3.2556 \(\int \frac{5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\)

Optimal. Leaf size=81 \[ -\frac{1194}{125 \sqrt{2 x+3}}-\frac{66}{25 (2 x+3)^{3/2}}-\frac{26}{25 (2 x+3)^{5/2}}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{306}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

-26/(25*(3 + 2*x)^(5/2)) - 66/(25*(3 + 2*x)^(3/2)) - 1194/(125*Sqrt[3 + 2*x]) +
12*ArcTanh[Sqrt[3 + 2*x]] - (306*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/125

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Rubi [A]  time = 0.239862, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ -\frac{1194}{125 \sqrt{2 x+3}}-\frac{66}{25 (2 x+3)^{3/2}}-\frac{26}{25 (2 x+3)^{5/2}}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{306}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-26/(25*(3 + 2*x)^(5/2)) - 66/(25*(3 + 2*x)^(3/2)) - 1194/(125*Sqrt[3 + 2*x]) +
12*ArcTanh[Sqrt[3 + 2*x]] - (306*Sqrt[3/5]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/125

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Rubi in Sympy [A]  time = 42.36, size = 71, normalized size = 0.88 \[ - \frac{306 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{625} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} - \frac{1194}{125 \sqrt{2 x + 3}} - \frac{66}{25 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{26}{25 \left (2 x + 3\right )^{\frac{5}{2}}} \]

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),x)

[Out]

-306*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/625 + 12*atanh(sqrt(2*x + 3)) - 11
94/(125*sqrt(2*x + 3)) - 66/(25*(2*x + 3)**(3/2)) - 26/(25*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.256679, size = 90, normalized size = 1.11 \[ \frac{2}{625} \left (\frac{5 \left (-2388 x^2-7494 x+375 (2 x+3)^{5/2} \log \left (\sqrt{2 x+3}+1\right )-5933\right )}{(2 x+3)^{5/2}}-1875 \log \left (1-\sqrt{2 x+3}\right )-153 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*(-153*Sqrt[15]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]] - 1875*Log[1 - Sqrt[3 + 2*x]]
+ (5*(-5933 - 7494*x - 2388*x^2 + 375*(3 + 2*x)^(5/2)*Log[1 + Sqrt[3 + 2*x]]))/
(3 + 2*x)^(5/2)))/625

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Maple [A]  time = 0.02, size = 71, normalized size = 0.9 \[ -6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{306\,\sqrt{15}}{625}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-{\frac{26}{25} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}}-{\frac{66}{25} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{1194}{125}{\frac{1}{\sqrt{3+2\,x}}}}+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

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),x)

[Out]

-6*ln(-1+(3+2*x)^(1/2))-306/625*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)-26/
25/(3+2*x)^(5/2)-66/25/(3+2*x)^(3/2)-1194/125/(3+2*x)^(1/2)+6*ln(1+(3+2*x)^(1/2)
)

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Maxima [A]  time = 0.789044, size = 113, normalized size = 1.4 \[ \frac{153}{625} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (597 \,{\left (2 \, x + 3\right )}^{2} + 330 \, x + 560\right )}}{125 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153/625*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3)))
- 2/125*(597*(2*x + 3)^2 + 330*x + 560)/(2*x + 3)^(5/2) + 6*log(sqrt(2*x + 3) +
1) - 6*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.29259, size = 212, normalized size = 2.62 \[ \frac{\sqrt{5}{\left (750 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 750 \, \sqrt{5}{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 153 \, \sqrt{3}{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3} \log \left (\frac{\sqrt{5}{\left (3 \, x + 7\right )} - 5 \, \sqrt{3} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) - 2 \, \sqrt{5}{\left (2388 \, x^{2} + 7494 \, x + 5933\right )}\right )}}{625 \,{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/625*sqrt(5)*(750*sqrt(5)*(4*x^2 + 12*x + 9)*sqrt(2*x + 3)*log(sqrt(2*x + 3) +
1) - 750*sqrt(5)*(4*x^2 + 12*x + 9)*sqrt(2*x + 3)*log(sqrt(2*x + 3) - 1) + 153*s
qrt(3)*(4*x^2 + 12*x + 9)*sqrt(2*x + 3)*log((sqrt(5)*(3*x + 7) - 5*sqrt(3)*sqrt(
2*x + 3))/(3*x + 2)) - 2*sqrt(5)*(2388*x^2 + 7494*x + 5933))/((4*x^2 + 12*x + 9)
*sqrt(2*x + 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{24 x^{5} \sqrt{2 x + 3} + 148 x^{4} \sqrt{2 x + 3} + 358 x^{3} \sqrt{2 x + 3} + 423 x^{2} \sqrt{2 x + 3} + 243 x \sqrt{2 x + 3} + 54 \sqrt{2 x + 3}}\, dx - \int \left (- \frac{5}{24 x^{5} \sqrt{2 x + 3} + 148 x^{4} \sqrt{2 x + 3} + 358 x^{3} \sqrt{2 x + 3} + 423 x^{2} \sqrt{2 x + 3} + 243 x \sqrt{2 x + 3} + 54 \sqrt{2 x + 3}}\right )\, dx \]

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),x)

[Out]

-Integral(x/(24*x**5*sqrt(2*x + 3) + 148*x**4*sqrt(2*x + 3) + 358*x**3*sqrt(2*x
+ 3) + 423*x**2*sqrt(2*x + 3) + 243*x*sqrt(2*x + 3) + 54*sqrt(2*x + 3)), x) - In
tegral(-5/(24*x**5*sqrt(2*x + 3) + 148*x**4*sqrt(2*x + 3) + 358*x**3*sqrt(2*x +
3) + 423*x**2*sqrt(2*x + 3) + 243*x*sqrt(2*x + 3) + 54*sqrt(2*x + 3)), x)

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GIAC/XCAS [A]  time = 0.270828, size = 119, normalized size = 1.47 \[ \frac{153}{625} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{2 \,{\left (597 \,{\left (2 \, x + 3\right )}^{2} + 330 \, x + 560\right )}}{125 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

153/625*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*
x + 3))) - 2/125*(597*(2*x + 3)^2 + 330*x + 560)/(2*x + 3)^(5/2) + 6*ln(sqrt(2*x
+ 3) + 1) - 6*ln(abs(sqrt(2*x + 3) - 1))