3.1418 \(\int \frac{(5-x) (3+2 x)^6}{\left (2+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}+\frac{(2427 x+158) (2 x+3)^3}{54 \sqrt{3 x^2+2}}-\frac{2639}{81} \sqrt{3 x^2+2} (2 x+3)^2-\frac{70}{243} (801 x+2167) \sqrt{3 x^2+2}+\frac{20720 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^5)/(18*(2 + 3*x^2)^(3/2)) + ((3 + 2*x)^3*(158 + 2427*x))
/(54*Sqrt[2 + 3*x^2]) - (2639*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/81 - (70*(2167 + 801*
x)*Sqrt[2 + 3*x^2])/243 + (20720*ArcSinh[Sqrt[3/2]*x])/(27*Sqrt[3])

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Rubi [A]  time = 0.22589, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{7 (2-7 x) (2 x+3)^5}{18 \left (3 x^2+2\right )^{3/2}}+\frac{(2427 x+158) (2 x+3)^3}{54 \sqrt{3 x^2+2}}-\frac{2639}{81} \sqrt{3 x^2+2} (2 x+3)^2-\frac{70}{243} (801 x+2167) \sqrt{3 x^2+2}+\frac{20720 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{27 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]

[Out]

(-7*(2 - 7*x)*(3 + 2*x)^5)/(18*(2 + 3*x^2)^(3/2)) + ((3 + 2*x)^3*(158 + 2427*x))
/(54*Sqrt[2 + 3*x^2]) - (2639*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/81 - (70*(2167 + 801*
x)*Sqrt[2 + 3*x^2])/243 + (20720*ArcSinh[Sqrt[3/2]*x])/(27*Sqrt[3])

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Rubi in Sympy [A]  time = 20.4543, size = 102, normalized size = 0.88 \[ - \frac{\left (- 98 x + 28\right ) \left (2 x + 3\right )^{5}}{36 \left (3 x^{2} + 2\right )^{\frac{3}{2}}} + \frac{\left (2 x + 3\right )^{3} \left (19416 x + 1264\right )}{432 \sqrt{3 x^{2} + 2}} - \frac{2639 \left (2 x + 3\right )^{2} \sqrt{3 x^{2} + 2}}{81} - \frac{\left (5382720 x + 14562240\right ) \sqrt{3 x^{2} + 2}}{23328} + \frac{20720 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**6/(3*x**2+2)**(5/2),x)

[Out]

-(-98*x + 28)*(2*x + 3)**5/(36*(3*x**2 + 2)**(3/2)) + (2*x + 3)**3*(19416*x + 12
64)/(432*sqrt(3*x**2 + 2)) - 2639*(2*x + 3)**2*sqrt(3*x**2 + 2)/81 - (5382720*x
+ 14562240)*sqrt(3*x**2 + 2)/23328 + 20720*sqrt(3)*asinh(sqrt(6)*x/2)/81

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Mathematica [A]  time = 0.135108, size = 65, normalized size = 0.56 \[ \frac{1}{486} \left (124320 \sqrt{3} \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )-\frac{3456 x^6+20736 x^5-130464 x^4-1125999 x^3+2363976 x^2-139815 x+1798610}{\left (3 x^2+2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(3 + 2*x)^6)/(2 + 3*x^2)^(5/2),x]

[Out]

(-((1798610 - 139815*x + 2363976*x^2 - 1125999*x^3 - 130464*x^4 + 20736*x^5 + 34
56*x^6)/(2 + 3*x^2)^(3/2)) + 124320*Sqrt[3]*ArcSinh[Sqrt[3/2]*x])/486

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Maple [A]  time = 0.03, size = 119, normalized size = 1. \[ -{\frac{3537\,x}{2} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{55517\,x}{54}{\frac{1}{\sqrt{3\,{x}^{2}+2}}}}-{\frac{899305}{243} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{131332\,{x}^{2}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{20720\,{x}^{3}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{20720\,\sqrt{3}}{81}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{2416\,{x}^{4}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{128\,{x}^{5}}{3} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{64\,{x}^{6}}{9} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(2*x+3)^6/(3*x^2+2)^(5/2),x)

[Out]

-3537/2*x/(3*x^2+2)^(3/2)+55517/54*x/(3*x^2+2)^(1/2)-899305/243/(3*x^2+2)^(3/2)-
131332/27*x^2/(3*x^2+2)^(3/2)-20720/27*x^3/(3*x^2+2)^(3/2)+20720/81*arcsinh(1/2*
x*6^(1/2))*3^(1/2)+2416/9*x^4/(3*x^2+2)^(3/2)-128/3*x^5/(3*x^2+2)^(3/2)-64/9*x^6
/(3*x^2+2)^(3/2)

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Maxima [A]  time = 0.759832, size = 180, normalized size = 1.55 \[ -\frac{64 \, x^{6}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{128 \, x^{5}}{3 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{2416 \, x^{4}}{9 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{20720}{81} \, x{\left (\frac{9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} + \frac{4}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\right )} + \frac{20720}{81} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{249431 \, x}{162 \, \sqrt{3 \, x^{2} + 2}} - \frac{131332 \, x^{2}}{27 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{3537 \, x}{2 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} - \frac{899305}{243 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^6*(x - 5)/(3*x^2 + 2)^(5/2),x, algorithm="maxima")

[Out]

-64/9*x^6/(3*x^2 + 2)^(3/2) - 128/3*x^5/(3*x^2 + 2)^(3/2) + 2416/9*x^4/(3*x^2 +
2)^(3/2) - 20720/81*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 20720/81
*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 249431/162*x/sqrt(3*x^2 + 2) - 131332/27*x^2/(
3*x^2 + 2)^(3/2) - 3537/2*x/(3*x^2 + 2)^(3/2) - 899305/243/(3*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.275465, size = 139, normalized size = 1.2 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (3456 \, x^{6} + 20736 \, x^{5} - 130464 \, x^{4} - 1125999 \, x^{3} + 2363976 \, x^{2} - 139815 \, x + 1798610\right )} \sqrt{3 \, x^{2} + 2} - 186480 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )}}{1458 \,{\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^6*(x - 5)/(3*x^2 + 2)^(5/2),x, algorithm="fricas")

[Out]

-1/1458*sqrt(3)*(sqrt(3)*(3456*x^6 + 20736*x^5 - 130464*x^4 - 1125999*x^3 + 2363
976*x^2 - 139815*x + 1798610)*sqrt(3*x^2 + 2) - 186480*(9*x^4 + 12*x^2 + 4)*log(
-sqrt(3)*(3*x^2 + 1) - 3*sqrt(3*x^2 + 2)*x))/(9*x^4 + 12*x^2 + 4)

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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)**6/(3*x**2+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.308886, size = 81, normalized size = 0.7 \[ -\frac{20720}{81} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) - \frac{9 \,{\left ({\left ({\left (96 \,{\left (4 \,{\left (x + 6\right )} x - 151\right )} x - 125111\right )} x + 262664\right )} x - 15535\right )} x + 1798610}{486 \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x + 3)^6*(x - 5)/(3*x^2 + 2)^(5/2),x, algorithm="giac")

[Out]

-20720/81*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/486*(9*(((96*(4*(x + 6)*x
- 151)*x - 125111)*x + 262664)*x - 15535)*x + 1798610)/(3*x^2 + 2)^(3/2)