3.1412 \(\int \frac{(1-2 x)^3}{(2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=50 \[ -\frac{3136}{9 (3 x+2)}-\frac{1331}{5 (5 x+3)}-\frac{343}{18 (3 x+2)^2}+2541 \log (3 x+2)-2541 \log (5 x+3) \]

[Out]

-343/(18*(2 + 3*x)^2) - 3136/(9*(2 + 3*x)) - 1331/(5*(3 + 5*x)) + 2541*Log[2 + 3*x] - 2541*Log[3 + 5*x]

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Rubi [A]  time = 0.0258822, antiderivative size = 50, 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} \[ -\frac{3136}{9 (3 x+2)}-\frac{1331}{5 (5 x+3)}-\frac{343}{18 (3 x+2)^2}+2541 \log (3 x+2)-2541 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-343/(18*(2 + 3*x)^2) - 3136/(9*(2 + 3*x)) - 1331/(5*(3 + 5*x)) + 2541*Log[2 + 3*x] - 2541*Log[3 + 5*x]

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{(1-2 x)^3}{(2+3 x)^3 (3+5 x)^2} \, dx &=\int \left (\frac{343}{3 (2+3 x)^3}+\frac{3136}{3 (2+3 x)^2}+\frac{7623}{2+3 x}+\frac{1331}{(3+5 x)^2}-\frac{12705}{3+5 x}\right ) \, dx\\ &=-\frac{343}{18 (2+3 x)^2}-\frac{3136}{9 (2+3 x)}-\frac{1331}{5 (3+5 x)}+2541 \log (2+3 x)-2541 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0331741, size = 47, normalized size = 0.94 \[ -\frac{686022 x^2+891911 x+289137}{90 (3 x+2)^2 (5 x+3)}+2541 \log (5 (3 x+2))-2541 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(289137 + 891911*x + 686022*x^2)/(90*(2 + 3*x)^2*(3 + 5*x)) + 2541*Log[5*(2 + 3*x)] - 2541*Log[3 + 5*x]

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Maple [A]  time = 0.009, size = 45, normalized size = 0.9 \begin{align*} -{\frac{343}{18\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{3136}{18+27\,x}}-{\frac{1331}{15+25\,x}}+2541\,\ln \left ( 2+3\,x \right ) -2541\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/18/(2+3*x)^2-3136/9/(2+3*x)-1331/5/(3+5*x)+2541*ln(2+3*x)-2541*ln(3+5*x)

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Maxima [A]  time = 1.02915, size = 62, normalized size = 1.24 \begin{align*} -\frac{686022 \, x^{2} + 891911 \, x + 289137}{90 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - 2541 \, \log \left (5 \, x + 3\right ) + 2541 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(686022*x^2 + 891911*x + 289137)/(45*x^3 + 87*x^2 + 56*x + 12) - 2541*log(5*x + 3) + 2541*log(3*x + 2)

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Fricas [A]  time = 1.20537, size = 235, normalized size = 4.7 \begin{align*} -\frac{686022 \, x^{2} + 228690 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 228690 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 891911 \, x + 289137}{90 \,{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(686022*x^2 + 228690*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 228690*(45*x^3 + 87*x^2 + 56*x + 12)*l
og(3*x + 2) + 891911*x + 289137)/(45*x^3 + 87*x^2 + 56*x + 12)

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Sympy [A]  time = 0.156002, size = 41, normalized size = 0.82 \begin{align*} - \frac{686022 x^{2} + 891911 x + 289137}{4050 x^{3} + 7830 x^{2} + 5040 x + 1080} - 2541 \log{\left (x + \frac{3}{5} \right )} + 2541 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(686022*x**2 + 891911*x + 289137)/(4050*x**3 + 7830*x**2 + 5040*x + 1080) - 2541*log(x + 3/5) + 2541*log(x +
2/3)

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Giac [A]  time = 2.9028, size = 66, normalized size = 1.32 \begin{align*} -\frac{1331}{5 \,{\left (5 \, x + 3\right )}} + \frac{245 \,{\left (\frac{66}{5 \, x + 3} + 163\right )}}{2 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{2}} + 2541 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/5/(5*x + 3) + 245/2*(66/(5*x + 3) + 163)/(1/(5*x + 3) + 3)^2 + 2541*log(abs(-1/(5*x + 3) - 3))