3.1464 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^5} \, dx\)

Optimal. Leaf size=65 \[ -\frac{242}{2401 (3 x+2)}-\frac{121}{686 (3 x+2)^2}+\frac{68}{1323 (3 x+2)^3}-\frac{1}{252 (3 x+2)^4}-\frac{484 \log (1-2 x)}{16807}+\frac{484 \log (3 x+2)}{16807} \]

[Out]

-1/(252*(2 + 3*x)^4) + 68/(1323*(2 + 3*x)^3) - 121/(686*(2 + 3*x)^2) - 242/(2401*(2 + 3*x)) - (484*Log[1 - 2*x
])/16807 + (484*Log[2 + 3*x])/16807

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Rubi [A]  time = 0.0254265, antiderivative size = 65, 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} \[ -\frac{242}{2401 (3 x+2)}-\frac{121}{686 (3 x+2)^2}+\frac{68}{1323 (3 x+2)^3}-\frac{1}{252 (3 x+2)^4}-\frac{484 \log (1-2 x)}{16807}+\frac{484 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(252*(2 + 3*x)^4) + 68/(1323*(2 + 3*x)^3) - 121/(686*(2 + 3*x)^2) - 242/(2401*(2 + 3*x)) - (484*Log[1 - 2*x
])/16807 + (484*Log[2 + 3*x])/16807

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)^5} \, dx &=\int \left (-\frac{968}{16807 (-1+2 x)}+\frac{1}{21 (2+3 x)^5}-\frac{68}{147 (2+3 x)^4}+\frac{363}{343 (2+3 x)^3}+\frac{726}{2401 (2+3 x)^2}+\frac{1452}{16807 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{252 (2+3 x)^4}+\frac{68}{1323 (2+3 x)^3}-\frac{121}{686 (2+3 x)^2}-\frac{242}{2401 (2+3 x)}-\frac{484 \log (1-2 x)}{16807}+\frac{484 \log (2+3 x)}{16807}\\ \end{align*}

Mathematica [A]  time = 0.02714, size = 47, normalized size = 0.72 \[ \frac{2 \left (-\frac{7 \left (705672 x^3+1822986 x^2+1449768 x+366413\right )}{8 (3 x+2)^4}-6534 \log (1-2 x)+6534 \log (6 x+4)\right )}{453789} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(366413 + 1449768*x + 1822986*x^2 + 705672*x^3))/(8*(2 + 3*x)^4) - 6534*Log[1 - 2*x] + 6534*Log[4 + 6*
x]))/453789

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Maple [A]  time = 0.007, size = 54, normalized size = 0.8 \begin{align*} -{\frac{484\,\ln \left ( 2\,x-1 \right ) }{16807}}-{\frac{1}{252\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{68}{1323\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{121}{686\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{242}{4802+7203\,x}}+{\frac{484\,\ln \left ( 2+3\,x \right ) }{16807}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-484/16807*ln(2*x-1)-1/252/(2+3*x)^4+68/1323/(2+3*x)^3-121/686/(2+3*x)^2-242/2401/(2+3*x)+484/16807*ln(2+3*x)

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Maxima [A]  time = 1.01592, size = 76, normalized size = 1.17 \begin{align*} -\frac{705672 \, x^{3} + 1822986 \, x^{2} + 1449768 \, x + 366413}{259308 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{484}{16807} \, \log \left (3 \, x + 2\right ) - \frac{484}{16807} \, \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)^5,x, algorithm="maxima")

[Out]

-1/259308*(705672*x^3 + 1822986*x^2 + 1449768*x + 366413)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 484/16807
*log(3*x + 2) - 484/16807*log(2*x - 1)

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Fricas [A]  time = 1.2822, size = 309, normalized size = 4.75 \begin{align*} -\frac{4939704 \, x^{3} + 12760902 \, x^{2} - 52272 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 52272 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (2 \, x - 1\right ) + 10148376 \, x + 2564891}{1815156 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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)^5,x, algorithm="fricas")

[Out]

-1/1815156*(4939704*x^3 + 12760902*x^2 - 52272*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(3*x + 2) + 52272*(
81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(2*x - 1) + 10148376*x + 2564891)/(81*x^4 + 216*x^3 + 216*x^2 + 96*
x + 16)

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Sympy [A]  time = 0.170928, size = 54, normalized size = 0.83 \begin{align*} - \frac{705672 x^{3} + 1822986 x^{2} + 1449768 x + 366413}{21003948 x^{4} + 56010528 x^{3} + 56010528 x^{2} + 24893568 x + 4148928} - \frac{484 \log{\left (x - \frac{1}{2} \right )}}{16807} + \frac{484 \log{\left (x + \frac{2}{3} \right )}}{16807} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(705672*x**3 + 1822986*x**2 + 1449768*x + 366413)/(21003948*x**4 + 56010528*x**3 + 56010528*x**2 + 24893568*x
+ 4148928) - 484*log(x - 1/2)/16807 + 484*log(x + 2/3)/16807

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Giac [A]  time = 1.24926, size = 70, normalized size = 1.08 \begin{align*} -\frac{242}{2401 \,{\left (3 \, x + 2\right )}} - \frac{121}{686 \,{\left (3 \, x + 2\right )}^{2}} + \frac{68}{1323 \,{\left (3 \, x + 2\right )}^{3}} - \frac{1}{252 \,{\left (3 \, x + 2\right )}^{4}} - \frac{484}{16807} \, \log \left ({\left | -\frac{7}{3 \, x + 2} + 2 \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)^5,x, algorithm="giac")

[Out]

-242/2401/(3*x + 2) - 121/686/(3*x + 2)^2 + 68/1323/(3*x + 2)^3 - 1/252/(3*x + 2)^4 - 484/16807*log(abs(-7/(3*
x + 2) + 2))