3.1392 \(\int \frac{(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\)

Optimal. Leaf size=133 \[ \frac{(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}+\frac{(403 x+202) \left (3 x^2+2\right )^{3/2}}{1568 (2 x+3)^4}+\frac{9 (5167 x+4373) \sqrt{3 x^2+2}}{109760 (2 x+3)^2}-\frac{159759 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{219520 \sqrt{35}}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(9*(4373 + 5167*x)*Sqrt[2 + 3*x^2])/(109760*(3 + 2*x)^2) + ((202 + 403*x)*(2 + 3
*x^2)^(3/2))/(1568*(3 + 2*x)^4) + ((11 + 159*x)*(2 + 3*x^2)^(5/2))/(420*(3 + 2*x
)^6) - (9*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (159759*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(219520*Sqrt[35])

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Rubi [A]  time = 0.238452, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{(159 x+11) \left (3 x^2+2\right )^{5/2}}{420 (2 x+3)^6}+\frac{(403 x+202) \left (3 x^2+2\right )^{3/2}}{1568 (2 x+3)^4}+\frac{9 (5167 x+4373) \sqrt{3 x^2+2}}{109760 (2 x+3)^2}-\frac{159759 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{219520 \sqrt{35}}-\frac{9}{128} \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

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

[Out]

(9*(4373 + 5167*x)*Sqrt[2 + 3*x^2])/(109760*(3 + 2*x)^2) + ((202 + 403*x)*(2 + 3
*x^2)^(3/2))/(1568*(3 + 2*x)^4) + ((11 + 159*x)*(2 + 3*x^2)^(5/2))/(420*(3 + 2*x
)^6) - (9*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/128 - (159759*ArcTanh[(4 - 9*x)/(Sqrt[35
]*Sqrt[2 + 3*x^2])])/(219520*Sqrt[35])

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Rubi in Sympy [A]  time = 24.0726, size = 121, normalized size = 0.91 \[ - \frac{9 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{128} - \frac{159759 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{7683200} + \frac{\left (223214400 x + 188913600\right ) \sqrt{3 x^{2} + 2}}{526848000 \left (2 x + 3\right )^{2}} + \frac{\left (725400 x + 363600\right ) \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{2822400 \left (2 x + 3\right )^{4}} + \frac{\left (1590 x + 110\right ) \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{4200 \left (2 x + 3\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9*sqrt(3)*asinh(sqrt(6)*x/2)/128 - 159759*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(3
5*sqrt(3*x**2 + 2)))/7683200 + (223214400*x + 188913600)*sqrt(3*x**2 + 2)/(52684
8000*(2*x + 3)**2) + (725400*x + 363600)*(3*x**2 + 2)**(3/2)/(2822400*(2*x + 3)*
*4) + (1590*x + 110)*(3*x**2 + 2)**(5/2)/(4200*(2*x + 3)**6)

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Mathematica [A]  time = 0.232995, size = 112, normalized size = 0.84 \[ \frac{-958554 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )+\frac{140 \sqrt{3 x^2+2} \left (4369608 x^5+18915336 x^4+47453802 x^3+59256588 x^2+39843609 x+10361807\right )}{(2 x+3)^6}+958554 \sqrt{35} \log (2 x+3)-3241350 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{46099200} \]

Antiderivative was successfully verified.

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

[Out]

((140*Sqrt[2 + 3*x^2]*(10361807 + 39843609*x + 59256588*x^2 + 47453802*x^3 + 189
15336*x^4 + 4369608*x^5))/(3 + 2*x)^6 - 3241350*Sqrt[3]*ArcSinh[Sqrt[3/2]*x] + 9
58554*Sqrt[35]*Log[3 + 2*x] - 958554*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 +
3*x^2])])/46099200

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Maple [B]  time = 0.025, size = 269, normalized size = 2. \[ -{\frac{41043}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}-{\frac{45711\,x}{3841600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}-{\frac{159759\,\sqrt{35}}{7683200}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) }+{\frac{123129\,x}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}}-{\frac{1}{3136} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-5}}-{\frac{9\,\sqrt{3}}{128}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }-{\frac{13}{13440} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-6}}-{\frac{113}{548800} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{1039}{9604000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{27009\,x}{67228000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}-{\frac{6561}{84035000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}+{\frac{53253}{33614000} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{3}{2}}}}+{\frac{159759}{7683200}\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}+{\frac{159759}{1470612500} \left ( 3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-41043/1470612500/(x+3/2)*(3*(x+3/2)^2-9*x-19/4)^(7/2)-45711/3841600*x*(3*(x+3/2
)^2-9*x-19/4)^(1/2)-159759/7683200*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x
+3/2)^2-36*x-19)^(1/2))+123129/1470612500*x*(3*(x+3/2)^2-9*x-19/4)^(5/2)-1/3136/
(x+3/2)^5*(3*(x+3/2)^2-9*x-19/4)^(7/2)-9/128*arcsinh(1/2*x*6^(1/2))*3^(1/2)-13/1
3440/(x+3/2)^6*(3*(x+3/2)^2-9*x-19/4)^(7/2)-113/548800/(x+3/2)^4*(3*(x+3/2)^2-9*
x-19/4)^(7/2)-1039/9604000/(x+3/2)^3*(3*(x+3/2)^2-9*x-19/4)^(7/2)-27009/67228000
*x*(3*(x+3/2)^2-9*x-19/4)^(3/2)-6561/84035000/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(
7/2)+53253/33614000*(3*(x+3/2)^2-9*x-19/4)^(3/2)+159759/7683200*(12*(x+3/2)^2-36
*x-19)^(1/2)+159759/1470612500*(3*(x+3/2)^2-9*x-19/4)^(5/2)

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Maxima [A]  time = 0.785652, size = 387, normalized size = 2.91 \[ \frac{19683}{84035000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} - \frac{13 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{210 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{98 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{113 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{34300 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{1039 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{1200500 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{6561 \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}}}{21008750 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{27009}{67228000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{53253}{33614000} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} - \frac{41043 \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}}}{84035000 \,{\left (2 \, x + 3\right )}} - \frac{45711}{3841600} \, \sqrt{3 \, x^{2} + 2} x - \frac{9}{128} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{159759}{7683200} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) + \frac{159759}{3841600} \, \sqrt{3 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

19683/84035000*(3*x^2 + 2)^(5/2) - 13/210*(3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 +
2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 1/98*(3*x^2 + 2)^(7/2)/(32*x^5
+ 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 113/34300*(3*x^2 + 2)^(7/2)/(16*
x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 1039/1200500*(3*x^2 + 2)^(7/2)/(8*x^3 + 3
6*x^2 + 54*x + 27) - 6561/21008750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) - 27009/
67228000*(3*x^2 + 2)^(3/2)*x + 53253/33614000*(3*x^2 + 2)^(3/2) - 41043/84035000
*(3*x^2 + 2)^(5/2)/(2*x + 3) - 45711/3841600*sqrt(3*x^2 + 2)*x - 9/128*sqrt(3)*a
rcsinh(1/2*sqrt(6)*x) + 159759/7683200*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x +
3) - 2/3*sqrt(6)/abs(2*x + 3)) + 159759/3841600*sqrt(3*x^2 + 2)

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Fricas [A]  time = 0.293267, size = 290, normalized size = 2.18 \[ \frac{\sqrt{35}{\left (46305 \, \sqrt{35} \sqrt{3}{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 4 \, \sqrt{35}{\left (4369608 \, x^{5} + 18915336 \, x^{4} + 47453802 \, x^{3} + 59256588 \, x^{2} + 39843609 \, x + 10361807\right )} \sqrt{3 \, x^{2} + 2} + 479277 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{46099200 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/46099200*sqrt(35)*(46305*sqrt(35)*sqrt(3)*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*
x^3 + 4860*x^2 + 2916*x + 729)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 4*sq
rt(35)*(4369608*x^5 + 18915336*x^4 + 47453802*x^3 + 59256588*x^2 + 39843609*x +
10361807)*sqrt(3*x^2 + 2) + 479277*(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 486
0*x^2 + 2916*x + 729)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35*sqrt(3*x^2 + 2)*(
9*x - 4))/(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+2)**(5/2)/(3+2*x)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.327281, size = 520, normalized size = 3.91 \[ \frac{9}{128} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) + \frac{159759}{7683200} \, \sqrt{35}{\rm ln}\left (-\frac{{\left | -2 \, \sqrt{3} x - \sqrt{35} - 3 \, \sqrt{3} + 2 \, \sqrt{3 \, x^{2} + 2} \right |}}{2 \, \sqrt{3} x - \sqrt{35} + 3 \, \sqrt{3} - 2 \, \sqrt{3 \, x^{2} + 2}}\right ) + \frac{3 \,{\left (1700928 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{11} + 16427322 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{10} + 212377560 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{9} + 421378065 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{8} + 732041442 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{7} - 879808433 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{6} - 1537837812 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{5} + 2079633300 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{4} - 2495803200 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{3} + 500387712 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} - 155311488 \, \sqrt{3} x + 7768192 \, \sqrt{3} + 155311488 \, \sqrt{3 \, x^{2} + 2}\right )}}{878080 \,{\left ({\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/128*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 159759/7683200*sqrt(35)*ln(-abs
(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35
) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2))) + 3/878080*(1700928*(sqrt(3)*x - sqrt(3*x^2
+ 2))^11 + 16427322*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 + 212377560*(sqrt(3
)*x - sqrt(3*x^2 + 2))^9 + 421378065*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 7
32041442*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 879808433*sqrt(3)*(sqrt(3)*x - sqrt(3
*x^2 + 2))^6 - 1537837812*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 2079633300*sqrt(3)*(
sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 2495803200*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 50
0387712*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^2 - 155311488*sqrt(3)*x + 7768192*
sqrt(3) + 155311488*sqrt(3*x^2 + 2))/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3
)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^6