3.1417 \(\int \frac{5-x}{(3+2 x)^5 \left (2+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac{41 x+26}{70 (2 x+3)^4 \sqrt{3 x^2+2}}-\frac{14944 \sqrt{3 x^2+2}}{1500625 (2 x+3)}-\frac{708 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}-\frac{298 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{58 \sqrt{3 x^2+2}}{1225 (2 x+3)^4}-\frac{30078 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

[Out]

(26 + 41*x)/(70*(3 + 2*x)^4*Sqrt[2 + 3*x^2]) + (58*Sqrt[2 + 3*x^2])/(1225*(3 + 2
*x)^4) - (298*Sqrt[2 + 3*x^2])/(18375*(3 + 2*x)^3) - (708*Sqrt[2 + 3*x^2])/(4287
5*(3 + 2*x)^2) - (14944*Sqrt[2 + 3*x^2])/(1500625*(3 + 2*x)) - (30078*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*Sqrt[35])

_______________________________________________________________________________________

Rubi [A]  time = 0.298165, antiderivative size = 148, 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 \[ \frac{41 x+26}{70 (2 x+3)^4 \sqrt{3 x^2+2}}-\frac{14944 \sqrt{3 x^2+2}}{1500625 (2 x+3)}-\frac{708 \sqrt{3 x^2+2}}{42875 (2 x+3)^2}-\frac{298 \sqrt{3 x^2+2}}{18375 (2 x+3)^3}+\frac{58 \sqrt{3 x^2+2}}{1225 (2 x+3)^4}-\frac{30078 \tanh ^{-1}\left (\frac{4-9 x}{\sqrt{35} \sqrt{3 x^2+2}}\right )}{1500625 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

(26 + 41*x)/(70*(3 + 2*x)^4*Sqrt[2 + 3*x^2]) + (58*Sqrt[2 + 3*x^2])/(1225*(3 + 2
*x)^4) - (298*Sqrt[2 + 3*x^2])/(18375*(3 + 2*x)^3) - (708*Sqrt[2 + 3*x^2])/(4287
5*(3 + 2*x)^2) - (14944*Sqrt[2 + 3*x^2])/(1500625*(3 + 2*x)) - (30078*ArcTanh[(4
- 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(1500625*Sqrt[35])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.5236, size = 133, normalized size = 0.9 \[ - \frac{30078 \sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 9 x + 4\right )}{35 \sqrt{3 x^{2} + 2}} \right )}}{52521875} - \frac{14944 \sqrt{3 x^{2} + 2}}{1500625 \left (2 x + 3\right )} - \frac{708 \sqrt{3 x^{2} + 2}}{42875 \left (2 x + 3\right )^{2}} - \frac{298 \sqrt{3 x^{2} + 2}}{18375 \left (2 x + 3\right )^{3}} + \frac{123 x + 78}{210 \left (2 x + 3\right )^{4} \sqrt{3 x^{2} + 2}} + \frac{58 \sqrt{3 x^{2} + 2}}{1225 \left (2 x + 3\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30078*sqrt(35)*atanh(sqrt(35)*(-9*x + 4)/(35*sqrt(3*x**2 + 2)))/52521875 - 1494
4*sqrt(3*x**2 + 2)/(1500625*(2*x + 3)) - 708*sqrt(3*x**2 + 2)/(42875*(2*x + 3)**
2) - 298*sqrt(3*x**2 + 2)/(18375*(2*x + 3)**3) + (123*x + 78)/(210*(2*x + 3)**4*
sqrt(3*x**2 + 2)) + 58*sqrt(3*x**2 + 2)/(1225*(2*x + 3)**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.216901, size = 95, normalized size = 0.64 \[ \frac{-180468 \sqrt{35} \log \left (2 \left (\sqrt{35} \sqrt{3 x^2+2}-9 x+4\right )\right )-\frac{35 \left (2151936 x^5+11467872 x^4+22188792 x^3+18957672 x^2+8562487 x+4197366\right )}{(2 x+3)^4 \sqrt{3 x^2+2}}+180468 \sqrt{35} \log (2 x+3)}{315131250} \]

Antiderivative was successfully verified.

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

[Out]

((-35*(4197366 + 8562487*x + 18957672*x^2 + 22188792*x^3 + 11467872*x^4 + 215193
6*x^5))/((3 + 2*x)^4*Sqrt[2 + 3*x^2]) + 180468*Sqrt[35]*Log[3 + 2*x] - 180468*Sq
rt[35]*Log[2*(4 - 9*x + Sqrt[35]*Sqrt[2 + 3*x^2])])/315131250

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 149, normalized size = 1. \[ -{\frac{13}{2240} \left ( x+{\frac{3}{2}} \right ) ^{-4}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{913}{117600} \left ( x+{\frac{3}{2}} \right ) ^{-3}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{9}{1000} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{2143}{171500} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}+{\frac{15039}{1500625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{22416\,x}{1500625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-9\,x-{\frac{19}{4}}}}}}-{\frac{30078\,\sqrt{35}}{52521875}{\it Artanh} \left ({\frac{ \left ( 8-18\,x \right ) \sqrt{35}}{35}{\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-36\,x-19}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/2240/(x+3/2)^4/(3*(x+3/2)^2-9*x-19/4)^(1/2)-913/117600/(x+3/2)^3/(3*(x+3/2)^
2-9*x-19/4)^(1/2)-9/1000/(x+3/2)^2/(3*(x+3/2)^2-9*x-19/4)^(1/2)-2143/171500/(x+3
/2)/(3*(x+3/2)^2-9*x-19/4)^(1/2)+15039/1500625/(3*(x+3/2)^2-9*x-19/4)^(1/2)-2241
6/1500625*x/(3*(x+3/2)^2-9*x-19/4)^(1/2)-30078/52521875*35^(1/2)*arctanh(2/35*(4
-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.765666, size = 343, normalized size = 2.32 \[ \frac{30078}{52521875} \, \sqrt{35} \operatorname{arsinh}\left (\frac{3 \, \sqrt{6} x}{2 \,{\left | 2 \, x + 3 \right |}} - \frac{2 \, \sqrt{6}}{3 \,{\left | 2 \, x + 3 \right |}}\right ) - \frac{22416 \, x}{1500625 \, \sqrt{3 \, x^{2} + 2}} + \frac{15039}{1500625 \, \sqrt{3 \, x^{2} + 2}} - \frac{13}{140 \,{\left (16 \, \sqrt{3 \, x^{2} + 2} x^{4} + 96 \, \sqrt{3 \, x^{2} + 2} x^{3} + 216 \, \sqrt{3 \, x^{2} + 2} x^{2} + 216 \, \sqrt{3 \, x^{2} + 2} x + 81 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{913}{14700 \,{\left (8 \, \sqrt{3 \, x^{2} + 2} x^{3} + 36 \, \sqrt{3 \, x^{2} + 2} x^{2} + 54 \, \sqrt{3 \, x^{2} + 2} x + 27 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{9}{250 \,{\left (4 \, \sqrt{3 \, x^{2} + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 2} x + 9 \, \sqrt{3 \, x^{2} + 2}\right )}} - \frac{2143}{85750 \,{\left (2 \, \sqrt{3 \, x^{2} + 2} x + 3 \, \sqrt{3 \, x^{2} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30078/52521875*sqrt(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x
+ 3)) - 22416/1500625*x/sqrt(3*x^2 + 2) + 15039/1500625/sqrt(3*x^2 + 2) - 13/14
0/(16*sqrt(3*x^2 + 2)*x^4 + 96*sqrt(3*x^2 + 2)*x^3 + 216*sqrt(3*x^2 + 2)*x^2 + 2
16*sqrt(3*x^2 + 2)*x + 81*sqrt(3*x^2 + 2)) - 913/14700/(8*sqrt(3*x^2 + 2)*x^3 +
36*sqrt(3*x^2 + 2)*x^2 + 54*sqrt(3*x^2 + 2)*x + 27*sqrt(3*x^2 + 2)) - 9/250/(4*s
qrt(3*x^2 + 2)*x^2 + 12*sqrt(3*x^2 + 2)*x + 9*sqrt(3*x^2 + 2)) - 2143/85750/(2*s
qrt(3*x^2 + 2)*x + 3*sqrt(3*x^2 + 2))

_______________________________________________________________________________________

Fricas [A]  time = 0.282777, size = 208, normalized size = 1.41 \[ -\frac{\sqrt{35}{\left (\sqrt{35}{\left (2151936 \, x^{5} + 11467872 \, x^{4} + 22188792 \, x^{3} + 18957672 \, x^{2} + 8562487 \, x + 4197366\right )} \sqrt{3 \, x^{2} + 2} - 90234 \,{\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\right )} \log \left (-\frac{\sqrt{35}{\left (93 \, x^{2} - 36 \, x + 43\right )} + 35 \, \sqrt{3 \, x^{2} + 2}{\left (9 \, x - 4\right )}}{4 \, x^{2} + 12 \, x + 9}\right )\right )}}{315131250 \,{\left (48 \, x^{6} + 288 \, x^{5} + 680 \, x^{4} + 840 \, x^{3} + 675 \, x^{2} + 432 \, x + 162\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/315131250*sqrt(35)*(sqrt(35)*(2151936*x^5 + 11467872*x^4 + 22188792*x^3 + 189
57672*x^2 + 8562487*x + 4197366)*sqrt(3*x^2 + 2) - 90234*(48*x^6 + 288*x^5 + 680
*x^4 + 840*x^3 + 675*x^2 + 432*x + 162)*log(-(sqrt(35)*(93*x^2 - 36*x + 43) + 35
*sqrt(3*x^2 + 2)*(9*x - 4))/(4*x^2 + 12*x + 9)))/(48*x^6 + 288*x^5 + 680*x^4 + 8
40*x^3 + 675*x^2 + 432*x + 162)

_______________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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