3.2571 \(\int \frac{5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=115 \[ -\frac{3 (47 x+37)}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac{2229 x+1888}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}+\frac{2667}{25 \sqrt{2 x+3}}+402 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{12717}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

2667/(25*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^2)
+ (1888 + 2229*x)/(10*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)) + 402*ArcTanh[Sqrt[3 + 2
*x]] - (12717*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

_______________________________________________________________________________________

Rubi [A]  time = 0.245144, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{3 (47 x+37)}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^2}+\frac{2229 x+1888}{10 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )}+\frac{2667}{25 \sqrt{2 x+3}}+402 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{12717}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

2667/(25*Sqrt[3 + 2*x]) - (3*(37 + 47*x))/(10*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)^2)
+ (1888 + 2229*x)/(10*Sqrt[3 + 2*x]*(2 + 5*x + 3*x^2)) + 402*ArcTanh[Sqrt[3 + 2
*x]] - (12717*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.4029, size = 100, normalized size = 0.87 \[ - \frac{12717 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{125} + 402 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{141 x + 111}{10 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{11145 x + 9440}{50 \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )} + \frac{2667}{25 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12717*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/125 + 402*atanh(sqrt(2*x + 3)) -
(141*x + 111)/(10*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**2) + (11145*x + 9440)/(50*s
qrt(2*x + 3)*(3*x**2 + 5*x + 2)) + 2667/(25*sqrt(2*x + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.263711, size = 128, normalized size = 1.11 \[ \frac{1}{125} \left (-\frac{15 \sqrt{2 x+3} (201 x+151)}{2 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (41253 x+34738)}{6 x^2+10 x+4}-\frac{416}{\sqrt{2 x+3}}-25125 \log \left (1-\sqrt{2 x+3}\right )+25125 \log \left (\sqrt{2 x+3}+1\right )-12717 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-416/Sqrt[3 + 2*x] - (15*Sqrt[3 + 2*x]*(151 + 201*x))/(2*(2 + 5*x + 3*x^2)^2) +
(Sqrt[3 + 2*x]*(34738 + 41253*x))/(4 + 10*x + 6*x^2) - 12717*Sqrt[15]*ArcTanh[S
qrt[3/5]*Sqrt[3 + 2*x]] - 25125*Log[1 - Sqrt[3 + 2*x]] + 25125*Log[1 + Sqrt[3 +
2*x]])/125

_______________________________________________________________________________________

Maple [A]  time = 0.03, size = 133, normalized size = 1.2 \[ 32\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}-201\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +{\frac{486}{125\, \left ( 4+6\,x \right ) ^{2}} \left ({\frac{213}{2} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{3365}{18}\sqrt{3+2\,x}} \right ) }-{\frac{12717\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{416}{125}{\frac{1}{\sqrt{3+2\,x}}}}+32\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+201\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/(-1+(3+2*x)^(1/2))+3/(-1+(3+2*x)^(1/2))^2-201*ln(-1+(3+2*x)^(1/2))+486/125*(2
13/2*(3+2*x)^(3/2)-3365/18*(3+2*x)^(1/2))/(4+6*x)^2-12717/125*arctanh(1/5*15^(1/
2)*(3+2*x)^(1/2))*15^(1/2)-416/125/(3+2*x)^(1/2)+32/(1+(3+2*x)^(1/2))-3/(1+(3+2*
x)^(1/2))^2+201*ln(1+(3+2*x)^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.795153, size = 193, normalized size = 1.68 \[ \frac{12717}{250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{24003 \,{\left (2 \, x + 3\right )}^{4} - 94581 \,{\left (2 \, x + 3\right )}^{3} + 117873 \,{\left (2 \, x + 3\right )}^{2} - 88030 \, x - 134125}{25 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + 25 \, \sqrt{2 \, x + 3}\right )}} + 201 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 201 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12717/250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)
)) + 1/25*(24003*(2*x + 3)^4 - 94581*(2*x + 3)^3 + 117873*(2*x + 3)^2 - 88030*x
- 134125)/(9*(2*x + 3)^(9/2) - 48*(2*x + 3)^(7/2) + 94*(2*x + 3)^(5/2) - 80*(2*x
+ 3)^(3/2) + 25*sqrt(2*x + 3)) + 201*log(sqrt(2*x + 3) + 1) - 201*log(sqrt(2*x
+ 3) - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.299072, size = 278, normalized size = 2.42 \[ \frac{\sqrt{5}{\left (10050 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 10050 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 12717 \, \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (48006 \, x^{4} + 193455 \, x^{3} + 281403 \, x^{2} + 175465 \, x + 39661\right )}\right )}}{250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/250*sqrt(5)*(10050*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt(2*x + 3)*
log(sqrt(2*x + 3) + 1) - 10050*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt
(2*x + 3)*log(sqrt(2*x + 3) - 1) + 12717*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x
+ 4)*sqrt(2*x + 3)*log((sqrt(5)*(3*x + 7) - 5*sqrt(3)*sqrt(2*x + 3))/(3*x + 2))
+ sqrt(5)*(48006*x^4 + 193455*x^3 + 281403*x^2 + 175465*x + 39661))/((9*x^4 + 3
0*x^3 + 37*x^2 + 20*x + 4)*sqrt(2*x + 3))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{54 x^{7} \sqrt{2 x + 3} + 351 x^{6} \sqrt{2 x + 3} + 963 x^{5} \sqrt{2 x + 3} + 1447 x^{4} \sqrt{2 x + 3} + 1287 x^{3} \sqrt{2 x + 3} + 678 x^{2} \sqrt{2 x + 3} + 196 x \sqrt{2 x + 3} + 24 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{5}{54 x^{7} \sqrt{2 x + 3} + 351 x^{6} \sqrt{2 x + 3} + 963 x^{5} \sqrt{2 x + 3} + 1447 x^{4} \sqrt{2 x + 3} + 1287 x^{3} \sqrt{2 x + 3} + 678 x^{2} \sqrt{2 x + 3} + 196 x \sqrt{2 x + 3} + 24 \sqrt{2 x + 3}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(54*x**7*sqrt(2*x + 3) + 351*x**6*sqrt(2*x + 3) + 963*x**5*sqrt(2*x
+ 3) + 1447*x**4*sqrt(2*x + 3) + 1287*x**3*sqrt(2*x + 3) + 678*x**2*sqrt(2*x + 3
) + 196*x*sqrt(2*x + 3) + 24*sqrt(2*x + 3)), x) - Integral(-5/(54*x**7*sqrt(2*x
+ 3) + 351*x**6*sqrt(2*x + 3) + 963*x**5*sqrt(2*x + 3) + 1447*x**4*sqrt(2*x + 3)
+ 1287*x**3*sqrt(2*x + 3) + 678*x**2*sqrt(2*x + 3) + 196*x*sqrt(2*x + 3) + 24*s
qrt(2*x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278173, size = 174, normalized size = 1.51 \[ \frac{12717}{250} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{416}{125 \, \sqrt{2 \, x + 3}} + \frac{123759 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 492873 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 628469 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 253355 \, \sqrt{2 \, x + 3}}{125 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 201 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 201 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12717/250*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(
2*x + 3))) - 416/125/sqrt(2*x + 3) + 1/125*(123759*(2*x + 3)^(7/2) - 492873*(2*x
+ 3)^(5/2) + 628469*(2*x + 3)^(3/2) - 253355*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16
*x - 19)^2 + 201*ln(sqrt(2*x + 3) + 1) - 201*ln(abs(sqrt(2*x + 3) - 1))