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

Optimal. Leaf size=57 \[ \frac {707}{3 x+2}+\frac {1133}{5 x+3}+\frac {49}{2 (3 x+2)^2}-\frac {121}{2 (5 x+3)^2}-6934 \log (3 x+2)+6934 \log (5 x+3) \]

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {707}{3 x+2}+\frac {1133}{5 x+3}+\frac {49}{2 (3 x+2)^2}-\frac {121}{2 (5 x+3)^2}-6934 \log (3 x+2)+6934 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2*(2 + 3*x)^2) + 707/(2 + 3*x) - 121/(2*(3 + 5*x)^2) + 1133/(3 + 5*x) - 6934*Log[2 + 3*x] + 6934*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)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {147}{(2+3 x)^3}-\frac {2121}{(2+3 x)^2}-\frac {20802}{2+3 x}+\frac {605}{(3+5 x)^3}-\frac {5665}{(3+5 x)^2}+\frac {34670}{3+5 x}\right ) \, dx\\ &=\frac {49}{2 (2+3 x)^2}+\frac {707}{2+3 x}-\frac {121}{2 (3+5 x)^2}+\frac {1133}{3+5 x}-6934 \log (2+3 x)+6934 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 59, normalized size = 1.04 \begin {gather*} \frac {707}{3 x+2}+\frac {1133}{5 x+3}+\frac {49}{2 (3 x+2)^2}-\frac {121}{2 (5 x+3)^2}-6934 \log (5 (3 x+2))+6934 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2*(2 + 3*x)^2) + 707/(2 + 3*x) - 121/(2*(3 + 5*x)^2) + 1133/(3 + 5*x) - 6934*Log[5*(2 + 3*x)] + 6934*Log[3
+ 5*x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 95, normalized size = 1.67 \begin {gather*} \frac {208020 \, x^{3} + 395238 \, x^{2} + 13868 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (5 \, x + 3\right ) - 13868 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (3 \, x + 2\right ) + 249932 \, x + 52601}{2 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(208020*x^3 + 395238*x^2 + 13868*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log(5*x + 3) - 13868*(225*x^4
+ 570*x^3 + 541*x^2 + 228*x + 36)*log(3*x + 2) + 249932*x + 52601)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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giac [A]  time = 0.93, size = 48, normalized size = 0.84 \begin {gather*} \frac {208020 \, x^{3} + 395238 \, x^{2} + 249932 \, x + 52601}{2 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}^{2}} + 6934 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 6934 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(208020*x^3 + 395238*x^2 + 249932*x + 52601)/(15*x^2 + 19*x + 6)^2 + 6934*log(abs(5*x + 3)) - 6934*log(abs
(3*x + 2))

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maple [A]  time = 0.01, size = 54, normalized size = 0.95 \begin {gather*} -6934 \ln \left (3 x +2\right )+6934 \ln \left (5 x +3\right )+\frac {49}{2 \left (3 x +2\right )^{2}}+\frac {707}{3 x +2}-\frac {121}{2 \left (5 x +3\right )^{2}}+\frac {1133}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/2/(3*x+2)^2+707/(3*x+2)-121/2/(5*x+3)^2+1133/(5*x+3)-6934*ln(3*x+2)+6934*ln(5*x+3)

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maxima [A]  time = 0.51, size = 56, normalized size = 0.98 \begin {gather*} \frac {208020 \, x^{3} + 395238 \, x^{2} + 249932 \, x + 52601}{2 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} + 6934 \, \log \left (5 \, x + 3\right ) - 6934 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(208020*x^3 + 395238*x^2 + 249932*x + 52601)/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36) + 6934*log(5*x + 3
) - 6934*log(3*x + 2)

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mupad [B]  time = 0.04, size = 45, normalized size = 0.79 \begin {gather*} \frac {\frac {6934\,x^3}{15}+\frac {65873\,x^2}{75}+\frac {124966\,x}{225}+\frac {52601}{450}}{x^4+\frac {38\,x^3}{15}+\frac {541\,x^2}{225}+\frac {76\,x}{75}+\frac {4}{25}}-13868\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((124966*x)/225 + (65873*x^2)/75 + (6934*x^3)/15 + 52601/450)/((76*x)/75 + (541*x^2)/225 + (38*x^3)/15 + x^4 +
4/25) - 13868*atanh(30*x + 19)

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sympy [A]  time = 0.17, size = 51, normalized size = 0.89 \begin {gather*} \frac {208020 x^{3} + 395238 x^{2} + 249932 x + 52601}{450 x^{4} + 1140 x^{3} + 1082 x^{2} + 456 x + 72} + 6934 \log {\left (x + \frac {3}{5} \right )} - 6934 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(208020*x**3 + 395238*x**2 + 249932*x + 52601)/(450*x**4 + 1140*x**3 + 1082*x**2 + 456*x + 72) + 6934*log(x +
3/5) - 6934*log(x + 2/3)

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