3.32 \(\int \frac{x^7 \left (2+3 x^2\right )}{\sqrt{5+x^4}} \, dx\)

Optimal. Leaf size=67 \[ \frac{1}{3} \sqrt{x^4+5} x^4+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{3}{8} \sqrt{x^4+5} x^6-\frac{5}{48} \left (27 x^2+32\right ) \sqrt{x^4+5} \]

[Out]

(x^4*Sqrt[5 + x^4])/3 + (3*x^6*Sqrt[5 + x^4])/8 - (5*(32 + 27*x^2)*Sqrt[5 + x^4]
)/48 + (225*ArcSinh[x^2/Sqrt[5]])/16

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Rubi [A]  time = 0.185368, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{3} \sqrt{x^4+5} x^4+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{3}{8} \sqrt{x^4+5} x^6-\frac{5}{48} \left (27 x^2+32\right ) \sqrt{x^4+5} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(2 + 3*x^2))/Sqrt[5 + x^4],x]

[Out]

(x^4*Sqrt[5 + x^4])/3 + (3*x^6*Sqrt[5 + x^4])/8 - (5*(32 + 27*x^2)*Sqrt[5 + x^4]
)/48 + (225*ArcSinh[x^2/Sqrt[5]])/16

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Rubi in Sympy [A]  time = 13.1133, size = 60, normalized size = 0.9 \[ \frac{3 x^{6} \sqrt{x^{4} + 5}}{8} + \frac{x^{4} \sqrt{x^{4} + 5}}{3} - \frac{\left (135 x^{2} + 160\right ) \sqrt{x^{4} + 5}}{48} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(3*x**2+2)/(x**4+5)**(1/2),x)

[Out]

3*x**6*sqrt(x**4 + 5)/8 + x**4*sqrt(x**4 + 5)/3 - (135*x**2 + 160)*sqrt(x**4 + 5
)/48 + 225*asinh(sqrt(5)*x**2/5)/16

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Mathematica [A]  time = 0.0423699, size = 44, normalized size = 0.66 \[ \frac{1}{48} \left (675 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\sqrt{x^4+5} \left (18 x^6+16 x^4-135 x^2-160\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(2 + 3*x^2))/Sqrt[5 + x^4],x]

[Out]

(Sqrt[5 + x^4]*(-160 - 135*x^2 + 16*x^4 + 18*x^6) + 675*ArcSinh[x^2/Sqrt[5]])/48

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Maple [A]  time = 0.022, size = 51, normalized size = 0.8 \[{\frac{{x}^{4}-10}{3}\sqrt{{x}^{4}+5}}+{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5}}-{\frac{45\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}+{\frac{225}{16}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(3*x^2+2)/(x^4+5)^(1/2),x)

[Out]

1/3*(x^4+5)^(1/2)*(x^4-10)+3/8*x^6*(x^4+5)^(1/2)-45/16*x^2*(x^4+5)^(1/2)+225/16*
arcsinh(1/5*5^(1/2)*x^2)

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Maxima [A]  time = 0.78087, size = 140, normalized size = 2.09 \[ \frac{1}{3} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{4} + 5} - \frac{75 \,{\left (\frac{5 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{16 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^7/sqrt(x^4 + 5),x, algorithm="maxima")

[Out]

1/3*(x^4 + 5)^(3/2) - 5*sqrt(x^4 + 5) - 75/16*(5*sqrt(x^4 + 5)/x^2 - 3*(x^4 + 5)
^(3/2)/x^6)/(2*(x^4 + 5)/x^4 - (x^4 + 5)^2/x^8 - 1) + 225/32*log(sqrt(x^4 + 5)/x
^2 + 1) - 225/32*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.261317, size = 224, normalized size = 3.34 \[ -\frac{144 \, x^{16} + 128 \, x^{14} - 320 \, x^{10} - 6300 \, x^{8} - 8000 \, x^{6} - 13500 \, x^{4} - 16000 \, x^{2} + 675 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (144 \, x^{14} + 128 \, x^{12} - 360 \, x^{10} - 640 \, x^{8} - 4950 \, x^{6} - 6000 \, x^{4} - 3375 \, x^{2} - 4000\right )} \sqrt{x^{4} + 5}}{48 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^7/sqrt(x^4 + 5),x, algorithm="fricas")

[Out]

-1/48*(144*x^16 + 128*x^14 - 320*x^10 - 6300*x^8 - 8000*x^6 - 13500*x^4 - 16000*
x^2 + 675*(8*x^8 + 40*x^4 - 4*(2*x^6 + 5*x^2)*sqrt(x^4 + 5) + 25)*log(-x^2 + sqr
t(x^4 + 5)) - (144*x^14 + 128*x^12 - 360*x^10 - 640*x^8 - 4950*x^6 - 6000*x^4 -
3375*x^2 - 4000)*sqrt(x^4 + 5))/(8*x^8 + 40*x^4 - 4*(2*x^6 + 5*x^2)*sqrt(x^4 + 5
) + 25)

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Sympy [A]  time = 15.6066, size = 85, normalized size = 1.27 \[ \frac{3 x^{10}}{8 \sqrt{x^{4} + 5}} - \frac{15 x^{6}}{16 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{3} - \frac{225 x^{2}}{16 \sqrt{x^{4} + 5}} - \frac{10 \sqrt{x^{4} + 5}}{3} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(3*x**2+2)/(x**4+5)**(1/2),x)

[Out]

3*x**10/(8*sqrt(x**4 + 5)) - 15*x**6/(16*sqrt(x**4 + 5)) + x**4*sqrt(x**4 + 5)/3
- 225*x**2/(16*sqrt(x**4 + 5)) - 10*sqrt(x**4 + 5)/3 + 225*asinh(sqrt(5)*x**2/5
)/16

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GIAC/XCAS [A]  time = 0.261964, size = 62, normalized size = 0.93 \[ \frac{1}{48} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left (9 \, x^{2} + 8\right )} x^{2} - 135\right )} x^{2} - 160\right )} - \frac{225}{16} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 2)*x^7/sqrt(x^4 + 5),x, algorithm="giac")

[Out]

1/48*sqrt(x^4 + 5)*((2*(9*x^2 + 8)*x^2 - 135)*x^2 - 160) - 225/16*ln(-x^2 + sqrt
(x^4 + 5))