3.1587 \(\int \frac{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^8} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.05431, antiderivative size = 109, normalized size of antiderivative = 1., 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} \[ \frac{21296}{5764801 (1-2 x)}-\frac{17424}{823543 (3 x+2)}-\frac{22506}{823543 (3 x+2)^2}-\frac{4840}{117649 (3 x+2)^3}-\frac{3267}{67228 (3 x+2)^4}+\frac{363}{12005 (3 x+2)^5}-\frac{101}{18522 (3 x+2)^6}+\frac{1}{3087 (3 x+2)^7}-\frac{307824 \log (1-2 x)}{40353607}+\frac{307824 \log (3 x+2)}{40353607} \]

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*}

Mathematica [A]  time = 0.0624825, size = 74, normalized size = 0.68 \[ \frac{4 \left (-\frac{7 \left (121177995840 x^7+494810149680 x^6+820756518120 x^5+677745912690 x^4+242725322763 x^3-18916696050 x^2-39853850134 x-8381276704\right )}{16 (2 x-1) (3 x+2)^7}-10389060 \log (1-2 x)[phone] \log (6 x+4)\right )}{5447736945} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 90, normalized size = 0.8 \begin{align*} -{\frac{21296}{11529602\,x-5764801}}-{\frac{307824\,\ln \left ( 2\,x-1 \right ) }{40353607}}+{\frac{1}{3087\, \left ( 2+3\,x \right ) ^{7}}}-{\frac{101}{18522\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{363}{12005\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{3267}{67228\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{4840}{117649\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{22506}{823543\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{17424}{1647086+2470629\,x}}+{\frac{307824\,\ln \left ( 2+3\,x \right ) }{40353607}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.11208, size = 130, normalized size = 1.19 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.3072, size = 675, normalized size = 6.19 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.222188, size = 95, normalized size = 0.87 \begin{align*} - \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{align*}

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

________________________________________________________________________________________

Giac [A]  time = 2.4825, size = 130, normalized size = 1.19 \begin{align*} -\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{align*}

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))