3.2557 \(\int \frac{(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx\)

Optimal. Leaf size=98 \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{3 \left (3 x^2+5 x+2\right )}+\frac{826}{27} (2 x+3)^{3/2}+\frac{1358}{27} \sqrt{2 x+3}-154 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

[Out]

(1358*Sqrt[3 + 2*x])/27 + (826*(3 + 2*x)^(3/2))/27 - ((3 + 2*x)^(5/2)*(121 + 139
*x))/(3*(2 + 5*x + 3*x^2)) - 154*ArcTanh[Sqrt[3 + 2*x]] + (2800*Sqrt[5/3]*ArcTan
h[Sqrt[3/5]*Sqrt[3 + 2*x]])/27

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Rubi [A]  time = 0.23137, antiderivative size = 98, 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 \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{3 \left (3 x^2+5 x+2\right )}+\frac{826}{27} (2 x+3)^{3/2}+\frac{1358}{27} \sqrt{2 x+3}-154 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(1358*Sqrt[3 + 2*x])/27 + (826*(3 + 2*x)^(3/2))/27 - ((3 + 2*x)^(5/2)*(121 + 139
*x))/(3*(2 + 5*x + 3*x^2)) - 154*ArcTanh[Sqrt[3 + 2*x]] + (2800*Sqrt[5/3]*ArcTan
h[Sqrt[3/5]*Sqrt[3 + 2*x]])/27

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Rubi in Sympy [A]  time = 42.2546, size = 83, normalized size = 0.85 \[ - \frac{\left (2 x + 3\right )^{\frac{5}{2}} \left (139 x + 121\right )}{3 \left (3 x^{2} + 5 x + 2\right )} + \frac{826 \left (2 x + 3\right )^{\frac{3}{2}}}{27} + \frac{1358 \sqrt{2 x + 3}}{27} + \frac{2800 \sqrt{15} \operatorname{atanh}{\left (\frac{\sqrt{15} \sqrt{2 x + 3}}{5} \right )}}{81} - 154 \operatorname{atanh}{\left (\sqrt{2 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x + 3)**(5/2)*(139*x + 121)/(3*(3*x**2 + 5*x + 2)) + 826*(2*x + 3)**(3/2)/27
+ 1358*sqrt(2*x + 3)/27 + 2800*sqrt(15)*atanh(sqrt(15)*sqrt(2*x + 3)/5)/81 - 15
4*atanh(sqrt(2*x + 3))

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Mathematica [A]  time = 0.19394, size = 100, normalized size = 1.02 \[ -\frac{\sqrt{2 x+3} \left (48 x^3-400 x^2+1843 x+2129\right )}{27 \left (3 x^2+5 x+2\right )}+77 \log \left (1-\sqrt{2 x+3}\right )-77 \log \left (\sqrt{2 x+3}+1\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[3 + 2*x]*(2129 + 1843*x - 400*x^2 + 48*x^3))/(27*(2 + 5*x + 3*x^2)) + (28
00*Sqrt[5/3]*ArcTanh[Sqrt[3/5]*Sqrt[3 + 2*x]])/27 + 77*Log[1 - Sqrt[3 + 2*x]] -
77*Log[1 + Sqrt[3 + 2*x]]

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Maple [A]  time = 0.028, size = 104, normalized size = 1.1 \[ -{\frac{8}{27} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{184}{27}\sqrt{3+2\,x}}-6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+77\,\ln \left ( -1+\sqrt{3+2\,x} \right ) -{\frac{4250}{81}\sqrt{3+2\,x} \left ({\frac{4}{3}}+2\,x \right ) ^{-1}}+{\frac{2800\,\sqrt{15}}{81}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-77\,\ln \left ( 1+\sqrt{3+2\,x} \right ) \]

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]

-8/27*(3+2*x)^(3/2)+184/27*(3+2*x)^(1/2)-6/(-1+(3+2*x)^(1/2))+77*ln(-1+(3+2*x)^(
1/2))-4250/81*(3+2*x)^(1/2)/(4/3+2*x)+2800/81*arctanh(1/5*15^(1/2)*(3+2*x)^(1/2)
)*15^(1/2)-6/(1+(3+2*x)^(1/2))-77*ln(1+(3+2*x)^(1/2))

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Maxima [A]  time = 0.785037, size = 157, normalized size = 1.6 \[ -\frac{8}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1400}{81} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{184}{27} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (2611 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 2935 \, \sqrt{2 \, x + 3}\right )}}{27 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 77 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/27*(2*x + 3)^(3/2) - 1400/81*sqrt(15)*log(-(sqrt(15) - 3*sqrt(2*x + 3))/(sqrt
(15) + 3*sqrt(2*x + 3))) + 184/27*sqrt(2*x + 3) - 2/27*(2611*(2*x + 3)^(3/2) - 2
935*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 77*log(sqrt(2*x + 3) + 1) + 77*
log(sqrt(2*x + 3) - 1)

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Fricas [A]  time = 0.292014, size = 189, normalized size = 1.93 \[ -\frac{\sqrt{3}{\left (2079 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 2079 \, \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 1400 \, \sqrt{5}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{3}{\left (3 \, x + 7\right )} + 3 \, \sqrt{5} \sqrt{2 \, x + 3}}{3 \, x + 2}\right ) + \sqrt{3}{\left (48 \, x^{3} - 400 \, x^{2} + 1843 \, x + 2129\right )} \sqrt{2 \, x + 3}\right )}}{81 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/81*sqrt(3)*(2079*sqrt(3)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) + 1) - 2079*sqrt
(3)*(3*x^2 + 5*x + 2)*log(sqrt(2*x + 3) - 1) - 1400*sqrt(5)*(3*x^2 + 5*x + 2)*lo
g((sqrt(3)*(3*x + 7) + 3*sqrt(5)*sqrt(2*x + 3))/(3*x + 2)) + sqrt(3)*(48*x^3 - 4
00*x^2 + 1843*x + 2129)*sqrt(2*x + 3))/(3*x^2 + 5*x + 2)

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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 = 0.279846, size = 162, normalized size = 1.65 \[ -\frac{8}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1400}{81} \, \sqrt{15}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) + \frac{184}{27} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (2611 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 2935 \, \sqrt{2 \, x + 3}\right )}}{27 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \,{\rm ln}\left (\sqrt{2 \, x + 3} + 1\right ) + 77 \,{\rm ln}\left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8/27*(2*x + 3)^(3/2) - 1400/81*sqrt(15)*ln(1/2*abs(-2*sqrt(15) + 6*sqrt(2*x + 3
))/(sqrt(15) + 3*sqrt(2*x + 3))) + 184/27*sqrt(2*x + 3) - 2/27*(2611*(2*x + 3)^(
3/2) - 2935*sqrt(2*x + 3))/(3*(2*x + 3)^2 - 16*x - 19) - 77*ln(sqrt(2*x + 3) + 1
) + 77*ln(abs(sqrt(2*x + 3) - 1))