3.1322 \(\int \frac{(1-2 x)^2 (2+3 x)^7}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{8748 x^7}{875}+\frac{13608 x^6}{625}+\frac{104247 x^5}{15625}-\frac{193833 x^4}{12500}-\frac{162612 x^3}{15625}+\frac{1390203 x^2}{390625}+\frac{9251661 x}{1953125}-\frac{2497}{9765625 (5 x+3)}-\frac{121}{19531250 (5 x+3)^2}+\frac{21949 \log (5 x+3)}{9765625} \]

[Out]

(9251661*x)/1953125 + (1390203*x^2)/390625 - (162612*x^3)/15625 - (193833*x^4)/12500 + (104247*x^5)/15625 + (1
3608*x^6)/625 + (8748*x^7)/875 - 121/(19531250*(3 + 5*x)^2) - 2497/(9765625*(3 + 5*x)) + (21949*Log[3 + 5*x])/
9765625

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Rubi [A]  time = 0.0423137, antiderivative size = 80, 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} \[ \frac{8748 x^7}{875}+\frac{13608 x^6}{625}+\frac{104247 x^5}{15625}-\frac{193833 x^4}{12500}-\frac{162612 x^3}{15625}+\frac{1390203 x^2}{390625}+\frac{9251661 x}{1953125}-\frac{2497}{9765625 (5 x+3)}-\frac{121}{19531250 (5 x+3)^2}+\frac{21949 \log (5 x+3)}{9765625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9251661*x)/1953125 + (1390203*x^2)/390625 - (162612*x^3)/15625 - (193833*x^4)/12500 + (104247*x^5)/15625 + (1
3608*x^6)/625 + (8748*x^7)/875 - 121/(19531250*(3 + 5*x)^2) - 2497/(9765625*(3 + 5*x)) + (21949*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)^3} \, dx &=\int \left (\frac{9251661}{1953125}+\frac{2780406 x}{390625}-\frac{487836 x^2}{15625}-\frac{193833 x^3}{3125}+\frac{104247 x^4}{3125}+\frac{81648 x^5}{625}+\frac{8748 x^6}{125}+\frac{121}{1953125 (3+5 x)^3}+\frac{2497}{1953125 (3+5 x)^2}+\frac{21949}{1953125 (3+5 x)}\right ) \, dx\\ &=\frac{9251661 x}{1953125}+\frac{1390203 x^2}{390625}-\frac{162612 x^3}{15625}-\frac{193833 x^4}{12500}+\frac{104247 x^5}{15625}+\frac{13608 x^6}{625}+\frac{8748 x^7}{875}-\frac{121}{19531250 (3+5 x)^2}-\frac{2497}{9765625 (3+5 x)}+\frac{21949 \log (3+5 x)}{9765625}\\ \end{align*}

Mathematica [A]  time = 0.0275327, size = 71, normalized size = 0.89 \[ \frac{341718750000 x^9+1154250000000 x^8+1244084062500 x^7+11543765625 x^6-909633768750 x^5-496018096875 x^4+179818432500 x^3+275860261575 x^2+103624499690 x+3072860 (5 x+3)^2 \log (5 x+3)[phone]}{1367187500 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13601177777 + 103624499690*x + 275860261575*x^2 + 179818432500*x^3 - 496018096875*x^4 - 909633768750*x^5 + 11
543765625*x^6 + 1244084062500*x^7 + 1154250000000*x^8 + 341718750000*x^9 + 3072860*(3 + 5*x)^2*Log[3 + 5*x])/(
1367187500*(3 + 5*x)^2)

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Maple [A]  time = 0.008, size = 61, normalized size = 0.8 \begin{align*}{\frac{9251661\,x}{1953125}}+{\frac{1390203\,{x}^{2}}{390625}}-{\frac{162612\,{x}^{3}}{15625}}-{\frac{193833\,{x}^{4}}{12500}}+{\frac{104247\,{x}^{5}}{15625}}+{\frac{13608\,{x}^{6}}{625}}+{\frac{8748\,{x}^{7}}{875}}-{\frac{121}{19531250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{2497}{29296875+48828125\,x}}+{\frac{21949\,\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)^3,x)

[Out]

9251661/1953125*x+1390203/390625*x^2-162612/15625*x^3-193833/12500*x^4+104247/15625*x^5+13608/625*x^6+8748/875
*x^7-121/19531250/(3+5*x)^2-2497/9765625/(3+5*x)+21949/9765625*ln(3+5*x)

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Maxima [A]  time = 2.7805, size = 82, normalized size = 1.02 \begin{align*} \frac{8748}{875} \, x^{7} + \frac{13608}{625} \, x^{6} + \frac{104247}{15625} \, x^{5} - \frac{193833}{12500} \, x^{4} - \frac{162612}{15625} \, x^{3} + \frac{1390203}{390625} \, x^{2} + \frac{9251661}{1953125} \, x - \frac{11 \,{\left (2270 \, x + 1373\right )}}{19531250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{21949}{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)^3,x, algorithm="maxima")

[Out]

8748/875*x^7 + 13608/625*x^6 + 104247/15625*x^5 - 193833/12500*x^4 - 162612/15625*x^3 + 1390203/390625*x^2 + 9
251661/1953125*x - 11/19531250*(2270*x + 1373)/(25*x^2 + 30*x + 9) + 21949/9765625*log(5*x + 3)

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Fricas [A]  time = 1.22276, size = 332, normalized size = 4.15 \begin{align*} \frac{68343750000 \, x^{9} + 230850000000 \, x^{8} + 248816812500 \, x^{7} + 2308753125 \, x^{6} - 181926753750 \, x^{5} - 99203619375 \, x^{4} + 35963686500 \, x^{3} + 47615255100 \, x^{2} + 614572 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 11656743280 \, x - 211442}{273437500 \,{\left (25 \, x^{2} + 30 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/273437500*(68343750000*x^9 + 230850000000*x^8 + 248816812500*x^7 + 2308753125*x^6 - 181926753750*x^5 - 99203
619375*x^4 + 35963686500*x^3 + 47615255100*x^2 + 614572*(25*x^2 + 30*x + 9)*log(5*x + 3) + 11656743280*x - 211
442)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.130017, size = 70, normalized size = 0.88 \begin{align*} \frac{8748 x^{7}}{875} + \frac{13608 x^{6}}{625} + \frac{104247 x^{5}}{15625} - \frac{193833 x^{4}}{12500} - \frac{162612 x^{3}}{15625} + \frac{1390203 x^{2}}{390625} + \frac{9251661 x}{1953125} - \frac{24970 x + 15103}{488281250 x^{2} + 585937500 x + 175781250} + \frac{21949 \log{\left (5 x + 3 \right )}}{9765625} \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)**3,x)

[Out]

8748*x**7/875 + 13608*x**6/625 + 104247*x**5/15625 - 193833*x**4/12500 - 162612*x**3/15625 + 1390203*x**2/3906
25 + 9251661*x/1953125 - (24970*x + 15103)/(488281250*x**2 + 585937500*x + 175781250) + 21949*log(5*x + 3)/976
5625

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Giac [A]  time = 1.49997, size = 77, normalized size = 0.96 \begin{align*} \frac{8748}{875} \, x^{7} + \frac{13608}{625} \, x^{6} + \frac{104247}{15625} \, x^{5} - \frac{193833}{12500} \, x^{4} - \frac{162612}{15625} \, x^{3} + \frac{1390203}{390625} \, x^{2} + \frac{9251661}{1953125} \, x - \frac{11 \,{\left (2270 \, x + 1373\right )}}{19531250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{21949}{9765625} \, \log \left ({\left | 5 \, x + 3 \right |}\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)^3,x, algorithm="giac")

[Out]

8748/875*x^7 + 13608/625*x^6 + 104247/15625*x^5 - 193833/12500*x^4 - 162612/15625*x^3 + 1390203/390625*x^2 + 9
251661/1953125*x - 11/19531250*(2270*x + 1373)/(5*x + 3)^2 + 21949/9765625*log(abs(5*x + 3))