3.1406 \(\int \frac{5-x}{(3+2 x)^5 \sqrt{2+3 x^2}} \, dx\)

Optimal. Leaf size=121 \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

[Out]

(-13*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^4) - (97*Sqrt[2 + 3*x^2])/(2100*(3 + 2*x)^3
) - (87*Sqrt[2 + 3*x^2])/(4900*(3 + 2*x)^2) - (991*Sqrt[2 + 3*x^2])/(171500*(3 +
2*x)) + (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42875*Sqrt[35])

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Rubi [A]  time = 0.237619, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{991 \sqrt{3 x^2+2}}{171500 (2 x+3)}-\frac{87 \sqrt{3 x^2+2}}{4900 (2 x+3)^2}-\frac{97 \sqrt{3 x^2+2}}{2100 (2 x+3)^3}-\frac{13 \sqrt{3 x^2+2}}{140 (2 x+3)^4}+\frac{27 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{42875 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(-13*Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^4) - (97*Sqrt[2 + 3*x^2])/(2100*(3 + 2*x)^3
) - (87*Sqrt[2 + 3*x^2])/(4900*(3 + 2*x)^2) - (991*Sqrt[2 + 3*x^2])/(171500*(3 +
2*x)) + (27*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(42875*Sqrt[35])

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Rubi in Sympy [A]  time = 22.5632, size = 109, normalized size = 0.9 \[ \frac{27 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{1500625} - \frac{991 \sqrt{3 x^{2} + 2}}{171500 \left (2 x + 3\right )} - \frac{87 \sqrt{3 x^{2} + 2}}{4900 \left (2 x + 3\right )^{2}} - \frac{97 \sqrt{3 x^{2} + 2}}{2100 \left (2 x + 3\right )^{3}} - \frac{13 \sqrt{3 x^{2} + 2}}{140 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**5/(3*x**2+2)**(1/2),x)

[Out]

27*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/1500625 - 991*sqrt(
3*x**2 + 2)/(171500*(2*x + 3)) - 87*sqrt(3*x**2 + 2)/(4900*(2*x + 3)**2) - 97*sq
rt(3*x**2 + 2)/(2100*(2*x + 3)**3) - 13*sqrt(3*x**2 + 2)/(140*(2*x + 3)**4)

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Mathematica [A]  time = 0.163943, size = 85, normalized size = 0.7 \[ \frac{81 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \sqrt{3 x^2+2} \left (5946 x^3+35892 x^2+79423 x+70389\right )}{(2 x+3)^4}-81 \sqrt{35} \log (2 x+3)}{4501875} \]

Antiderivative was successfully verified.

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

[Out]

((-35*Sqrt[2 + 3*x^2]*(70389 + 79423*x + 35892*x^2 + 5946*x^3))/(3 + 2*x)^4 - 81
*Sqrt[35]*Log[3 + 2*x] + 81*Sqrt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])]
)/4501875

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Maple [A]  time = 0.016, size = 116, normalized size = 1. \[ -{\frac{13}{2240}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-4}}-{\frac{97}{16800}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-3}}-{\frac{87}{19600}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-2}}-{\frac{991}{343000}\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}} \left ( x+{\frac{3}{2}} \right ) ^{-1}}+{\frac{27\,\sqrt{35}}{1500625}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(2*x+3)^5/(3*x^2+2)^(1/2),x)

[Out]

-13/2240/(x+3/2)^4*(3*(x+3/2)^2-9*x-19/4)^(1/2)-97/16800/(x+3/2)^3*(3*(x+3/2)^2-
9*x-19/4)^(1/2)-87/19600/(x+3/2)^2*(3*(x+3/2)^2-9*x-19/4)^(1/2)-991/343000/(x+3/
2)*(3*(x+3/2)^2-9*x-19/4)^(1/2)+27/1500625*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2
)/(12*(x+3/2)^2-36*x-19)^(1/2))

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Maxima [A]  time = 0.756217, size = 185, normalized size = 1.53 \[ -\frac{27}{1500625} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{13 \, \sqrt{3 \, x^{2} + 2}}{140 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{97 \, \sqrt{3 \, x^{2} + 2}}{2100 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{87 \, \sqrt{3 \, x^{2} + 2}}{4900 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac{991 \, \sqrt{3 \, x^{2} + 2}}{171500 \,{\left (2 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27/1500625*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x +
3)) - 13/140*sqrt(3*x^2 + 2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 97/2100*
sqrt(3*x^2 + 2)/(8*x^3 + 36*x^2 + 54*x + 27) - 87/4900*sqrt(3*x^2 + 2)/(4*x^2 +
12*x + 9) - 991/171500*sqrt(3*x^2 + 2)/(2*x + 3)

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Fricas [A]  time = 0.283041, size = 169, normalized size = 1.4 \[ -\frac{\sqrt{35}{\left (2 \, \sqrt{35}{\left (5946 \, x^{3} + 35892 \, x^{2} + 79423 \, x + 70389\right )} \sqrt{3 \, x^{2} + 2} - 81 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} - 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{9003750 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9003750*sqrt(35)*(2*sqrt(35)*(5946*x^3 + 35892*x^2 + 79423*x + 70389)*sqrt(3*
x^2 + 2) - 81*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(-(sqrt(35)*(93*x^2 -
36*x + 43) - 35*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(16*x^4 + 96*x^3
+ 216*x^2 + 216*x + 81)

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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+2*x)**5/(3*x**2+2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{\sqrt{3 \, x^{2} + 2}{\left (2 \, x + 3\right )}^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(x - 5)/(sqrt(3*x^2 + 2)*(2*x + 3)^5), x)