2*x-1)-307824/40353607*ln(2*x-1)

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maxima [A]  time = 0.53, size = 96, normalized size = 0.88 \begin {gather*} -\frac {121177995840 \, x^{7} + 494810149680 \, x^{6} + 820756518120 \, x^{5} + 677745912690 \, x^{4} + 242725322763 \, x^{3} - 18916696050 \, x^{2} - 39853850134 \, x - 8381276704}{3112992540 \, {\left (4374 \, x^{8} + 18225 \, x^{7} + 30618 \, x^{6} + 24948 \, x^{5} + 7560 \, x^{4} - 3024 \, x^{3} - 3360 \, x^{2} - 1088 \, x - 128\right )}} + \frac {307824}{40353607} \, \log \left (3 \, x + 2\right ) - \frac {307824}{40353607} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/3112992540*(121177995840*x^7 + 494810149680*x^6 + 820756518120*x^5 + 677745912690*x^4 + 242725322763*x^3 -
18916696050*x^2 - 39853850134*x - 8381276704)/(4374*x^8 + 18225*x^7 + 30618*x^6 + 24948*x^5 + 7560*x^4 - 3024*
x^3 - 3360*x^2 - 1088*x - 128) + 307824/40353607*log(3*x + 2) - 307824/40353607*log(2*x - 1)

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mupad [B]  time = 0.05, size = 86, normalized size = 0.79 \begin {gather*} \frac {615648\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{40353607}-\frac {\frac {51304\,x^7}{5764801}+\frac {12826\,x^6}{352947}+\frac {1340317\,x^5}{22235661}+\frac {13281323\,x^4}{266827932}+\frac {47565221\,x^3}{2668279320}-\frac {90079505\,x^2}{64839187476}-\frac {2846703581\,x}{972587812140}-\frac {1047659588}{1702028671245}}{x^8+\frac {25\,x^7}{6}+7\,x^6+\frac {154\,x^5}{27}+\frac {140\,x^4}{81}-\frac {56\,x^3}{81}-\frac {560\,x^2}{729}-\frac {544\,x}{2187}-\frac {64}{2187}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/((2*x - 1)^2*(3*x + 2)^8),x)

[Out]

(615648*atanh((12*x)/7 + 1/7))/40353607 - ((47565221*x^3)/2668279320 - (90079505*x^2)/64839187476 - (284670358
1*x)/972587812140 + (13281323*x^4)/266827932 + (1340317*x^5)/22235661 + (12826*x^6)/352947 + (51304*x^7)/57648
01 - 1047659588/1702028671245)/((140*x^4)/81 - (560*x^2)/729 - (56*x^3)/81 - (544*x)/2187 + (154*x^5)/27 + 7*x
^6 + (25*x^7)/6 + x^8 - 64/2187)

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sympy [A]  time = 0.23, size = 95, normalized size = 0.87 \begin {gather*} \frac {- 121177995840 x^{7} - 494810149680 x^{6} - 820756518120 x^{5} - 677745912690 x^{4} - 242725322763 x^{3} + 18916696050 x^{2} + 39853850134 x + 8381276704}{13616229369960 x^{8} + 56734289041500 x^{7} + 95313605589720 x^{6} + 77662937887920 x^{5} + 23534223602400 x^{4} - 9413689440960 x^{3} - 10459654934400 x^{2} - 3386935883520 x - 398463045120} - \frac {307824 \log {\left (x - \frac {1}{2} \right )}}{40353607} + \frac {307824 \log {\left (x + \frac {2}{3} \right )}}{40353607} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)**2/(2+3*x)**8,x)

[Out]

(-121177995840*x**7 - 494810149680*x**6 - 820756518120*x**5 - 677745912690*x**4 - 242725322763*x**3 + 18916696
050*x**2 + 39853850134*x + 8381276704)/(13616229369960*x**8 + 56734289041500*x**7 + 95313605589720*x**6 + 7766
2937887920*x**5 + 23534223602400*x**4 - 9413689440960*x**3 - 10459654934400*x**2 - 3386935883520*x - 398463045
120) - 307824*log(x - 1/2)/40353607 + 307824*log(x + 2/3)/40353607

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