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

Optimal. Leaf size=62 \[ \frac {162 x^6}{25}+\frac {5508 x^5}{625}-\frac {8721 x^4}{2500}-\frac {25332 x^3}{3125}+\frac {1893 x^2}{6250}+\frac {277174 x}{78125}-\frac {121}{390625 (5 x+3)}+\frac {1771 \log (5 x+3)}{390625} \]

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Rubi [A]  time = 0.03, antiderivative size = 62, 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 {162 x^6}{25}+\frac {5508 x^5}{625}-\frac {8721 x^4}{2500}-\frac {25332 x^3}{3125}+\frac {1893 x^2}{6250}+\frac {277174 x}{78125}-\frac {121}{390625 (5 x+3)}+\frac {1771 \log (5 x+3)}{390625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(277174*x)/78125 + (1893*x^2)/6250 - (25332*x^3)/3125 - (8721*x^4)/2500 + (5508*x^5)/625 + (162*x^6)/25 - 121/
(390625*(3 + 5*x)) + (1771*Log[3 + 5*x])/390625

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)^5}{(3+5 x)^2} \, dx &=\int \left (\frac {277174}{78125}+\frac {1893 x}{3125}-\frac {75996 x^2}{3125}-\frac {8721 x^3}{625}+\frac {5508 x^4}{125}+\frac {972 x^5}{25}+\frac {121}{78125 (3+5 x)^2}+\frac {1771}{78125 (3+5 x)}\right ) \, dx\\ &=\frac {277174 x}{78125}+\frac {1893 x^2}{6250}-\frac {25332 x^3}{3125}-\frac {8721 x^4}{2500}+\frac {5508 x^5}{625}+\frac {162 x^6}{25}-\frac {121}{390625 (3+5 x)}+\frac {1771 \log (3+5 x)}{390625}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 61, normalized size = 0.98 \begin {gather*} \frac {253125000 x^7+496125000 x^6+70284375 x^5-398409375 x^4-178158750 x^3+145685750 x^2+126267855 x+35420 (5 x+3) \log (6 (5 x+3))[phone]}{7812500 (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(25866973 + 126267855*x + 145685750*x^2 - 178158750*x^3 - 398409375*x^4 + 70284375*x^5 + 496125000*x^6 + 25312
5000*x^7 + 35420*(3 + 5*x)*Log[6*(3 + 5*x)])/(7812500*(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)^5}{(3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.39, size = 57, normalized size = 0.92 \begin {gather*} \frac {50625000 \, x^{7} + 99225000 \, x^{6} + 14056875 \, x^{5} - 79681875 \, x^{4} - 35631750 \, x^{3} + 29137150 \, x^{2} + 7084 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 16630440 \, x - 484}{1562500 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1562500*(50625000*x^7 + 99225000*x^6 + 14056875*x^5 - 79681875*x^4 - 35631750*x^3 + 29137150*x^2 + 7084*(5*x
+ 3)*log(5*x + 3) + 16630440*x - 484)/(5*x + 3)

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giac [A]  time = 0.99, size = 84, normalized size = 1.35 \begin {gather*} -\frac {1}{7812500} \, {\left (5 \, x + 3\right )}^{6} {\left (\frac {36288}{5 \, x + 3} - \frac {63315}{{\left (5 \, x + 3\right )}^{2}} - \frac {249900}{{\left (5 \, x + 3\right )}^{3}} - \frac {287700}{{\left (5 \, x + 3\right )}^{4}} - \frac {204680}{{\left (5 \, x + 3\right )}^{5}} - 3240\right )} - \frac {121}{390625 \, {\left (5 \, x + 3\right )}} - \frac {1771}{390625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7812500*(5*x + 3)^6*(36288/(5*x + 3) - 63315/(5*x + 3)^2 - 249900/(5*x + 3)^3 - 287700/(5*x + 3)^4 - 204680
/(5*x + 3)^5 - 3240) - 121/390625/(5*x + 3) - 1771/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

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maple [A]  time = 0.01, size = 47, normalized size = 0.76 \begin {gather*} \frac {162 x^{6}}{25}+\frac {5508 x^{5}}{625}-\frac {8721 x^{4}}{2500}-\frac {25332 x^{3}}{3125}+\frac {1893 x^{2}}{6250}+\frac {277174 x}{78125}+\frac {1771 \ln \left (5 x +3\right )}{390625}-\frac {121}{390625 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

277174/78125*x+1893/6250*x^2-25332/3125*x^3-8721/2500*x^4+5508/625*x^5+162/25*x^6-121/390625/(5*x+3)+1771/3906
25*ln(5*x+3)

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maxima [A]  time = 0.55, size = 46, normalized size = 0.74 \begin {gather*} \frac {162}{25} \, x^{6} + \frac {5508}{625} \, x^{5} - \frac {8721}{2500} \, x^{4} - \frac {25332}{3125} \, x^{3} + \frac {1893}{6250} \, x^{2} + \frac {277174}{78125} \, x - \frac {121}{390625 \, {\left (5 \, x + 3\right )}} + \frac {1771}{390625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

162/25*x^6 + 5508/625*x^5 - 8721/2500*x^4 - 25332/3125*x^3 + 1893/6250*x^2 + 277174/78125*x - 121/390625/(5*x
+ 3) + 1771/390625*log(5*x + 3)

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mupad [B]  time = 0.04, size = 44, normalized size = 0.71 \begin {gather*} \frac {277174\,x}{78125}+\frac {1771\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {121}{1953125\,\left (x+\frac {3}{5}\right )}+\frac {1893\,x^2}{6250}-\frac {25332\,x^3}{3125}-\frac {8721\,x^4}{2500}+\frac {5508\,x^5}{625}+\frac {162\,x^6}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(277174*x)/78125 + (1771*log(x + 3/5))/390625 - 121/(1953125*(x + 3/5)) + (1893*x^2)/6250 - (25332*x^3)/3125 -
(8721*x^4)/2500 + (5508*x^5)/625 + (162*x^6)/25

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sympy [A]  time = 0.12, size = 54, normalized size = 0.87 \begin {gather*} \frac {162 x^{6}}{25} + \frac {5508 x^{5}}{625} - \frac {8721 x^{4}}{2500} - \frac {25332 x^{3}}{3125} + \frac {1893 x^{2}}{6250} + \frac {277174 x}{78125} + \frac {1771 \log {\left (5 x + 3 \right )}}{390625} - \frac {121}{1953125 x + 1171875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

162*x**6/25 + 5508*x**5/625 - 8721*x**4/2500 - 25332*x**3/3125 + 1893*x**2/6250 + 277174*x/78125 + 1771*log(5*
x + 3)/390625 - 121/(1953125*x + 1171875)

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