3.1432 \(\int \frac{(1-2 x)^3}{(2+3 x)^7 (3+5 x)^3} \, dx\)

Optimal. Leaf size=97 \[ \frac{6618975}{3 x+2}+\frac{3584625}{5 x+3}+\frac{953535}{2 (3 x+2)^2}-\frac{166375}{2 (5 x+3)^2}+\frac{42878}{(3 x+2)^3}+\frac{3927}{(3 x+2)^4}+\frac{1617}{5 (3 x+2)^5}+\frac{343}{18 (3 x+2)^6}-43848750 \log (3 x+2)[phone] \log (5 x+3) \]

[Out]

343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) +
6618975/(2 + 3*x) - 166375/(2*(3 + 5*x)^2) + 3584625/(3 + 5*x) - 43848750*Log[2 + 3*x] + 43848750*Log[3 + 5*x]

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Rubi [A]  time = 0.054347, antiderivative size = 97, 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} \[ \frac{6618975}{3 x+2}+\frac{3584625}{5 x+3}+\frac{953535}{2 (3 x+2)^2}-\frac{166375}{2 (5 x+3)^2}+\frac{42878}{(3 x+2)^3}+\frac{3927}{(3 x+2)^4}+\frac{1617}{5 (3 x+2)^5}+\frac{343}{18 (3 x+2)^6}-43848750 \log (3 x+2)[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^3),x]

[Out]

343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) +
6618975/(2 + 3*x) - 166375/(2*(3 + 5*x)^2) + 3584625/(3 + 5*x) - 43848750*Log[2 + 3*x] + 43848750*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)^3}{(2+3 x)^7 (3+5 x)^3} \, dx &=\int \left (-\frac{343}{(2+3 x)^7}-\frac{4851}{(2+3 x)^6}-\frac{47124}{(2+3 x)^5}-\frac{385902}{(2+3 x)^4}-\frac{2860605}{(2+3 x)^3}-\frac{19856925}{(2+3 x)^2}-\frac{131546250}{2+3 x}+\frac{831875}{(3+5 x)^3}-\frac{17923125}{(3+5 x)^2}+\frac{219243750}{3+5 x}\right ) \, dx\\ &=\frac{343}{18 (2+3 x)^6}+\frac{1617}{5 (2+3 x)^5}+\frac{3927}{(2+3 x)^4}+\frac{42878}{(2+3 x)^3}+\frac{953535}{2 (2+3 x)^2}+\frac{6618975}{2+3 x}-\frac{166375}{2 (3+5 x)^2}+\frac{3584625}{3+5 x}-43848750 \log (2+3 x)[phone] \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.097724, size = 99, normalized size = 1.02 \[ \frac{6618975}{3 x+2}+\frac{3584625}{5 x+3}+\frac{953535}{2 (3 x+2)^2}-\frac{166375}{2 (5 x+3)^2}+\frac{42878}{(3 x+2)^3}+\frac{3927}{(3 x+2)^4}+\frac{1617}{5 (3 x+2)^5}+\frac{343}{18 (3 x+2)^6}-43848750 \log (5 (3 x+2))[phone] \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^3),x]

[Out]

343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) +
6618975/(2 + 3*x) - 166375/(2*(3 + 5*x)^2) + 3584625/(3 + 5*x) - 43848750*Log[5*(2 + 3*x)] + 43848750*Log[3 +
5*x]

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Maple [A]  time = 0.008, size = 90, normalized size = 0.9 \begin{align*}{\frac{343}{18\, \left ( 2+3\,x \right ) ^{6}}}+{\frac{1617}{5\, \left ( 2+3\,x \right ) ^{5}}}+3927\, \left ( 2+3\,x \right ) ^{-4}+42878\, \left ( 2+3\,x \right ) ^{-3}+{\frac{953535}{2\, \left ( 2+3\,x \right ) ^{2}}}+6618975\, \left ( 2+3\,x \right ) ^{-1}-{\frac{166375}{2\, \left ( 3+5\,x \right ) ^{2}}}+3584625\, \left ( 3+5\,x \right ) ^{-1}-43848750\,\ln \left ( 2+3\,x \right ) [phone]\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3/(2+3*x)^7/(3+5*x)^3,x)

[Out]

343/18/(2+3*x)^6+1617/5/(2+3*x)^5+3927/(2+3*x)^4+42878/(2+3*x)^3+953535/2/(2+3*x)^2+6618975/(2+3*x)-166375/2/(
3+5*x)^2+3584625/(3+5*x)-43848750*ln(2+3*x)[phone]*ln(3+5*x)

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Maxima [A]  time = 1.07315, size = 130, normalized size = 1.34 \begin{align*} \frac{4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} + 43848750 \, \log \left (5 \, x + 3\right ) - 43848750 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +
11860532030465*x^2 + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340
*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576) + 43848750*log(5*x + 3) - 43848750*log(3*x + 2)

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Fricas [A]  time = 1.2836, size = 709, normalized size = 7.31 \begin{align*} \frac{4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 3946387500 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (5 \, x + 3\right ) - 3946387500 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (3 \, x + 2\right ) + 2574943269792 \, x + 239497011063}{90 \,{\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +
11860532030465*x^2 + 3946387500*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 3
8320*x^2 + 7104*x + 576)*log(5*x + 3) - 3946387500*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x
^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)*log(3*x + 2) + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*
x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)

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Sympy [A]  time = 0.231774, size = 92, normalized size = 0.95 \begin{align*} \frac{4794860812500 x^{7} + 21896531043750 x^{6} + 42841193422500 x^{5} + 46551705341625 x^{4} + 30340145968110 x^{3} + 11860532030465 x^{2} + 2574943269792 x + 239497011063}{1640250 x^{8} + 8529300 x^{7} + 19398690 x^{6} + 25203960 x^{5} + 20460600 x^{4} + 10627200 x^{3} + 3448800 x^{2} + 639360 x + 51840} + 43848750 \log{\left (x + \frac{3}{5} \right )} - 43848750 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3/(2+3*x)**7/(3+5*x)**3,x)

[Out]

(4794860812500*x**7 + 21896531043750*x**6 + 42841193422500*x**5 + 46551705341625*x**4 + 30340145968110*x**3 +
11860532030465*x**2 + 2574943269792*x + 239497011063)/(1640250*x**8 + 8529300*x**7 + 19398690*x**6 + 25203960*
x**5 + 20460600*x**4 + 10627200*x**3 + 3448800*x**2 + 639360*x + 51840) + 43848750*log(x + 3/5) - 43848750*log
(x + 2/3)

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Giac [A]  time = 3.02193, size = 95, normalized size = 0.98 \begin{align*} \frac{4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \,{\left (5 \, x + 3\right )}^{2}{\left (3 \, x + 2\right )}^{6}} + 43848750 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 43848750 \, \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)^3/(2+3*x)^7/(3+5*x)^3,x, algorithm="giac")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +
11860532030465*x^2 + 2574943269792*x + 239497011063)/((5*x + 3)^2*(3*x + 2)^6) + 43848750*log(abs(5*x + 3)) -
43848750*log(abs(3*x + 2))