3.2456 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^6} \, dx\)

Optimal. Leaf size=197 \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]

[Out]

(-21*(47145 + 21974*x)*Sqrt[2 + 5*x + 3*x^2])/(10240*(3 + 2*x)) + (7*(42733 + 33
142*x)*(2 + 5*x + 3*x^2)^(3/2))/(7680*(3 + 2*x)^3) + (7*(1003 + 548*x)*(2 + 5*x
+ 3*x^2)^(5/2))/(960*(3 + 2*x)^4) - ((27 + 5*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3
+ 2*x)^5) + (30275*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])
/1024 - (2345091*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(20480*Sq
rt[5])

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Rubi [A]  time = 0.3804, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{(5 x+27) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^5}+\frac{7 (548 x+1003) \left (3 x^2+5 x+2\right )^{5/2}}{960 (2 x+3)^4}+\frac{7 (33142 x+42733) \left (3 x^2+5 x+2\right )^{3/2}}{7680 (2 x+3)^3}-\frac{21 (21974 x+47145) \sqrt{3 x^2+5 x+2}}{10240 (2 x+3)}+\frac{30275 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}-\frac{2345091 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{20480 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

(-21*(47145 + 21974*x)*Sqrt[2 + 5*x + 3*x^2])/(10240*(3 + 2*x)) + (7*(42733 + 33
142*x)*(2 + 5*x + 3*x^2)^(3/2))/(7680*(3 + 2*x)^3) + (7*(1003 + 548*x)*(2 + 5*x
+ 3*x^2)^(5/2))/(960*(3 + 2*x)^4) - ((27 + 5*x)*(2 + 5*x + 3*x^2)^(7/2))/(30*(3
+ 2*x)^5) + (30275*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])
/1024 - (2345091*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(20480*Sq
rt[5])

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Rubi in Sympy [A]  time = 51.2613, size = 182, normalized size = 0.92 \[ \frac{30275 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024} + \frac{2345091 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{102400} - \frac{7 \left (3164256 x + 6788880\right ) \sqrt{3 x^{2} + 5 x + 2}}{491520 \left (2 x + 3\right )} + \frac{7 \left (795408 x + 1025592\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{184320 \left (2 x + 3\right )^{3}} + \frac{7 \left (2192 x + 4012\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{3840 \left (2 x + 3\right )^{4}} - \frac{\left (10 x + 54\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{60 \left (2 x + 3\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**6,x)

[Out]

30275*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/1024 + 2345091
*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/102400 - 7*(31642
56*x + 6788880)*sqrt(3*x**2 + 5*x + 2)/(491520*(2*x + 3)) + 7*(795408*x + 102559
2)*(3*x**2 + 5*x + 2)**(3/2)/(184320*(2*x + 3)**3) + 7*(2192*x + 4012)*(3*x**2 +
5*x + 2)**(5/2)/(3840*(2*x + 3)**4) - (10*x + 54)*(3*x**2 + 5*x + 2)**(7/2)/(60
*(2*x + 3)**5)

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Mathematica [A]  time = 0.171422, size = 139, normalized size = 0.71 \[ \frac{2345091 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )+3027500 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (46080 x^7-257280 x^6+483840 x^5+27897856 x^4+127665096 x^3+242016116 x^2+213122626 x+72189541\right )}{(2 x+3)^5}-2345091 \sqrt{5} \log (2 x+3)}{102400} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^6,x]

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(72189541 + 213122626*x + 242016116*x^2 + 127665096*
x^3 + 27897856*x^4 + 483840*x^5 - 257280*x^6 + 46080*x^7))/(3 + 2*x)^5 - 2345091
*Sqrt[5]*Log[3 + 2*x] + 2345091*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt[5]*Sqrt[2 + 5*x +
3*x^2]] + 3027500*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*x^2]])/102400

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Maple [A]  time = 0.022, size = 316, normalized size = 1.6 \[ -{\frac{335013}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}-{\frac{2345091}{400000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{781697}{64000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{2345091}{102400}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}-{\frac{13}{800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{27}{8000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{251}{5000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}+{\frac{10023}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{19059}{12500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{9}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{95295+114354\,x}{25000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}}}+{\frac{614355+737226\,x}{100000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}+{\frac{185185+222222\,x}{16000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{186165+223398\,x}{6400}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}+{\frac{30275\,\sqrt{3}}{1024}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}} \right ) }+{\frac{2345091\,\sqrt{5}}{102400}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^6,x)

[Out]

-335013/100000*(3*(x+3/2)^2-4*x-19/4)^(7/2)-2345091/400000*(3*(x+3/2)^2-4*x-19/4
)^(5/2)-781697/64000*(3*(x+3/2)^2-4*x-19/4)^(3/2)-2345091/102400*(12*(x+3/2)^2-1
6*x-19)^(1/2)-13/800/(x+3/2)^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-27/8000/(x+3/2)^4*(3
*(x+3/2)^2-4*x-19/4)^(9/2)-251/5000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)+10023
/100000/(x+3/2)^2*(3*(x+3/2)^2-4*x-19/4)^(9/2)-19059/12500/(x+3/2)*(3*(x+3/2)^2-
4*x-19/4)^(9/2)+19059/25000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)+122871/100000*(
5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(5/2)+37037/16000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(
3/2)+37233/6400*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)+30275/1024*ln(1/3*(5/2+3*x)
*3^(1/2)+(3*(x+3/2)^2-4*x-19/4)^(1/2))*3^(1/2)+2345091/102400*5^(1/2)*arctanh(2/
5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.823595, size = 440, normalized size = 2.23 \[ -\frac{30069}{100000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{500 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{251 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{625 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac{10023 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{25000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{368613}{50000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{112329}{400000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{19059 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{5000 \,{\left (2 \, x + 3\right )}} + \frac{111111}{8000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{40957}{64000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{111699}{3200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{30275}{1024} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{2345091}{102400} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{855771}{51200} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^6,x, algorithm="maxima")

[Out]

-30069/100000*(3*x^2 + 5*x + 2)^(7/2) - 13/25*(3*x^2 + 5*x + 2)^(9/2)/(32*x^5 +
240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 27/500*(3*x^2 + 5*x + 2)^(9/2)/(16
*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 251/625*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 +
36*x^2 + 54*x + 27) + 10023/25000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) +
368613/50000*(3*x^2 + 5*x + 2)^(5/2)*x + 112329/400000*(3*x^2 + 5*x + 2)^(5/2) -
19059/5000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 111111/8000*(3*x^2 + 5*x + 2)^(3
/2)*x - 40957/64000*(3*x^2 + 5*x + 2)^(3/2) + 111699/3200*sqrt(3*x^2 + 5*x + 2)*
x + 30275/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 2345091/
102400*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3)
- 2) - 855771/51200*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.294765, size = 305, normalized size = 1.55 \[ \frac{\sqrt{5}{\left (605500 \, \sqrt{5} \sqrt{3}{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 4 \, \sqrt{5}{\left (46080 \, x^{7} - 257280 \, x^{6} + 483840 \, x^{5} + 27897856 \, x^{4} + 127665096 \, x^{3} + 242016116 \, x^{2} + 213122626 \, x + 72189541\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 2345091 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} - 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{204800 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^6,x, algorithm="fricas")

[Out]

1/204800*sqrt(5)*(605500*sqrt(5)*sqrt(3)*(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2
+ 810*x + 243)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x +
49) - 4*sqrt(5)*(46080*x^7 - 257280*x^6 + 483840*x^5 + 27897856*x^4 + 127665096*
x^3 + 242016116*x^2 + 213122626*x + 72189541)*sqrt(3*x^2 + 5*x + 2) + 2345091*(3
2*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)*log((sqrt(5)*(124*x^2 + 212*
x + 89) - 20*sqrt(3*x^2 + 5*x + 2)*(8*x + 7))/(4*x^2 + 12*x + 9)))/(32*x^5 + 240
*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{40 \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{292 x \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{870 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{1339 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{1090 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \left (- \frac{396 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\right )\, dx - \int \frac{27 x^{7} \sqrt{3 x^{2} + 5 x + 2}}{64 x^{6} + 576 x^{5} + 2160 x^{4} + 4320 x^{3} + 4860 x^{2} + 2916 x + 729}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5-x)*(3*x**2+5*x+2)**(7/2)/(3+2*x)**6,x)

[Out]

-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3
+ 4860*x**2 + 2916*x + 729), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(64*x*
*6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integral
(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 +
4860*x**2 + 2916*x + 729), x) - Integral(-1339*x**3*sqrt(3*x**2 + 5*x + 2)/(64*x
**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integra
l(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3
+ 4860*x**2 + 2916*x + 729), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(64*
x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 + 4860*x**2 + 2916*x + 729), x) - Integr
al(27*x**7*sqrt(3*x**2 + 5*x + 2)/(64*x**6 + 576*x**5 + 2160*x**4 + 4320*x**3 +
4860*x**2 + 2916*x + 729), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(7/2)*(x - 5)/(2*x + 3)^6,x, algorithm="giac")

[Out]

undef