3.1302 \(\int \frac {(1-2 x)^2}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=70 \[ \frac {3025}{3 x+2}+\frac {605}{2 (3 x+2)^2}+\frac {121}{3 (3 x+2)^3}+\frac {217}{36 (3 x+2)^4}+\frac {49}{45 (3 x+2)^5}-15125 \log (3 x+2)+15125 \log (5 x+3) \]

[Out]

49/45/(2+3*x)^5+217/36/(2+3*x)^4+121/3/(2+3*x)^3+605/2/(2+3*x)^2+3025/(2+3*x)-15125*ln(2+3*x)+15125*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 70, 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} \[ \frac {3025}{3 x+2}+\frac {605}{2 (3 x+2)^2}+\frac {121}{3 (3 x+2)^3}+\frac {217}{36 (3 x+2)^4}+\frac {49}{45 (3 x+2)^5}-15125 \log (3 x+2)+15125 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(45*(2 + 3*x)^5) + 217/(36*(2 + 3*x)^4) + 121/(3*(2 + 3*x)^3) + 605/(2*(2 + 3*x)^2) + 3025/(2 + 3*x) - 1512
5*Log[2 + 3*x] + 15125*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)} \, dx &=\int \left (-\frac {49}{3 (2+3 x)^6}-\frac {217}{3 (2+3 x)^5}-\frac {363}{(2+3 x)^4}-\frac {1815}{(2+3 x)^3}-\frac {9075}{(2+3 x)^2}-\frac {45375}{2+3 x}+\frac {75625}{3+5 x}\right ) \, dx\\ &=\frac {49}{45 (2+3 x)^5}+\frac {217}{36 (2+3 x)^4}+\frac {121}{3 (2+3 x)^3}+\frac {605}{2 (2+3 x)^2}+\frac {3025}{2+3 x}-15125 \log (2+3 x)+15125 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 57, normalized size = 0.81 \[ \frac {44104500 x^4+119082150 x^3+120617640 x^2+54322575 x+2722500 (3 x+2)^5 \log (5 x+3)+9179006}{180 (3 x+2)^5}-15125 \log (5 (3 x+2)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-15125*Log[5*(2 + 3*x)] + (9179006 + 54322575*x + 120617640*x^2 + 119082150*x^3 + 44104500*x^4 + 2722500*(2 +
3*x)^5*Log[3 + 5*x])/(180*(2 + 3*x)^5)

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fricas [A]  time = 0.56, size = 115, normalized size = 1.64 \[ \frac {44104500 \, x^{4} + 119082150 \, x^{3} + 120617640 \, x^{2} + 2722500 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (5 \, x + 3\right ) - 2722500 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 54322575 \, x + 9179006}{180 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(44104500*x^4 + 119082150*x^3 + 120617640*x^2 + 2722500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32)*log(5*x + 3) - 2722500*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(3*x + 2) + 54322575*x +
9179006)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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giac [A]  time = 0.90, size = 48, normalized size = 0.69 \[ \frac {44104500 \, x^{4} + 119082150 \, x^{3} + 120617640 \, x^{2} + 54322575 \, x + 9179006}{180 \, {\left (3 \, x + 2\right )}^{5}} + 15125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 15125 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(44104500*x^4 + 119082150*x^3 + 120617640*x^2 + 54322575*x + 9179006)/(3*x + 2)^5 + 15125*log(abs(5*x +
3)) - 15125*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.90 \[ -15125 \ln \left (3 x +2\right )+15125 \ln \left (5 x +3\right )+\frac {49}{45 \left (3 x +2\right )^{5}}+\frac {217}{36 \left (3 x +2\right )^{4}}+\frac {121}{3 \left (3 x +2\right )^{3}}+\frac {605}{2 \left (3 x +2\right )^{2}}+\frac {3025}{3 x +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/45/(3*x+2)^5+217/36/(3*x+2)^4+121/3/(3*x+2)^3+605/2/(3*x+2)^2+3025/(3*x+2)-15125*ln(3*x+2)+15125*ln(5*x+3)

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maxima [A]  time = 0.57, size = 66, normalized size = 0.94 \[ \frac {44104500 \, x^{4} + 119082150 \, x^{3} + 120617640 \, x^{2} + 54322575 \, x + 9179006}{180 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + 15125 \, \log \left (5 \, x + 3\right ) - 15125 \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(44104500*x^4 + 119082150*x^3 + 120617640*x^2 + 54322575*x + 9179006)/(243*x^5 + 810*x^4 + 1080*x^3 + 72
0*x^2 + 240*x + 32) + 15125*log(5*x + 3) - 15125*log(3*x + 2)

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mupad [B]  time = 1.12, size = 55, normalized size = 0.79 \[ \frac {\frac {3025\,x^4}{3}+\frac {5445\,x^3}{2}+\frac {223366\,x^2}{81}+\frac {3621505\,x}{2916}+\frac {4589503}{21870}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}}-30250\,\mathrm {atanh}\left (30\,x+19\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3621505*x)/2916 + (223366*x^2)/81 + (5445*x^3)/2 + (3025*x^4)/3 + 4589503/21870)/((80*x)/81 + (80*x^2)/27 +
(40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243) - 30250*atanh(30*x + 19)

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sympy [A]  time = 0.18, size = 61, normalized size = 0.87 \[ \frac {44104500 x^{4} + 119082150 x^{3} + 120617640 x^{2} + 54322575 x + 9179006}{43740 x^{5} + 145800 x^{4} + 194400 x^{3} + 129600 x^{2} + 43200 x + 5760} + 15125 \log {\left (x + \frac {3}{5} \right )} - 15125 \log {\left (x + \frac {2}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(44104500*x**4 + 119082150*x**3 + 120617640*x**2 + 54322575*x + 9179006)/(43740*x**5 + 145800*x**4 + 194400*x*
*3 + 129600*x**2 + 43200*x + 5760) + 15125*log(x + 3/5) - 15125*log(x + 2/3)

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