3.1331 \(\int \frac{(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{49}{3 x+2}+\frac{154}{5 x+3}-\frac{121}{10 (5 x+3)^2}-707 \log (3 x+2)+707 \log (5 x+3) \]

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[2 + 3*x] + 707*Log[3 + 5*x]

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Rubi [A]  time = 0.0228606, antiderivative size = 46, 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} \[ \frac{49}{3 x+2}+\frac{154}{5 x+3}-\frac{121}{10 (5 x+3)^2}-707 \log (3 x+2)+707 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[2 + 3*x] + 707*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)^2}{(2+3 x)^2 (3+5 x)^3} \, dx &=\int \left (-\frac{147}{(2+3 x)^2}-\frac{2121}{2+3 x}+\frac{121}{(3+5 x)^3}-\frac{770}{(3+5 x)^2}+\frac{3535}{3+5 x}\right ) \, dx\\ &=\frac{49}{2+3 x}-\frac{121}{10 (3+5 x)^2}+\frac{154}{3+5 x}-707 \log (2+3 x)+707 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0249397, size = 48, normalized size = 1.04 \[ \frac{49}{3 x+2}+\frac{154}{5 x+3}-\frac{121}{10 (5 x+3)^2}-707 \log (5 (3 x+2))+707 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[5*(2 + 3*x)] + 707*Log[3 + 5*x]

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Maple [A]  time = 0.008, size = 45, normalized size = 1. \begin{align*} 49\, \left ( 2+3\,x \right ) ^{-1}-{\frac{121}{10\, \left ( 3+5\,x \right ) ^{2}}}+154\, \left ( 3+5\,x \right ) ^{-1}-707\,\ln \left ( 2+3\,x \right ) +707\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/(2+3*x)-121/10/(3+5*x)^2+154/(3+5*x)-707*ln(2+3*x)+707*ln(3+5*x)

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Maxima [A]  time = 1.09746, size = 62, normalized size = 1.35 \begin{align*} \frac{35350 \, x^{2} + 43597 \, x + 13408}{10 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + 707 \, \log \left (5 \, x + 3\right ) - 707 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(35350*x^2 + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18) + 707*log(5*x + 3) - 707*log(3*x + 2)

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Fricas [A]  time = 1.3798, size = 228, normalized size = 4.96 \begin{align*} \frac{35350 \, x^{2} + 7070 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 7070 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) + 43597 \, x + 13408}{10 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(35350*x^2 + 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(3
*x + 2) + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18)

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Sympy [A]  time = 0.148441, size = 41, normalized size = 0.89 \begin{align*} \frac{35350 x^{2} + 43597 x + 13408}{750 x^{3} + 1400 x^{2} + 870 x + 180} + 707 \log{\left (x + \frac{3}{5} \right )} - 707 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35350*x**2 + 43597*x + 13408)/(750*x**3 + 1400*x**2 + 870*x + 180) + 707*log(x + 3/5) - 707*log(x + 2/3)

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Giac [A]  time = 1.5823, size = 66, normalized size = 1.43 \begin{align*} \frac{49}{3 \, x + 2} - \frac{33 \,{\left (\frac{206}{3 \, x + 2} - 865\right )}}{2 \,{\left (\frac{1}{3 \, x + 2} - 5\right )}^{2}} + 707 \, \log \left ({\left | -\frac{1}{3 \, x + 2} + 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/(3*x + 2) - 33/2*(206/(3*x + 2) - 865)/(1/(3*x + 2) - 5)^2 + 707*log(abs(-1/(3*x + 2) + 5))