3.1305 \(\int \frac{(1-2 x)^2 (2+3 x)^7}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=76 \[ \frac{2187 x^8}{50}+\frac{107892 x^7}{875}+\frac{116397 x^6}{1250}-\frac{656424 x^5}{15625}-\frac{213867 x^4}{2500}-\frac{1512378 x^3}{78125}+\frac{17592879 x^2}{781250}+\frac{27776932 x}{1953125}-\frac{121}{9765625 (5 x+3)}+\frac{2497 \log (5 x+3)}{9765625} \]

[Out]

(27776932*x)/1953125 + (17592879*x^2)/781250 - (1512378*x^3)/78125 - (213867*x^4)/2500 - (656424*x^5)/15625 +
(116397*x^6)/1250 + (107892*x^7)/875 + (2187*x^8)/50 - 121/(9765625*(3 + 5*x)) + (2497*Log[3 + 5*x])/9765625

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Rubi [A]  time = 0.044442, antiderivative size = 76, 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} \[ \frac{2187 x^8}{50}+\frac{107892 x^7}{875}+\frac{116397 x^6}{1250}-\frac{656424 x^5}{15625}-\frac{213867 x^4}{2500}-\frac{1512378 x^3}{78125}+\frac{17592879 x^2}{781250}+\frac{27776932 x}{1953125}-\frac{121}{9765625 (5 x+3)}+\frac{2497 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27776932*x)/1953125 + (17592879*x^2)/781250 - (1512378*x^3)/78125 - (213867*x^4)/2500 - (656424*x^5)/15625 +
(116397*x^6)/1250 + (107892*x^7)/875 + (2187*x^8)/50 - 121/(9765625*(3 + 5*x)) + (2497*Log[3 + 5*x])/9765625

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)^2} \, dx &=\int \left (\frac{27776932}{1953125}+\frac{17592879 x}{390625}-\frac{4537134 x^2}{78125}-\frac{213867 x^3}{625}-\frac{656424 x^4}{3125}+\frac{349191 x^5}{625}+\frac{107892 x^6}{125}+\frac{8748 x^7}{25}+\frac{121}{1953125 (3+5 x)^2}+\frac{2497}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac{27776932 x}{1953125}+\frac{17592879 x^2}{781250}-\frac{1512378 x^3}{78125}-\frac{213867 x^4}{2500}-\frac{656424 x^5}{15625}+\frac{116397 x^6}{1250}+\frac{107892 x^7}{875}+\frac{2187 x^8}{50}-\frac{121}{9765625 (3+5 x)}+\frac{2497 \log (3+5 x)}{9765625}\\ \end{align*}

Mathematica [A]  time = 0.0272867, size = 69, normalized size = 0.91 \[ \frac{299003906250 x^9+1022308593750 x^8+1142289843750 x^7+94742156250 x^6-757103878125 x^5-483208621875 x^4+74537846250 x^3+189581876750 x^2+74994343395 x+349580 (5 x+3) \log (5 x+3)[phone]}{1367187500 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9997654777 + 74994343395*x + 189581876750*x^2 + 74537846250*x^3 - 483208621875*x^4 - 757103878125*x^5 + 94742
156250*x^6 + 1142289843750*x^7 + 1022308593750*x^8 + 299003906250*x^9 + 349580*(3 + 5*x)*Log[3 + 5*x])/(136718
7500*(3 + 5*x))

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Maple [A]  time = 0.006, size = 57, normalized size = 0.8 \begin{align*}{\frac{27776932\,x}{1953125}}+{\frac{17592879\,{x}^{2}}{781250}}-{\frac{1512378\,{x}^{3}}{78125}}-{\frac{213867\,{x}^{4}}{2500}}-{\frac{656424\,{x}^{5}}{15625}}+{\frac{116397\,{x}^{6}}{1250}}+{\frac{107892\,{x}^{7}}{875}}+{\frac{2187\,{x}^{8}}{50}}-{\frac{121}{29296875+48828125\,x}}+{\frac{2497\,\ln \left ( 3+5\,x \right ) }{9765625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27776932/1953125*x+17592879/781250*x^2-1512378/78125*x^3-213867/2500*x^4-656424/15625*x^5+116397/1250*x^6+1078
92/875*x^7+2187/50*x^8-121/9765625/(3+5*x)+2497/9765625*ln(3+5*x)

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Maxima [A]  time = 1.12497, size = 76, normalized size = 1. \begin{align*} \frac{2187}{50} \, x^{8} + \frac{107892}{875} \, x^{7} + \frac{116397}{1250} \, x^{6} - \frac{656424}{15625} \, x^{5} - \frac{213867}{2500} \, x^{4} - \frac{1512378}{78125} \, x^{3} + \frac{17592879}{781250} \, x^{2} + \frac{27776932}{1953125} \, x - \frac{121}{9765625 \,{\left (5 \, x + 3\right )}} + \frac{2497}{9765625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2187/50*x^8 + 107892/875*x^7 + 116397/1250*x^6 - 656424/15625*x^5 - 213867/2500*x^4 - 1512378/78125*x^3 + 1759
2879/781250*x^2 + 27776932/1953125*x - 121/9765625/(5*x + 3) + 2497/9765625*log(5*x + 3)

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Fricas [A]  time = 1.44952, size = 302, normalized size = 3.97 \begin{align*} \frac{59800781250 \, x^{9} + 204461718750 \, x^{8} + 228457968750 \, x^{7} + 18948431250 \, x^{6} - 151420775625 \, x^{5} - 96641724375 \, x^{4} + 14907569250 \, x^{3} + 37916375350 \, x^{2} + 69916 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 11666311440 \, x - 3388}{273437500 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/273437500*(59800781250*x^9 + 204461718750*x^8 + 228457968750*x^7 + 18948431250*x^6 - 151420775625*x^5 - 9664
1724375*x^4 + 14907569250*x^3 + 37916375350*x^2 + 69916*(5*x + 3)*log(5*x + 3) + 11666311440*x - 3388)/(5*x +
3)

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Sympy [A]  time = 0.114421, size = 68, normalized size = 0.89 \begin{align*} \frac{2187 x^{8}}{50} + \frac{107892 x^{7}}{875} + \frac{116397 x^{6}}{1250} - \frac{656424 x^{5}}{15625} - \frac{213867 x^{4}}{2500} - \frac{1512378 x^{3}}{78125} + \frac{17592879 x^{2}}{781250} + \frac{27776932 x}{1953125} + \frac{2497 \log{\left (5 x + 3 \right )}}{9765625} - \frac{121}{48828125 x + 29296875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2187*x**8/50 + 107892*x**7/875 + 116397*x**6/1250 - 656424*x**5/15625 - 213867*x**4/2500 - 1512378*x**3/78125
+ 17592879*x**2/781250 + 27776932*x/1953125 + 2497*log(5*x + 3)/9765625 - 121/(48828125*x + 29296875)

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Giac [A]  time = 2.66023, size = 138, normalized size = 1.82 \begin{align*} -\frac{1}{1367187500} \,{\left (5 \, x + 3\right )}^{8}{\left (\frac{1516320}{5 \, x + 3} - \frac{1411830}{{\left (5 \, x + 3\right )}^{2}} - \frac{11319588}{{\left (5 \, x + 3\right )}^{3}} - \frac{17377605}{{\left (5 \, x + 3\right )}^{4}} - \frac{14103180}{{\left (5 \, x + 3\right )}^{5}} - \frac{7427910}{{\left (5 \, x + 3\right )}^{6}} - \frac{3072860}{{\left (5 \, x + 3\right )}^{7}} - 153090\right )} - \frac{121}{9765625 \,{\left (5 \, x + 3\right )}} - \frac{2497}{9765625} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1367187500*(5*x + 3)^8*(1516320/(5*x + 3) - 1411830/(5*x + 3)^2 - 11319588/(5*x + 3)^3 - 17377605/(5*x + 3)
^4 - 14103180/(5*x + 3)^5 - 7427910/(5*x + 3)^6 - 3072860/(5*x + 3)^7 - 153090) - 121/9765625/(5*x + 3) - 2497
/9765625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)