3.1466 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^7} \, dx\)

Optimal. Leaf size=87 \[ -\frac{968}{117649 (3 x+2)}-\frac{242}{16807 (3 x+2)^2}-\frac{242}{7203 (3 x+2)^3}-\frac{121}{1372 (3 x+2)^4}+\frac{68}{2205 (3 x+2)^5}-\frac{1}{378 (3 x+2)^6}-\frac{1936 \log (1-2 x)}{823543}+\frac{1936 \log (3 x+2)}{823543} \]

[Out]

-1/(378*(2 + 3*x)^6) + 68/(2205*(2 + 3*x)^5) - 121/(1372*(2 + 3*x)^4) - 242/(7203*(2 + 3*x)^3) - 242/(16807*(2
+ 3*x)^2) - 968/(117649*(2 + 3*x)) - (1936*Log[1 - 2*x])/823543 + (1936*Log[2 + 3*x])/823543

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Rubi [A]  time = 0.0312858, antiderivative size = 87, 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} \[ -\frac{968}{117649 (3 x+2)}-\frac{242}{16807 (3 x+2)^2}-\frac{242}{7203 (3 x+2)^3}-\frac{121}{1372 (3 x+2)^4}+\frac{68}{2205 (3 x+2)^5}-\frac{1}{378 (3 x+2)^6}-\frac{1936 \log (1-2 x)}{823543}+\frac{1936 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

-1/(378*(2 + 3*x)^6) + 68/(2205*(2 + 3*x)^5) - 121/(1372*(2 + 3*x)^4) - 242/(7203*(2 + 3*x)^3) - 242/(16807*(2
+ 3*x)^2) - 968/(117649*(2 + 3*x)) - (1936*Log[1 - 2*x])/823543 + (1936*Log[2 + 3*x])/823543

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)^2}{(1-2 x) (2+3 x)^7} \, dx &=\int \left (-\frac{3872}{823543 (-1+2 x)}+\frac{1}{21 (2+3 x)^7}-\frac{68}{147 (2+3 x)^6}+\frac{363}{343 (2+3 x)^5}+\frac{726}{2401 (2+3 x)^4}+\frac{1452}{16807 (2+3 x)^3}+\frac{2904}{117649 (2+3 x)^2}+\frac{5808}{823543 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{378 (2+3 x)^6}+\frac{68}{2205 (2+3 x)^5}-\frac{121}{1372 (2+3 x)^4}-\frac{242}{7203 (2+3 x)^3}-\frac{242}{16807 (2+3 x)^2}-\frac{968}{117649 (2+3 x)}-\frac{1936 \log (1-2 x)}{823543}+\frac{1936 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0416945, size = 57, normalized size = 0.66 \[ \frac{4 \left (-\frac{7 \left (127020960 x^5+497498760 x^4+819755640 x^3+739632465 x^2+351466812 x+67099978\right )}{16 (3 x+2)^6}-65340 \log (1-2 x)+65340 \log (6 x+4)\right )}{111178305} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^2/((1 - 2*x)*(2 + 3*x)^7),x]

[Out]

(4*((-7*(67099978 + 351466812*x + 739632465*x^2 + 819755640*x^3 + 497498760*x^4 + 127020960*x^5))/(16*(2 + 3*x
)^6) - 65340*Log[1 - 2*x] + 65340*Log[4 + 6*x]))/111178305

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Maple [A]  time = 0.009, size = 72, normalized size = 0.8 \begin{align*} -{\frac{1936\,\ln \left ( 2\,x-1 \right ) }{823543}}-{\frac{1}{378\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{68}{2205\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{121}{1372\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{242}{7203\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{242}{16807\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{968}{235298+352947\,x}}+{\frac{1936\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^2/(1-2*x)/(2+3*x)^7,x)

[Out]

-1936/823543*ln(2*x-1)-1/378/(2+3*x)^6+68/2205/(2+3*x)^5-121/1372/(2+3*x)^4-242/7203/(2+3*x)^3-242/16807/(2+3*
x)^2-968/117649/(2+3*x)+1936/823543*ln(2+3*x)

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Maxima [A]  time = 1.03476, size = 103, normalized size = 1.18 \begin{align*} -\frac{127020960 \, x^{5} + 497498760 \, x^{4} + 819755640 \, x^{3} + 739632465 \, x^{2} + 351466812 \, x + 67099978}{63530460 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{1936}{823543} \, \log \left (3 \, x + 2\right ) - \frac{1936}{823543} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/63530460*(127020960*x^5 + 497498760*x^4 + 819755640*x^3 + 739632465*x^2 + 351466812*x + 67099978)/(729*x^6
+ 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 1936/823543*log(3*x + 2) - 1936/823543*log(2*x - 1
)

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Fricas [A]  time = 1.1546, size = 479, normalized size = 5.51 \begin{align*} -\frac{889146720 \, x^{5} + 3482491320 \, x^{4} + 5738289480 \, x^{3} + 5177427255 \, x^{2} - 1045440 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 1045440 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (2 \, x - 1\right ) + 2460267684 \, x + 469699846}{444713220 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/444713220*(889146720*x^5 + 3482491320*x^4 + 5738289480*x^3 + 5177427255*x^2 - 1045440*(729*x^6 + 2916*x^5 +
4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 1045440*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3
+ 2160*x^2 + 576*x + 64)*log(2*x - 1) + 2460267684*x + 469699846)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

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Sympy [A]  time = 0.202721, size = 75, normalized size = 0.86 \begin{align*} - \frac{127020960 x^{5} + 497498760 x^{4} + 819755640 x^{3} + 739632465 x^{2} + 351466812 x + 67099978}{46313705340 x^{6} + 185254821360 x^{5} + 308758035600 x^{4} + 274451587200 x^{3} + 137225793600 x^{2} + 36593544960 x + 4065949440} - \frac{1936 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{1936 \log{\left (x + \frac{2}{3} \right )}}{823543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**2/(1-2*x)/(2+3*x)**7,x)

[Out]

-(127020960*x**5 + 497498760*x**4 + 819755640*x**3 + 739632465*x**2 + 351466812*x + 67099978)/(46313705340*x**
6 + 185254821360*x**5 + 308758035600*x**4 + 274451587200*x**3 + 137225793600*x**2 + 36593544960*x + 4065949440
) - 1936*log(x - 1/2)/823543 + 1936*log(x + 2/3)/823543

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Giac [A]  time = 1.27556, size = 72, normalized size = 0.83 \begin{align*} -\frac{127020960 \, x^{5} + 497498760 \, x^{4} + 819755640 \, x^{3} + 739632465 \, x^{2} + 351466812 \, x + 67099978}{63530460 \,{\left (3 \, x + 2\right )}^{6}} + \frac{1936}{823543} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1936}{823543} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^2/(1-2*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/63530460*(127020960*x^5 + 497498760*x^4 + 819755640*x^3 + 739632465*x^2 + 351466812*x + 67099978)/(3*x + 2)
^6 + 1936/823543*log(abs(3*x + 2)) - 1936/823543*log(abs(2*x - 1))