3.2550 \(\int \frac{(5-x) (3+2 x)^{5/2}}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2}{15} (2 x+3)^{5/2}+\frac{62}{27} (2 x+3)^{3/2}+\frac{526}{27} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{850}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(526*Sqrt[3 + 2*x])/27 + (62*(3 + 2*x)^(3/2))/27 - (2*(3 + 2*x)^(5/2))/15 + 12*A
rcTanh[Sqrt[3 + 2*x]] - (850*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/27

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Rubi [A]  time = 0.224785, 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 \[ -\frac{2}{15} (2 x+3)^{5/2}+\frac{62}{27} (2 x+3)^{3/2}+\frac{526}{27} \sqrt{2 x+3}+12 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{850}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(526*Sqrt[3 + 2*x])/27 + (62*(3 + 2*x)^(3/2))/27 - (2*(3 + 2*x)^(5/2))/15 + 12*A
rcTanh[Sqrt[3 + 2*x]] - (850*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/27

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Rubi in Sympy [A]  time = 41.7542, size = 71, normalized size = 0.88 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}}}{15} + \frac{62 \left (2 x + 3\right )^{\frac{3}{2}}}{27} + \frac{526 \sqrt{2 x + 3}}{27} - \frac{850 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{81} + 12 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(5/2)/15 + 62*(2*x + 3)**(3/2)/27 + 526*sqrt(2*x + 3)/27 - 850*sqr
t(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/81 + 12*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.0679861, size = 99, normalized size = 1.22 \[ -\frac{2}{15} (2 x+3)^{5/2}+\frac{62}{27} (2 x+3)^{3/2}+\frac{526}{27} \sqrt{2 x+3}-6 \log \left (1-\sqrt{2 x+3}\right )+6 \log \left (\sqrt{2 x+3}+1\right )-\frac{850}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(526*Sqrt[3 + 2*x])/27 + (62*(3 + 2*x)^(3/2))/27 - (2*(3 + 2*x)^(5/2))/15 - (850
*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/27 - 6*Log[1 - Sqrt[3 + 2*x]] + 6*L
og[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.014, size = 71, normalized size = 0.9 \[ -{\frac{2}{15} \left ( 3+2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{62}{27} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{526}{27}\sqrt{3+2\,x}}-6\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{850\,\sqrt{15}}{81}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }+6\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3+2*x)^(5/2)+62/27*(3+2*x)^(3/2)+526/27*(3+2*x)^(1/2)-6*ln(-1+(3+2*x)^(1/
2))-850/81*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2))*15^(1/2)+6*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.791909, size = 119, normalized size = 1.47 \[ -\frac{2}{15} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{62}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{425}{81} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{526}{27} \, \sqrt{2 \, x + 3} + 6 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 6 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(2*x + 3)^(5/2) + 62/27*(2*x + 3)^(3/2) + 425/81*sqrt(15)*log(-(sqrt(15) -
3*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 526/27*sqrt(2*x + 3) + 6*log(s
qrt(2*x + 3) + 1) - 6*log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.290533, size = 127, normalized size = 1.57 \[ -\frac{1}{405} \, \sqrt{3}{\left (2 \, \sqrt{3}{\left (36 \, x^{2} - 202 \, x - 1699\right )} \sqrt{2 \, x + 3} - 810 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 810 \, \sqrt{3} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 2125 \, \sqrt{5} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} - 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/405*sqrt(3)*(2*sqrt(3)*(36*x^2 - 202*x - 1699)*sqrt(2*x + 3) - 810*sqrt(3)*lo
g(sqrt(2*x + 3) + 1) + 810*sqrt(3)*log(sqrt(2*x + 3) - 1) - 2125*sqrt(5)*log((sq
rt(3)*(3*x + 7) - 3*sqrt(5)*sqrt(2*x + 3))/(3*x + 2)))

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Sympy [A]  time = 49.9889, size = 126, normalized size = 1.56 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}}}{15} + \frac{62 \left (2 x + 3\right )^{\frac{3}{2}}}{27} + \frac{526 \sqrt{2 x + 3}}{27} + \frac{4250 \left (\begin{cases} - \frac{\sqrt{15} \operatorname{acoth}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 > \frac{5}{3} \\- \frac{\sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{15} & \text{for}\: 2 x + 3 < \frac{5}{3} \end{cases}\right )}{27} - 6 \log{\left (\sqrt{2 x + 3} - 1 \right )} + 6 \log{\left (\sqrt{2 x + 3} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(5/2)/15 + 62*(2*x + 3)**(3/2)/27 + 526*sqrt(2*x + 3)/27 + 4250*Pi
ecewise((-sqrt(15)*acoth(sqrt(15)*sqrt(2*x + 3)/5)/15, 2*x + 3 > 5/3), (-sqrt(15
)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/15, 2*x + 3 < 5/3))/27 - 6*log(sqrt(2*x + 3) -
1) + 6*log(sqrt(2*x + 3) + 1)

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GIAC/XCAS [A]  time = 0.273718, size = 124, normalized size = 1.53 \[ -\frac{2}{15} \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + \frac{62}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} + \frac{425}{81} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{526}{27} \, \sqrt{2 \, x + 3} + 6 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) - 6 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(2*x + 3)^(5/2) + 62/27*(2*x + 3)^(3/2) + 425/81*sqrt(15)*ln(1/2*abs(-2*sq
rt(15) + 6*sqrt(2*x + 3))/(sqrt(15) + 3*sqrt(2*x + 3))) + 526/27*sqrt(2*x + 3) +
6*ln(sqrt(2*x + 3) + 1) - 6*ln(abs(sqrt(2*x + 3) - 1))