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) - 1512
5*Log[2 + 3*x] + 15125*Log[3 + 5*x]

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Rubi [A]  time = 0.028003, antiderivative size = 70, 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} \[ \frac{3025}{3 x+2}+\frac{605}{2 (3 x+2)^2}+\frac{121}{3 (3 x+2)^3}+\frac{217}{36 (3 x+2)^4}+\frac{49}{45 (3 x+2)^5}-15125 \log (3 x+2)+15125 \log (5 x+3) \]

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*}

Mathematica [A]  time = 0.032357, 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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Maple [A]  time = 0.006, size = 63, normalized size = 0.9 \begin{align*}{\frac{49}{45\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{217}{36\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{121}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{605}{2\, \left ( 2+3\,x \right ) ^{2}}}+3025\, \left ( 2+3\,x \right ) ^{-1}-15125\,\ln \left ( 2+3\,x \right ) +15125\,\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)^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)-15125*ln(2+3*x)+15125*ln(3+5*x)

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Maxima [A]  time = 2.14474, size = 89, normalized size = 1.27 \begin{align*} \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 ) \end{align*}

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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Fricas [A]  time = 1.51863, size = 385, normalized size = 5.5 \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 0.177324, size = 61, normalized size = 0.87 \begin{align*} \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 )} \end{align*}

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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Giac [A]  time = 2.55107, size = 65, normalized size = 0.93 \begin{align*} \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 ) \end{align*}

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))