3.1381 \(\int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac{49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac{6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac{4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac{4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac{4991}{12} x \sqrt{3 x^2+2}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

[Out]

(4991*x*Sqrt[2 + 3*x^2])/12 + (4991*x*(2 + 3*x^2)^(3/2))/36 + (4991*x*(2 + 3*x^2
)^(5/2))/90 + (6433*(3 + 2*x)^2*(2 + 3*x^2)^(7/2))/4455 + (49*(3 + 2*x)^3*(2 + 3
*x^2)^(7/2))/165 - ((3 + 2*x)^4*(2 + 3*x^2)^(7/2))/33 + (2*(181243 + 62244*x)*(2
+ 3*x^2)^(7/2))/13365 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi [A]  time = 0.247145, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{1}{33} \left (3 x^2+2\right )^{7/2} (2 x+3)^4+\frac{49}{165} \left (3 x^2+2\right )^{7/2} (2 x+3)^3+\frac{6433 \left (3 x^2+2\right )^{7/2} (2 x+3)^2}{4455}+\frac{2 (62244 x+181243) \left (3 x^2+2\right )^{7/2}}{13365}+\frac{4991}{90} x \left (3 x^2+2\right )^{5/2}+\frac{4991}{36} x \left (3 x^2+2\right )^{3/2}+\frac{4991}{12} x \sqrt{3 x^2+2}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)*(3 + 2*x)^4*(2 + 3*x^2)^(5/2),x]

[Out]

(4991*x*Sqrt[2 + 3*x^2])/12 + (4991*x*(2 + 3*x^2)^(3/2))/36 + (4991*x*(2 + 3*x^2
)^(5/2))/90 + (6433*(3 + 2*x)^2*(2 + 3*x^2)^(7/2))/4455 + (49*(3 + 2*x)^3*(2 + 3
*x^2)^(7/2))/165 - ((3 + 2*x)^4*(2 + 3*x^2)^(7/2))/33 + (2*(181243 + 62244*x)*(2
+ 3*x^2)^(7/2))/13365 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Rubi in Sympy [A]  time = 20.1153, size = 139, normalized size = 0.9 \[ \frac{4991 x \left (3 x^{2} + 2\right )^{\frac{5}{2}}}{90} + \frac{4991 x \left (3 x^{2} + 2\right )^{\frac{3}{2}}}{36} + \frac{4991 x \sqrt{3 x^{2} + 2}}{12} - \frac{\left (2 x + 3\right )^{4} \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{33} + \frac{49 \left (2 x + 3\right )^{3} \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{165} + \frac{6433 \left (2 x + 3\right )^{2} \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{4455} + \frac{\left (41827968 x + 121795296\right ) \left (3 x^{2} + 2\right )^{\frac{7}{2}}}{4490640} + \frac{4991 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3+2*x)**4*(3*x**2+2)**(5/2),x)

[Out]

4991*x*(3*x**2 + 2)**(5/2)/90 + 4991*x*(3*x**2 + 2)**(3/2)/36 + 4991*x*sqrt(3*x*
*2 + 2)/12 - (2*x + 3)**4*(3*x**2 + 2)**(7/2)/33 + 49*(2*x + 3)**3*(3*x**2 + 2)*
*(7/2)/165 + 6433*(2*x + 3)**2*(3*x**2 + 2)**(7/2)/4455 + (41827968*x + 12179529
6)*(3*x**2 + 2)**(7/2)/4490640 + 4991*sqrt(3)*asinh(sqrt(6)*x/2)/18

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Mathematica [A]  time = 0.110252, size = 85, normalized size = 0.55 \[ \frac{\sqrt{3 x^2+2} \left (-699840 x^{10}-769824 x^9+12921120 x^8+50615928 x^7+93646260 x^6+129966606 x^5+150762600 x^4+127123425 x^3+92160240 x^2+64370295 x+19537120\right )}{53460}+\frac{4991 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)*(3 + 2*x)^4*(2 + 3*x^2)^(5/2),x]

[Out]

(Sqrt[2 + 3*x^2]*(19537120 + 64370295*x + 92160240*x^2 + 127123425*x^3 + 1507626
00*x^4 + 129966606*x^5 + 93646260*x^6 + 50615928*x^7 + 12921120*x^8 - 769824*x^9
- 699840*x^10))/53460 + (4991*ArcSinh[Sqrt[3/2]*x])/(6*Sqrt[3])

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Maple [A]  time = 0.02, size = 115, normalized size = 0.8 \[{\frac{4991\,x}{90} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{4991\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{4991\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{4991\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) }+{\frac{122107}{2673} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{542\,x}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{8840\,{x}^{2}}{891} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{x}^{3}}{15} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}-{\frac{16\,{x}^{4}}{33} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)*(2*x+3)^4*(3*x^2+2)^(5/2),x)

[Out]

4991/90*x*(3*x^2+2)^(5/2)+4991/36*x*(3*x^2+2)^(3/2)+4991/12*x*(3*x^2+2)^(1/2)+49
91/18*arcsinh(1/2*x*6^(1/2))*3^(1/2)+122107/2673*(3*x^2+2)^(7/2)+542/15*x*(3*x^2
+2)^(7/2)+8840/891*x^2*(3*x^2+2)^(7/2)-8/15*x^3*(3*x^2+2)^(7/2)-16/33*x^4*(3*x^2
+2)^(7/2)

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Maxima [A]  time = 0.777357, size = 154, normalized size = 1. \[ -\frac{16}{33} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{4} - \frac{8}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{3} + \frac{8840}{891} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{2} + \frac{542}{15} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{122107}{2673} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{4991}{90} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{4991}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{4991}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{4991}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(2*x + 3)^4*(x - 5),x, algorithm="maxima")

[Out]

-16/33*(3*x^2 + 2)^(7/2)*x^4 - 8/15*(3*x^2 + 2)^(7/2)*x^3 + 8840/891*(3*x^2 + 2)
^(7/2)*x^2 + 542/15*(3*x^2 + 2)^(7/2)*x + 122107/2673*(3*x^2 + 2)^(7/2) + 4991/9
0*(3*x^2 + 2)^(5/2)*x + 4991/36*(3*x^2 + 2)^(3/2)*x + 4991/12*sqrt(3*x^2 + 2)*x
+ 4991/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)

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Fricas [A]  time = 0.295164, size = 131, normalized size = 0.85 \[ -\frac{1}{160380} \, \sqrt{3}{\left (\sqrt{3}{\left (699840 \, x^{10} + 769824 \, x^{9} - 12921120 \, x^{8} - 50615928 \, x^{7} - 93646260 \, x^{6} - 129966606 \, x^{5} - 150762600 \, x^{4} - 127123425 \, x^{3} - 92160240 \, x^{2} - 64370295 \, x - 19537120\right )} \sqrt{3 \, x^{2} + 2} - 22234905 \, \log \left (-\sqrt{3}{\left (3 \, x^{2} + 1\right )} - 3 \, \sqrt{3 \, x^{2} + 2} x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(2*x + 3)^4*(x - 5),x, algorithm="fricas")

[Out]

-1/160380*sqrt(3)*(sqrt(3)*(699840*x^10 + 769824*x^9 - 12921120*x^8 - 50615928*x
^7 - 93646260*x^6 - 129966606*x^5 - 150762600*x^4 - 127123425*x^3 - 92160240*x^2
- 64370295*x - 19537120)*sqrt(3*x^2 + 2) - 22234905*log(-sqrt(3)*(3*x^2 + 1) -
3*sqrt(3*x^2 + 2)*x))

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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)**4*(3*x**2+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.329348, size = 111, normalized size = 0.72 \[ -\frac{1}{53460} \,{\left (3 \,{\left ({\left (9 \,{\left (2 \,{\left ({\left (2 \,{\left (6 \,{\left (4 \,{\left (27 \,{\left (10 \, x + 11\right )} x - 4985\right )} x - 78111\right )} x - 867095\right )} x - 2406789\right )} x - 2791900\right )} x - 4708275\right )} x - 30720080\right )} x - 21456765\right )} x - 19537120\right )} \sqrt{3 \, x^{2} + 2} - \frac{4991}{18} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 2)^(5/2)*(2*x + 3)^4*(x - 5),x, algorithm="giac")

[Out]

-1/53460*(3*((9*(2*((2*(6*(4*(27*(10*x + 11)*x - 4985)*x - 78111)*x - 867095)*x
- 2406789)*x - 2791900)*x - 4708275)*x - 30720080)*x - 21456765)*x - 19537120)*s
qrt(3*x^2 + 2) - 4991/18*sqrt(3)*ln(-sqrt(3)*x + sqrt(3*x^2 + 2))