3.12.79 \(\int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=77 \[ -\frac {46475}{3 x+2}-\frac {15125}{5 x+3}-\frac {3740}{(3 x+2)^2}-\frac {1133}{3 (3 x+2)^3}-\frac {77}{2 (3 x+2)^4}-\frac {49}{15 (3 x+2)^5}+277750 \log (3 x+2)-277750 \log (5 x+3) \]

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Rubi [A]  time = 0.04, antiderivative size = 77, 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 {46475}{3 x+2}-\frac {15125}{5 x+3}-\frac {3740}{(3 x+2)^2}-\frac {1133}{3 (3 x+2)^3}-\frac {77}{2 (3 x+2)^4}-\frac {49}{15 (3 x+2)^5}+277750 \log (3 x+2)-277750 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

-49/(15*(2 + 3*x)^5) - 77/(2*(2 + 3*x)^4) - 1133/(3*(2 + 3*x)^3) - 3740/(2 + 3*x)^2 - 46475/(2 + 3*x) - 15125/
(3 + 5*x) + 277750*Log[2 + 3*x] - 277750*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)^6 (3+5 x)^2} \, dx &=\int \left (\frac {49}{(2+3 x)^6}+\frac {462}{(2+3 x)^5}+\frac {3399}{(2+3 x)^4}+\frac {22440}{(2+3 x)^3}+\frac {139425}{(2+3 x)^2}+\frac {833250}{2+3 x}+\frac {75625}{(3+5 x)^2}-\frac {1388750}{3+5 x}\right ) \, dx\\ &=-\frac {49}{15 (2+3 x)^5}-\frac {77}{2 (2+3 x)^4}-\frac {1133}{3 (2+3 x)^3}-\frac {3740}{(2+3 x)^2}-\frac {46475}{2+3 x}-\frac {15125}{3+5 x}+277750 \log (2+3 x)-277750 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 62, normalized size = 0.81 \begin {gather*} -\frac {674932500 x^5+2227277250 x^4+2939206050 x^3+1938789435 x^2+639246515 x+84279984}{30 (3 x+2)^5 (5 x+3)}+277750 \log (5 (3 x+2))-277750 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

-1/30*(84279984 + 639246515*x + 1938789435*x^2 + 2939206050*x^3 + 2227277250*x^4 + 674932500*x^5)/((2 + 3*x)^5
*(3 + 5*x)) + 277750*Log[5*(2 + 3*x)] - 277750*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)^6 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^6*(3 + 5*x)^2), x]

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fricas [A]  time = 1.24, size = 135, normalized size = 1.75 \begin {gather*} -\frac {674932500 \, x^{5} + 2227277250 \, x^{4} + 2939206050 \, x^{3} + 1938789435 \, x^{2} + 8332500 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 8332500 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 639246515 \, x + 84279984}{30 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/30*(674932500*x^5 + 2227277250*x^4 + 2939206050*x^3 + 1938789435*x^2 + 8332500*(1215*x^6 + 4779*x^5 + 7830*
x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 8332500*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 33
60*x^2 + 880*x + 96)*log(3*x + 2) + 639246515*x + 84279984)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*
x^2 + 880*x + 96)

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giac [A]  time = 0.92, size = 76, normalized size = 0.99 \begin {gather*} -\frac {15125}{5 \, x + 3} + \frac {125 \, {\left (\frac {2338497}{5 \, x + 3} + \frac {1317834}{{\left (5 \, x + 3\right )}^{2}} + \frac {338628}{{\left (5 \, x + 3\right )}^{3}} + \frac {33998}{{\left (5 \, x + 3\right )}^{4}} + 1583793\right )}}{2 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{5}} + 277750 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-15125/(5*x + 3) + 125/2*(2338497/(5*x + 3) + 1317834/(5*x + 3)^2 + 338628/(5*x + 3)^3 + 33998/(5*x + 3)^4 + 1
583793)/(1/(5*x + 3) + 3)^5 + 277750*log(abs(-1/(5*x + 3) - 3))

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maple [A]  time = 0.01, size = 72, normalized size = 0.94 \begin {gather*} 277750 \ln \left (3 x +2\right )-277750 \ln \left (5 x +3\right )-\frac {49}{15 \left (3 x +2\right )^{5}}-\frac {77}{2 \left (3 x +2\right )^{4}}-\frac {1133}{3 \left (3 x +2\right )^{3}}-\frac {3740}{\left (3 x +2\right )^{2}}-\frac {46475}{3 x +2}-\frac {15125}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2/(3*x+2)^6/(5*x+3)^2,x)

[Out]

-49/15/(3*x+2)^5-77/2/(3*x+2)^4-1133/3/(3*x+2)^3-3740/(3*x+2)^2-46475/(3*x+2)-15125/(5*x+3)+277750*ln(3*x+2)-2
77750*ln(5*x+3)

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maxima [A]  time = 0.63, size = 76, normalized size = 0.99 \begin {gather*} -\frac {674932500 \, x^{5} + 2227277250 \, x^{4} + 2939206050 \, x^{3} + 1938789435 \, x^{2} + 639246515 \, x + 84279984}{30 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - 277750 \, \log \left (5 \, x + 3\right ) + 277750 \, \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)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/30*(674932500*x^5 + 2227277250*x^4 + 2939206050*x^3 + 1938789435*x^2 + 639246515*x + 84279984)/(1215*x^6 +
4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 277750*log(5*x + 3) + 277750*log(3*x + 2)

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mupad [B]  time = 1.08, size = 66, normalized size = 0.86 \begin {gather*} 555500\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {55550\,x^5}{3}+61105\,x^4+\frac {6531569\,x^3}{81}+\frac {129252629\,x^2}{2430}+\frac {127849303\,x}{7290}+\frac {14046664}{6075}}{x^6+\frac {59\,x^5}{15}+\frac {58\,x^4}{9}+\frac {152\,x^3}{27}+\frac {224\,x^2}{81}+\frac {176\,x}{243}+\frac {32}{405}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2/((3*x + 2)^6*(5*x + 3)^2),x)

[Out]

555500*atanh(30*x + 19) - ((127849303*x)/7290 + (129252629*x^2)/2430 + (6531569*x^3)/81 + 61105*x^4 + (55550*x
^5)/3 + 14046664/6075)/((176*x)/243 + (224*x^2)/81 + (152*x^3)/27 + (58*x^4)/9 + (59*x^5)/15 + x^6 + 32/405)

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sympy [A]  time = 0.21, size = 73, normalized size = 0.95 \begin {gather*} \frac {- 674932500 x^{5} - 2227277250 x^{4} - 2939206050 x^{3} - 1938789435 x^{2} - 639246515 x - 84279984}{36450 x^{6} + 143370 x^{5} + 234900 x^{4} + 205200 x^{3} + 100800 x^{2} + 26400 x + 2880} - 277750 \log {\left (x + \frac {3}{5} \right )} + 277750 \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)**6/(3+5*x)**2,x)

[Out]

(-674932500*x**5 - 2227277250*x**4 - 2939206050*x**3 - 1938789435*x**2 - 639246515*x - 84279984)/(36450*x**6 +
143370*x**5 + 234900*x**4 + 205200*x**3 + 100800*x**2 + 26400*x + 2880) - 277750*log(x + 3/5) + 277750*log(x
+ 2/3)

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