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

Optimal. Leaf size=81 \[ \frac {15125}{3 x+2}+\frac {3025}{2 (3 x+2)^2}+\frac {605}{3 (3 x+2)^3}+\frac {121}{4 (3 x+2)^4}+\frac {217}{45 (3 x+2)^5}+\frac {49}{54 (3 x+2)^6}-75625 \log (3 x+2)+75625 \log (5 x+3) \]

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Rubi [A]  time = 0.03, antiderivative size = 81, 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 {15125}{3 x+2}+\frac {3025}{2 (3 x+2)^2}+\frac {605}{3 (3 x+2)^3}+\frac {121}{4 (3 x+2)^4}+\frac {217}{45 (3 x+2)^5}+\frac {49}{54 (3 x+2)^6}-75625 \log (3 x+2)+75625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(54*(2 + 3*x)^6) + 217/(45*(2 + 3*x)^5) + 121/(4*(2 + 3*x)^4) + 605/(3*(2 + 3*x)^3) + 3025/(2*(2 + 3*x)^2)
+ 15125/(2 + 3*x) - 75625*Log[2 + 3*x] + 75625*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)^7 (3+5 x)} \, dx &=\int \left (-\frac {49}{3 (2+3 x)^7}-\frac {217}{3 (2+3 x)^6}-\frac {363}{(2+3 x)^5}-\frac {1815}{(2+3 x)^4}-\frac {9075}{(2+3 x)^3}-\frac {45375}{(2+3 x)^2}-\frac {226875}{2+3 x}+\frac {378125}{3+5 x}\right ) \, dx\\ &=\frac {49}{54 (2+3 x)^6}+\frac {217}{45 (2+3 x)^5}+\frac {121}{4 (2+3 x)^4}+\frac {605}{3 (2+3 x)^3}+\frac {3025}{2 (2+3 x)^2}+\frac {15125}{2+3 x}-75625 \log (2+3 x)+75625 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 55, normalized size = 0.68 \begin {gather*} \frac {1984702500 x^5+6681831750 x^4+9000258300 x^3+6063045615 x^2+2042732232 x+275370238}{540 (3 x+2)^6}-75625 \log (5 (3 x+2))+75625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(275370238 + 2042732232*x + 6063045615*x^2 + 9000258300*x^3 + 6681831750*x^4 + 1984702500*x^5)/(540*(2 + 3*x)^
6) - 75625*Log[5*(2 + 3*x)] + 75625*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)^7 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.34, size = 135, normalized size = 1.67 \begin {gather*} \frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 40837500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 40837500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 2042732232 \, x + 275370238}{540 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 40837500*(729*x^6 + 2916*x^5 + 4860
*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 40837500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2
160*x^2 + 576*x + 64)*log(3*x + 2) + 2042732232*x + 275370238)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216
0*x^2 + 576*x + 64)

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giac [A]  time = 0.87, size = 53, normalized size = 0.65 \begin {gather*} \frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 2042732232 \, x + 275370238}{540 \, {\left (3 \, x + 2\right )}^{6}} + 75625 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 75625 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 2042732232*x + 275370238)/(3*x + 2)
^6 + 75625*log(abs(5*x + 3)) - 75625*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 72, normalized size = 0.89 \begin {gather*} -75625 \ln \left (3 x +2\right )+75625 \ln \left (5 x +3\right )+\frac {49}{54 \left (3 x +2\right )^{6}}+\frac {217}{45 \left (3 x +2\right )^{5}}+\frac {121}{4 \left (3 x +2\right )^{4}}+\frac {605}{3 \left (3 x +2\right )^{3}}+\frac {3025}{2 \left (3 x +2\right )^{2}}+\frac {15125}{3 x +2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/54/(3*x+2)^6+217/45/(3*x+2)^5+121/4/(3*x+2)^4+605/3/(3*x+2)^3+3025/2/(3*x+2)^2+15125/(3*x+2)-75625*ln(3*x+2
)+75625*ln(5*x+3)

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maxima [A]  time = 0.53, size = 76, normalized size = 0.94 \begin {gather*} \frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 2042732232 \, x + 275370238}{540 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 75625 \, \log \left (5 \, x + 3\right ) - 75625 \, \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)^7/(3+5*x),x, algorithm="maxima")

[Out]

1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 2042732232*x + 275370238)/(729*x^6
+ 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 75625*log(5*x + 3) - 75625*log(3*x + 2)

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mupad [B]  time = 1.10, size = 65, normalized size = 0.80 \begin {gather*} \frac {\frac {15125\,x^5}{3}+\frac {305525\,x^4}{18}+\frac {1851905\,x^3}{81}+\frac {1663387\,x^2}{108}+\frac {56742562\,x}{10935}+\frac {137685119}{196830}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}}-151250\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((56742562*x)/10935 + (1663387*x^2)/108 + (1851905*x^3)/81 + (305525*x^4)/18 + (15125*x^5)/3 + 137685119/19683
0)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729) - 151250*atanh(30*x + 19)

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sympy [A]  time = 0.20, size = 71, normalized size = 0.88 \begin {gather*} \frac {1984702500 x^{5} + 6681831750 x^{4} + 9000258300 x^{3} + 6063045615 x^{2} + 2042732232 x + 275370238}{393660 x^{6} + 1574640 x^{5} + 2624400 x^{4} + 2332800 x^{3} + 1166400 x^{2} + 311040 x + 34560} + 75625 \log {\left (x + \frac {3}{5} \right )} - 75625 \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)**7/(3+5*x),x)

[Out]

(1984702500*x**5 + 6681831750*x**4 + 9000258300*x**3 + 6063045615*x**2 + 2042732232*x + 275370238)/(393660*x**
6 + 1574640*x**5 + 2624400*x**4 + 2332800*x**3 + 1166400*x**2 + 311040*x + 34560) + 75625*log(x + 3/5) - 75625
*log(x + 2/3)

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