3.2564 \(\int \frac{5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=111 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24626}{625 \sqrt{2 x+3}}-\frac{7042}{375 (2 x+3)^{3/2}}-\frac{2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{15876}{625} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-2114/(125*(3 + 2*x)^(5/2)) - 7042/(375*(3 + 2*x)^(3/2)) - 24626/(625*Sqrt[3 + 2
*x]) - (3*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 14*ArcTanh[Sqrt[3
+ 2*x]] + (15876*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/625

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Rubi [A]  time = 0.294932, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24626}{625 \sqrt{2 x+3}}-\frac{7042}{375 (2 x+3)^{3/2}}-\frac{2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{15876}{625} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2114/(125*(3 + 2*x)^(5/2)) - 7042/(375*(3 + 2*x)^(3/2)) - 24626/(625*Sqrt[3 + 2
*x]) - (3*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)) + 14*ArcTanh[Sqrt[3
+ 2*x]] + (15876*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/625

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Rubi in Sympy [A]  time = 49.5718, size = 97, normalized size = 0.87 \[ \frac{15876 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{3125} + 14 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{24626}{625 \sqrt{2 x + 3}} - \frac{7042}{375 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{141 x + 111}{5 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )} - \frac{2114}{125 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15876*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/3125 + 14*atanh(sqrt(2*x + 3)) -
24626/(625*sqrt(2*x + 3)) - 7042/(375*(2*x + 3)**(3/2)) - (141*x + 111)/(5*(2*x
+ 3)**(5/2)*(3*x**2 + 5*x + 2)) - 2114/(125*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.272778, size = 121, normalized size = 1.09 \[ \frac{-\frac{45 \sqrt{2 x+3} (4209 x+2959)}{3 x^2+5 x+2}-\frac{243120}{\sqrt{2 x+3}}-\frac{40600}{(2 x+3)^{3/2}}-\frac{7800}{(2 x+3)^{5/2}}-65625 \log \left (1-\sqrt{2 x+3}\right )+65625 \log \left (\sqrt{2 x+3}+1\right )+47628 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{9375} \]

Antiderivative was successfully verified.

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

[Out]

(-7800/(3 + 2*x)^(5/2) - 40600/(3 + 2*x)^(3/2) - 243120/Sqrt[3 + 2*x] - (45*Sqrt
[3 + 2*x]*(2959 + 4209*x))/(2 + 5*x + 3*x^2) + 47628*Sqrt[15]*ArcTanh[Sqrt[3/5]*
Sqrt[3 + 2*x]] - 65625*Log[1 - Sqrt[3 + 2*x]] + 65625*Log[1 + Sqrt[3 + 2*x]])/93
75

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Maple [A]  time = 0.031, size = 113, normalized size = 1. \[ -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-7\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{918}{625}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{15876\,\sqrt{15}}{3125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{104}{125} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}}-{\frac{1624}{375} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{16208}{625}{\frac{1}{\sqrt{3+2\,x}}}}-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+7\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6/(-1+(3+2*x)^(1/2))-7*ln(-1+(3+2*x)^(1/2))-918/625*(3+2*x)^(1/2)/(4/3+2*x)+158
76/3125*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-104/125/(3+2*x)^(5/2)-1624/
375/(3+2*x)^(3/2)-16208/625/(3+2*x)^(1/2)-6/(1+(3+2*x)^(1/2))+7*ln(1+(3+2*x)^(1/
2))

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Maxima [A]  time = 0.790961, size = 169, normalized size = 1.52 \[ -\frac{7938}{3125} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (110817 \,{\left (2 \, x + 3\right )}^{4} - 242697 \,{\left (2 \, x + 3\right )}^{3} + 91420 \,{\left (2 \, x + 3\right )}^{2} + 28120 \, x + 46080\right )}}{1875 \,{\left (3 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 8 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 5 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}\right )}} + 7 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 7 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7938/3125*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3
))) - 2/1875*(110817*(2*x + 3)^4 - 242697*(2*x + 3)^3 + 91420*(2*x + 3)^2 + 2812
0*x + 46080)/(3*(2*x + 3)^(9/2) - 8*(2*x + 3)^(7/2) + 5*(2*x + 3)^(5/2)) + 7*log
(sqrt(2*x + 3) + 1) - 7*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.292388, size = 279, normalized size = 2.51 \[ \frac{\sqrt{5}{\left (13125 \, \sqrt{5}{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 13125 \, \sqrt{5}{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 23814 \, \sqrt{3}{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} + 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) - \sqrt{5}{\left (886536 \, x^{4} + 4348428 \, x^{3} + 7782530 \, x^{2} + 5977997 \, x + 1646109\right )}\right )}}{9375 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9375*sqrt(5)*(13125*sqrt(5)*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*sqrt(2*x +
3)*log(sqrt(2*x + 3) + 1) - 13125*sqrt(5)*(12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)
*sqrt(2*x + 3)*log(sqrt(2*x + 3) - 1) + 23814*sqrt(3)*(12*x^4 + 56*x^3 + 95*x^2
+ 69*x + 18)*sqrt(2*x + 3)*log((sqrt(5)*(3*x + 7) + 5*sqrt(3)*sqrt(2*x + 3))/(3*
x + 2)) - sqrt(5)*(886536*x^4 + 4348428*x^3 + 7782530*x^2 + 5977997*x + 1646109)
)/((12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*sqrt(2*x + 3))

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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)**(7/2)/(3*x**2+5*x+2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.281012, size = 169, normalized size = 1.52 \[ -\frac{7938}{3125} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{6 \,{\left (4209 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 6709 \, \sqrt{2 \, x + 3}\right )}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - \frac{16 \,{\left (3039 \,{\left (2 \, x + 3\right )}^{2} + 1015 \, x + 1620\right )}}{1875 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 7 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 7 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7938/3125*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt
(2*x + 3))) - 6/625*(4209*(2*x + 3)^(3/2) - 6709*sqrt(2*x + 3))/(3*(2*x + 3)^2 -
16*x - 19) - 16/1875*(3039*(2*x + 3)^2 + 1015*x + 1620)/(2*x + 3)^(5/2) + 7*ln(
sqrt(2*x + 3) + 1) - 7*ln(abs(sqrt(2*x + 3) - 1))