3.2570 \(\int \frac{5-x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^3} \, dx\)

Optimal. Leaf size=102 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (11739 x+9734)}{50 \left (3 x^2+5 x+2\right )}+542 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(-3*Sqrt[3 + 2*x]*(37 + 47*x))/(10*(2 + 5*x + 3*x^2)^2) + (Sqrt[3 + 2*x]*(9734 +
11739*x))/(50*(2 + 5*x + 3*x^2)) + 542*ArcTanh[Sqrt[3 + 2*x]] - (17463*Sqrt[3/5
]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi [A]  time = 0.195585, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{3 \sqrt{2 x+3} (47 x+37)}{10 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (11739 x+9734)}{50 \left (3 x^2+5 x+2\right )}+542 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[3 + 2*x]*(37 + 47*x))/(10*(2 + 5*x + 3*x^2)^2) + (Sqrt[3 + 2*x]*(9734 +
11739*x))/(50*(2 + 5*x + 3*x^2)) + 542*ArcTanh[Sqrt[3 + 2*x]] - (17463*Sqrt[3/5
]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25

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Rubi in Sympy [A]  time = 34.5164, size = 87, normalized size = 0.85 \[ - \frac{\sqrt{2 x + 3} \left (141 x + 111\right )}{10 \left (3 x^{2} + 5 x + 2\right )^{2}} + \frac{\sqrt{2 x + 3} \left (11739 x + 9734\right )}{50 \left (3 x^{2} + 5 x + 2\right )} - \frac{17463 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{125} + 542 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2*x + 3)*(141*x + 111)/(10*(3*x**2 + 5*x + 2)**2) + sqrt(2*x + 3)*(11739*x
+ 9734)/(50*(3*x**2 + 5*x + 2)) - 17463*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5
)/125 + 542*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.264685, size = 100, normalized size = 0.98 \[ \frac{\sqrt{2 x+3} \left (35217 x^3+87897 x^2+71443 x+18913\right )}{50 \left (3 x^2+5 x+2\right )^2}-271 \log \left (1-\sqrt{2 x+3}\right )+271 \log \left (\sqrt{2 x+3}+1\right )-\frac{17463}{25} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 2*x]*(18913 + 71443*x + 87897*x^2 + 35217*x^3))/(50*(2 + 5*x + 3*x^2)^
2) - (17463*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/25 - 271*Log[1 - Sqrt[3
+ 2*x]] + 271*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.029, size = 124, normalized size = 1.2 \[ 3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+44\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-271\,\ln \left ( -1+\sqrt{3+2\,x} \right ) +486\,{\frac{1}{ \left ( 4+6\,x \right ) ^{2}} \left ({\frac{571\, \left ( 3+2\,x \right ) ^{3/2}}{450}}-{\frac{121\,\sqrt{3+2\,x}}{54}} \right ) }-{\frac{17463\,\sqrt{15}}{125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+44\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+271\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/(-1+(3+2*x)^(1/2))^2+44/(-1+(3+2*x)^(1/2))-271*ln(-1+(3+2*x)^(1/2))+486*(571/4
50*(3+2*x)^(3/2)-121/54*(3+2*x)^(1/2))/(4+6*x)^2-17463/125*arctanh(1/5*15^(1/2)*
(3+2*x)^(1/2))*15^(1/2)-3/(1+(3+2*x)^(1/2))^2+44/(1+(3+2*x)^(1/2))+271*ln(1+(3+2
*x)^(1/2))

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Maxima [A]  time = 0.788812, size = 181, normalized size = 1.77 \[ \frac{17463}{250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{35217 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 141159 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 181867 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 74725 \, \sqrt{2 \, x + 3}}{25 \,{\left (9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 271 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 271 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17463/250*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)
)) + 1/25*(35217*(2*x + 3)^(7/2) - 141159*(2*x + 3)^(5/2) + 181867*(2*x + 3)^(3/
2) - 74725*sqrt(2*x + 3))/(9*(2*x + 3)^4 - 48*(2*x + 3)^3 + 94*(2*x + 3)^2 - 160
*x - 215) + 271*log(sqrt(2*x + 3) + 1) - 271*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.293432, size = 243, normalized size = 2.38 \[ \frac{\sqrt{5}{\left (13550 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 13550 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 17463 \, \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{5}{\left (35217 \, x^{3} + 87897 \, x^{2} + 71443 \, x + 18913\right )} \sqrt{2 \, x + 3}\right )}}{250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/250*sqrt(5)*(13550*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x +
3) + 1) - 13550*sqrt(5)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(sqrt(2*x + 3)
- 1) + 17463*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log((sqrt(5)*(3*x + 7)
- 5*sqrt(3)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(5)*(35217*x^3 + 87897*x^2 + 71443*
x + 18913)*sqrt(2*x + 3))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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Sympy [A]  time = 106.697, size = 396, normalized size = 3.88 \[ \frac{35424 \left (\begin{cases} \frac{\sqrt{15} \left (- \frac{\log{\left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{4} + \frac{\log{\left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{4} - \frac{1}{4 \left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )} - \frac{1}{4 \left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )}\right )}{45} & \text{for}\: x > - \frac{2}{3} \end{cases}\right )}{25} + \frac{11016 \left (\begin{cases} \frac{\sqrt{15} \left (\frac{3 \log{\left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{16} - \frac{3 \log{\left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{16} + \frac{3}{16 \left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )} + \frac{1}{16 \left (1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )^{2}} + \frac{3}{16 \left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )} - \frac{1}{16 \left (-1 + \frac{\sqrt{15}}{3 \sqrt{2 x + 3}}\right )^{2}}\right )}{135} & \text{for}\: x > - \frac{2}{3} \end{cases}\right )}{25} + \frac{57834 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} > \frac{3}{5} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15}}{3 \sqrt{2 x + 3}} \right )}}{15} & \text{for}\: \frac{1}{2 x + 3} < \frac{3}{5} \end{cases}\right )}{25} - 271 \log{\left (-1 + \frac{1}{\sqrt{2 x + 3}} \right )} + 271 \log{\left (1 + \frac{1}{\sqrt{2 x + 3}} \right )} - \frac{38}{1 + \frac{1}{\sqrt{2 x + 3}}} - \frac{3}{\left (1 + \frac{1}{\sqrt{2 x + 3}}\right )^{2}} - \frac{38}{-1 + \frac{1}{\sqrt{2 x + 3}}} + \frac{3}{\left (-1 + \frac{1}{\sqrt{2 x + 3}}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35424*Piecewise((sqrt(15)*(-log(-1 + sqrt(15)/(3*sqrt(2*x + 3)))/4 + log(1 + sqr
t(15)/(3*sqrt(2*x + 3)))/4 - 1/(4*(1 + sqrt(15)/(3*sqrt(2*x + 3)))) - 1/(4*(-1 +
sqrt(15)/(3*sqrt(2*x + 3)))))/45, x > -2/3))/25 + 11016*Piecewise((sqrt(15)*(3*
log(-1 + sqrt(15)/(3*sqrt(2*x + 3)))/16 - 3*log(1 + sqrt(15)/(3*sqrt(2*x + 3)))/
16 + 3/(16*(1 + sqrt(15)/(3*sqrt(2*x + 3)))) + 1/(16*(1 + sqrt(15)/(3*sqrt(2*x +
3)))**2) + 3/(16*(-1 + sqrt(15)/(3*sqrt(2*x + 3)))) - 1/(16*(-1 + sqrt(15)/(3*s
qrt(2*x + 3)))**2))/135, x > -2/3))/25 + 57834*Piecewise((-sqrt(15)*acoth(sqrt(1
5)/(3*sqrt(2*x + 3)))/15, 1/(2*x + 3) > 3/5), (-sqrt(15)*atanh(sqrt(15)/(3*sqrt(
2*x + 3)))/15, 1/(2*x + 3) < 3/5))/25 - 271*log(-1 + 1/sqrt(2*x + 3)) + 271*log(
1 + 1/sqrt(2*x + 3)) - 38/(1 + 1/sqrt(2*x + 3)) - 3/(1 + 1/sqrt(2*x + 3))**2 - 3
8/(-1 + 1/sqrt(2*x + 3)) + 3/(-1 + 1/sqrt(2*x + 3))**2

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GIAC/XCAS [A]  time = 0.270392, size = 162, normalized size = 1.59 \[ \frac{17463}{250} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{35217 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 141159 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 181867 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 74725 \, \sqrt{2 \, x + 3}}{25 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 271 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 271 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

17463/250*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(
2*x + 3))) + 1/25*(35217*(2*x + 3)^(7/2) - 141159*(2*x + 3)^(5/2) + 181867*(2*x
+ 3)^(3/2) - 74725*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19)^2 + 271*ln(sqrt(2*
x + 3) + 1) - 271*ln(abs(sqrt(2*x + 3) - 1))