3.2457 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=197 \[ -\frac{(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac{7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac{7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac{63 (20678 x+44365) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}-\frac{8547 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}+\frac{6620481 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

[Out]

(63*(44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) - (7*(40201 + 31
174*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600*(3 + 2*x)^3) - (7*(1301 + 1046*x)*(2 + 5*
x + 3*x^2)^(5/2))/(1920*(3 + 2*x)^5) - ((11 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(12*
(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
])/1024 + (6620481*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800
*Sqrt[5])

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Rubi [A]  time = 0.390515, 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 \[ -\frac{(3 x+11) \left (3 x^2+5 x+2\right )^{7/2}}{12 (2 x+3)^6}-\frac{7 (1046 x+1301) \left (3 x^2+5 x+2\right )^{5/2}}{1920 (2 x+3)^5}-\frac{7 (31174 x+40201) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}+\frac{63 (20678 x+44365) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}-\frac{8547 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{1024}+\frac{6620481 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]

Antiderivative was successfully verified.

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

[Out]

(63*(44365 + 20678*x)*Sqrt[2 + 5*x + 3*x^2])/(102400*(3 + 2*x)) - (7*(40201 + 31
174*x)*(2 + 5*x + 3*x^2)^(3/2))/(25600*(3 + 2*x)^3) - (7*(1301 + 1046*x)*(2 + 5*
x + 3*x^2)^(5/2))/(1920*(3 + 2*x)^5) - ((11 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(12*
(3 + 2*x)^6) - (8547*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
])/1024 + (6620481*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/(204800
*Sqrt[5])

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Rubi in Sympy [A]  time = 51.0584, size = 182, normalized size = 0.92 \[ - \frac{8547 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x + 5\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024} - \frac{6620481 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \left (- 8 x - 7\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} \right )}}{1024000} + \frac{7 \left (8932896 x + 19165680\right ) \sqrt{3 x^{2} + 5 x + 2}}{4915200 \left (2 x + 3\right )} - \frac{7 \left (2244528 x + 2894472\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1843200 \left (2 x + 3\right )^{3}} - \frac{7 \left (20920 x + 26020\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{38400 \left (2 x + 3\right )^{5}} - \frac{\left (12 x + 44\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{7}{2}}}{48 \left (2 x + 3\right )^{6}} \]

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)**7,x)

[Out]

-8547*sqrt(3)*atanh(sqrt(3)*(6*x + 5)/(6*sqrt(3*x**2 + 5*x + 2)))/1024 - 6620481
*sqrt(5)*atanh(sqrt(5)*(-8*x - 7)/(10*sqrt(3*x**2 + 5*x + 2)))/1024000 + 7*(8932
896*x + 19165680)*sqrt(3*x**2 + 5*x + 2)/(4915200*(2*x + 3)) - 7*(2244528*x + 28
94472)*(3*x**2 + 5*x + 2)**(3/2)/(1843200*(2*x + 3)**3) - 7*(20920*x + 26020)*(3
*x**2 + 5*x + 2)**(5/2)/(38400*(2*x + 3)**5) - (12*x + 44)*(3*x**2 + 5*x + 2)**(
7/2)/(48*(2*x + 3)**6)

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Mathematica [A]  time = 0.20861, size = 141, normalized size = 0.72 \[ \frac{-6620481 \sqrt{5} \log \left (2 \sqrt{5} \sqrt{3 x^2+5 x+2}-8 x-7\right )-8547000 \sqrt{3} \log \left (-2 \sqrt{9 x^2+15 x+6}-6 x-5\right )-\frac{10 \sqrt{3 x^2+5 x+2} \left (2073600 x^7-23155200 x^6-550079616 x^5-2968126160 x^4-7425343520 x^3-9799959120 x^2-6648875480 x-1835461379\right )}{3 (2 x+3)^6}+6620481 \sqrt{5} \log (2 x+3)}{1024000} \]

Antiderivative was successfully verified.

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

[Out]

((-10*Sqrt[2 + 5*x + 3*x^2]*(-1835461379 - 6648875480*x - 9799959120*x^2 - 74253
43520*x^3 - 2968126160*x^4 - 550079616*x^5 - 23155200*x^6 + 2073600*x^7))/(3*(3
+ 2*x)^6) + 6620481*Sqrt[5]*Log[3 + 2*x] - 6620481*Sqrt[5]*Log[-7 - 8*x + 2*Sqrt
[5]*Sqrt[2 + 5*x + 3*x^2]] - 8547000*Sqrt[3]*Log[-5 - 6*x - 2*Sqrt[6 + 15*x + 9*
x^2]])/1024000

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Maple [B]  time = 0.026, size = 337, normalized size = 1.7 \[ \text{result too large to display} \]

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)^7,x)

[Out]

945783/1000000*(3*(x+3/2)^2-4*x-19/4)^(7/2)+6620481/4000000*(3*(x+3/2)^2-4*x-19/
4)^(5/2)+2206827/640000*(3*(x+3/2)^2-4*x-19/4)^(3/2)+6620481/1024000*(12*(x+3/2)
^2-16*x-19)^(1/2)-13/1920/(x+3/2)^6*(3*(x+3/2)^2-4*x-19/4)^(9/2)-21/4000/(x+3/2)
^5*(3*(x+3/2)^2-4*x-19/4)^(9/2)-1143/80000/(x+3/2)^4*(3*(x+3/2)^2-4*x-19/4)^(9/2
)-459/50000/(x+3/2)^3*(3*(x+3/2)^2-4*x-19/4)^(9/2)-63693/1000000/(x+3/2)^2*(3*(x
+3/2)^2-4*x-19/4)^(9/2)+47169/125000/(x+3/2)*(3*(x+3/2)^2-4*x-19/4)^(9/2)-47169/
250000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(7/2)-349461/1000000*(5+6*x)*(3*(x+3/2)^2-
4*x-19/4)^(5/2)-104517/160000*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(3/2)-210231/128000
*(5+6*x)*(3*(x+3/2)^2-4*x-19/4)^(1/2)-8547/1024*ln(1/3*(5/2+3*x)*3^(1/2)+(3*(x+3
/2)^2-4*x-19/4)^(1/2))*3^(1/2)-6620481/1024000*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 = 0.826787, size = 502, normalized size = 2.55 \[ \frac{191079}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{30 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{21 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{1143 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{5000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{459 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{6250 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{63693 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{250000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{1048383}{500000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{368739}{4000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{47169 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{50000 \,{\left (2 \, x + 3\right )}} - \frac{313551}{80000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{116487}{640000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{630693}{64000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x - \frac{8547}{1024} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) - \frac{6620481}{1024000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) + \frac{2415861}{512000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \]

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)^7,x, algorithm="maxima")

[Out]

191079/1000000*(3*x^2 + 5*x + 2)^(7/2) - 13/30*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 +
576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 21/125*(3*x^2 + 5*x
+ 2)^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 1143/5000*(3*
x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 459/6250*(3*x^2
+ 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 63693/250000*(3*x^2 + 5*x + 2)^(
9/2)/(4*x^2 + 12*x + 9) - 1048383/500000*(3*x^2 + 5*x + 2)^(5/2)*x - 368739/4000
000*(3*x^2 + 5*x + 2)^(5/2) + 47169/50000*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) - 31
3551/80000*(3*x^2 + 5*x + 2)^(3/2)*x + 116487/640000*(3*x^2 + 5*x + 2)^(3/2) - 6
30693/64000*sqrt(3*x^2 + 5*x + 2)*x - 8547/1024*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 +
5*x + 2) + 3*x + 5/2) - 6620481/1024000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x +
2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 2415861/512000*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]  time = 0.297704, size = 325, normalized size = 1.65 \[ \frac{\sqrt{5}{\left (5128200 \, \sqrt{5} \sqrt{3}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 4 \, \sqrt{5}{\left (2073600 \, x^{7} - 23155200 \, x^{6} - 550079616 \, x^{5} - 2968126160 \, x^{4} - 7425343520 \, x^{3} - 9799959120 \, x^{2} - 6648875480 \, x - 1835461379\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + 19861443 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac{\sqrt{5}{\left (124 \, x^{2} + 212 \, x + 89\right )} + 20 \, \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{6144000 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

1/6144000*sqrt(5)*(5128200*sqrt(5)*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x
^3 + 4860*x^2 + 2916*x + 729)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 7
2*x^2 + 120*x + 49) - 4*sqrt(5)*(2073600*x^7 - 23155200*x^6 - 550079616*x^5 - 29
68126160*x^4 - 7425343520*x^3 - 9799959120*x^2 - 6648875480*x - 1835461379)*sqrt
(3*x^2 + 5*x + 2) + 19861443*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2
+ 2916*x + 729)*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)))/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^
2 + 2916*x + 729)

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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)**7,x)

[Out]

Timed out

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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)^7,x, algorithm="giac")

[Out]

undef