3.1304 \(\int \frac{(1-2 x)^2}{(2+3 x)^8 (3+5 x)} \, dx\)

Optimal. Leaf size=92 \[ \frac{75625}{3 x+2}+\frac{15125}{2 (3 x+2)^2}+\frac{3025}{3 (3 x+2)^3}+\frac{605}{4 (3 x+2)^4}+\frac{121}{5 (3 x+2)^5}+\frac{217}{54 (3 x+2)^6}+\frac{7}{9 (3 x+2)^7}-378125 \log (3 x+2)+378125 \log (5 x+3) \]

[Out]

7/(9*(2 + 3*x)^7) + 217/(54*(2 + 3*x)^6) + 121/(5*(2 + 3*x)^5) + 605/(4*(2 + 3*x)^4) + 3025/(3*(2 + 3*x)^3) +
15125/(2*(2 + 3*x)^2) + 75625/(2 + 3*x) - 378125*Log[2 + 3*x] + 378125*Log[3 + 5*x]

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Rubi [A]  time = 0.034797, antiderivative size = 92, 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} \[ \frac{75625}{3 x+2}+\frac{15125}{2 (3 x+2)^2}+\frac{3025}{3 (3 x+2)^3}+\frac{605}{4 (3 x+2)^4}+\frac{121}{5 (3 x+2)^5}+\frac{217}{54 (3 x+2)^6}+\frac{7}{9 (3 x+2)^7}-378125 \log (3 x+2)+378125 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)),x]

[Out]

7/(9*(2 + 3*x)^7) + 217/(54*(2 + 3*x)^6) + 121/(5*(2 + 3*x)^5) + 605/(4*(2 + 3*x)^4) + 3025/(3*(2 + 3*x)^3) +
15125/(2*(2 + 3*x)^2) + 75625/(2 + 3*x) - 378125*Log[2 + 3*x] + 378125*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)^8 (3+5 x)} \, dx &=\int \left (-\frac{49}{3 (2+3 x)^8}-\frac{217}{3 (2+3 x)^7}-\frac{363}{(2+3 x)^6}-\frac{1815}{(2+3 x)^5}-\frac{9075}{(2+3 x)^4}-\frac{45375}{(2+3 x)^3}-\frac{226875}{(2+3 x)^2}-\frac{1134375}{2+3 x}+\frac{1890625}{3+5 x}\right ) \, dx\\ &=\frac{7}{9 (2+3 x)^7}+\frac{217}{54 (2+3 x)^6}+\frac{121}{5 (2+3 x)^5}+\frac{605}{4 (2+3 x)^4}+\frac{3025}{3 (2+3 x)^3}+\frac{15125}{2 (2+3 x)^2}+\frac{75625}{2+3 x}-378125 \log (2+3 x)+378125 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0600072, size = 84, normalized size = 0.91 \[ \frac{40837500 (3 x+2)^6+4083750 (3 x+2)^5+544500 (3 x+2)^4+81675 (3 x+2)^3+13068 (3 x+2)^2+2170 (3 x+2)+420}{540 (3 x+2)^7}-378125 \log (5 (3 x+2))+378125 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^8*(3 + 5*x)),x]

[Out]

(420 + 2170*(2 + 3*x) + 13068*(2 + 3*x)^2 + 81675*(2 + 3*x)^3 + 544500*(2 + 3*x)^4 + 4083750*(2 + 3*x)^5 + 408
37500*(2 + 3*x)^6)/(540*(2 + 3*x)^7) - 378125*Log[5*(2 + 3*x)] + 378125*Log[3 + 5*x]

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Maple [A]  time = 0.007, size = 81, normalized size = 0.9 \begin{align*}{\frac{7}{9\, \left ( 2+3\,x \right ) ^{7}}}+{\frac{217}{54\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{121}{5\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{605}{4\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{3025}{3\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{15125}{2\, \left ( 2+3\,x \right ) ^{2}}}+75625\, \left ( 2+3\,x \right ) ^{-1}-378125\,\ln \left ( 2+3\,x \right ) +378125\,\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)^8/(3+5*x),x)

[Out]

7/9/(2+3*x)^7+217/54/(2+3*x)^6+121/5/(2+3*x)^5+605/4/(2+3*x)^4+3025/3/(2+3*x)^3+15125/2/(2+3*x)^2+75625/(2+3*x
)-378125*ln(2+3*x)+378125*ln(3+5*x)

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Maxima [A]  time = 1.17076, size = 116, normalized size = 1.26 \begin{align*} \frac{29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 24557875626 \, x + 2753702432}{540 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + 378125 \, \log \left (5 \, x + 3\right ) - 378125 \, \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)^8/(3+5*x),x, algorithm="maxima")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 2455787562
6*x + 2753702432)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128) + 37812
5*log(5*x + 3) - 378125*log(3*x + 2)

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Fricas [A]  time = 1.48029, size = 586, normalized size = 6.37 \begin{align*} \frac{29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 204187500 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (5 \, x + 3\right ) - 204187500 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 24557875626 \, x + 2753702432}{540 \,{\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^8/(3+5*x),x, algorithm="fricas")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 204187500*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(5*x + 3) - 204187500*
(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 2455787562
6*x + 2753702432)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

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Sympy [A]  time = 0.204901, size = 82, normalized size = 0.89 \begin{align*} \frac{29770537500 x^{6} + 120074501250 x^{5} + 201822192000 x^{4} + 180948267225 x^{3} + 91271440062 x^{2} + 24557875626 x + 2753702432}{1180980 x^{7} + 5511240 x^{6} + 11022480 x^{5} + 12247200 x^{4} + 8164800 x^{3} + 3265920 x^{2} + 725760 x + 69120} + 378125 \log{\left (x + \frac{3}{5} \right )} - 378125 \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)**8/(3+5*x),x)

[Out]

(29770537500*x**6 + 120074501250*x**5 + 201822192000*x**4 + 180948267225*x**3 + 91271440062*x**2 + 24557875626
*x + 2753702432)/(1180980*x**7 + 5511240*x**6 + 11022480*x**5 + 12247200*x**4 + 8164800*x**3 + 3265920*x**2 +
725760*x + 69120) + 378125*log(x + 3/5) - 378125*log(x + 2/3)

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Giac [A]  time = 2.74745, size = 78, normalized size = 0.85 \begin{align*} \frac{29770537500 \, x^{6} + 120074501250 \, x^{5} + 201822192000 \, x^{4} + 180948267225 \, x^{3} + 91271440062 \, x^{2} + 24557875626 \, x + 2753702432}{540 \,{\left (3 \, x + 2\right )}^{7}} + 378125 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 378125 \, \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)^8/(3+5*x),x, algorithm="giac")

[Out]

1/540*(29770537500*x^6 + 120074501250*x^5 + 201822192000*x^4 + 180948267225*x^3 + 91271440062*x^2 + 2455787562
6*x + 2753702432)/(3*x + 2)^7 + 378125*log(abs(5*x + 3)) - 378125*log(abs(3*x + 2))