3.1655 \(\int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{319}{2401 (1-2 x)}+\frac{64}{2401 (3 x+2)}+\frac{121}{686 (1-2 x)^2}-\frac{1}{686 (3 x+2)^2}-\frac{829 \log (1-2 x)}{16807}+\frac{829 \log (3 x+2)}{16807} \]

[Out]

121/(686*(1 - 2*x)^2) + 319/(2401*(1 - 2*x)) - 1/(686*(2 + 3*x)^2) + 64/(2401*(2 + 3*x)) - (829*Log[1 - 2*x])/
16807 + (829*Log[2 + 3*x])/16807

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Rubi [A]  time = 0.0374913, 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} \[ \frac{319}{2401 (1-2 x)}+\frac{64}{2401 (3 x+2)}+\frac{121}{686 (1-2 x)^2}-\frac{1}{686 (3 x+2)^2}-\frac{829 \log (1-2 x)}{16807}+\frac{829 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(686*(1 - 2*x)^2) + 319/(2401*(1 - 2*x)) - 1/(686*(2 + 3*x)^2) + 64/(2401*(2 + 3*x)) - (829*Log[1 - 2*x])/
16807 + (829*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)^3 (2+3 x)^3} \, dx &=\int \left (-\frac{242}{343 (-1+2 x)^3}+\frac{638}{2401 (-1+2 x)^2}-\frac{1658}{16807 (-1+2 x)}+\frac{3}{343 (2+3 x)^3}-\frac{192}{2401 (2+3 x)^2}+\frac{2487}{16807 (2+3 x)}\right ) \, dx\\ &=\frac{121}{686 (1-2 x)^2}+\frac{319}{2401 (1-2 x)}-\frac{1}{686 (2+3 x)^2}+\frac{64}{2401 (2+3 x)}-\frac{829 \log (1-2 x)}{16807}+\frac{829 \log (2+3 x)}{16807}\\ \end{align*}

Mathematica [A]  time = 0.0312384, size = 48, normalized size = 0.74 \[ \frac{-\frac{7 \left (9948 x^3+2487 x^2-12104 x-6189\right )}{\left (6 x^2+x-2\right )^2}-1658 \log (1-2 x)+1658 \log (3 x+2)}{33614} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(-6189 - 12104*x + 2487*x^2 + 9948*x^3))/(-2 + x + 6*x^2)^2 - 1658*Log[1 - 2*x] + 1658*Log[2 + 3*x])/3361
4

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Maple [A]  time = 0.01, size = 54, normalized size = 0.8 \begin{align*}{\frac{121}{686\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{319}{4802\,x-2401}}-{\frac{829\,\ln \left ( 2\,x-1 \right ) }{16807}}-{\frac{1}{686\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{64}{4802+7203\,x}}+{\frac{829\,\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)^3/(2+3*x)^3,x)

[Out]

121/686/(2*x-1)^2-319/2401/(2*x-1)-829/16807*ln(2*x-1)-1/686/(2+3*x)^2+64/2401/(2+3*x)+829/16807*ln(2+3*x)

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Maxima [A]  time = 1.11393, size = 76, normalized size = 1.17 \begin{align*} -\frac{9948 \, x^{3} + 2487 \, x^{2} - 12104 \, x - 6189}{4802 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} + \frac{829}{16807} \, \log \left (3 \, x + 2\right ) - \frac{829}{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)^3/(2+3*x)^3,x, algorithm="maxima")

[Out]

-1/4802*(9948*x^3 + 2487*x^2 - 12104*x - 6189)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) + 829/16807*log(3*x + 2) -
829/16807*log(2*x - 1)

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Fricas [A]  time = 1.57702, size = 274, normalized size = 4.22 \begin{align*} -\frac{69636 \, x^{3} + 17409 \, x^{2} - 1658 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 1658 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 84728 \, x - 43323}{33614 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/33614*(69636*x^3 + 17409*x^2 - 1658*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(3*x + 2) + 1658*(36*x^4 + 12*x
^3 - 23*x^2 - 4*x + 4)*log(2*x - 1) - 84728*x - 43323)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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Sympy [A]  time = 0.169199, size = 54, normalized size = 0.83 \begin{align*} - \frac{9948 x^{3} + 2487 x^{2} - 12104 x - 6189}{172872 x^{4} + 57624 x^{3} - 110446 x^{2} - 19208 x + 19208} - \frac{829 \log{\left (x - \frac{1}{2} \right )}}{16807} + \frac{829 \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)**3/(2+3*x)**3,x)

[Out]

-(9948*x**3 + 2487*x**2 - 12104*x - 6189)/(172872*x**4 + 57624*x**3 - 110446*x**2 - 19208*x + 19208) - 829*log
(x - 1/2)/16807 + 829*log(x + 2/3)/16807

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Giac [A]  time = 2.90543, size = 62, normalized size = 0.95 \begin{align*} -\frac{9948 \, x^{3} + 2487 \, x^{2} - 12104 \, x - 6189}{4802 \,{\left (6 \, x^{2} + x - 2\right )}^{2}} + \frac{829}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{829}{16807} \, \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)^3/(2+3*x)^3,x, algorithm="giac")

[Out]

-1/4802*(9948*x^3 + 2487*x^2 - 12104*x - 6189)/(6*x^2 + x - 2)^2 + 829/16807*log(abs(3*x + 2)) - 829/16807*log
(abs(2*x - 1))