Optimal. Leaf size=197 \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.392961, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]
Antiderivative was successfully verified.
[In] Int[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 50.8971, size = 180, normalized size = 0.91 \[ - \frac{27 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{512} - \frac{1673211 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{40960000} + \frac{\left (156277440 x + 201542760\right ) \sqrt{3 x^{2} + 5 x + 2}}{491520000 \left (2 x + 3\right )^{2}} + \frac{\left (3077280 x + 3772620\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{18432000 \left (2 x + 3\right )^{4}} + \frac{\left (19920 x + 26430\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{192000 \left (2 x + 3\right )^{6}} + \frac{\left (808 x + 757\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{1120 \left (2 x + 3\right )^{8}} \]
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)**9,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.266222, size = 139, normalized size = 0.71 \[ \frac{-11712477 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-15120000 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )+\frac{10 \sqrt{3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{(2 x+3)^8}+11712477 \sqrt{5} \log (2 x+3)}{286720000} \]
Antiderivative was successfully verified.
[In] Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^9,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.041, size = 379, normalized size = 1.9 \[ \text{result too large to display} \]
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)^9,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.825794, size = 647, normalized size = 3.28 \[ \text{result too large to display} \]
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)^9,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.299465, size = 366, normalized size = 1.86 \[ \frac{\sqrt{5}{\left (3024000 \, \sqrt{5} \sqrt{3}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 4 \, \sqrt{5}{\left (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 11712477 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{573440000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} \]
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)^9,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**9,x)
[Out]
_______________________________________________________________________________________
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)^9,x, algorithm="giac")
[Out]