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

Optimal. Leaf size=58 \[ -\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (3 x^2+2\right ) x^4}{2 \sqrt{x^4+5}}+\frac{1}{4} \left (9 x^2+8\right ) \sqrt{x^4+5} \]

[Out]

-(x^4*(2 + 3*x^2))/(2*Sqrt[5 + x^4]) + ((8 + 9*x^2)*Sqrt[5 + x^4])/4 - (45*ArcSi
nh[x^2/Sqrt[5]])/4

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Rubi [A]  time = 0.145792, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (3 x^2+2\right ) x^4}{2 \sqrt{x^4+5}}+\frac{1}{4} \left (9 x^2+8\right ) \sqrt{x^4+5} \]

Antiderivative was successfully verified.

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

[Out]

-(x^4*(2 + 3*x^2))/(2*Sqrt[5 + x^4]) + ((8 + 9*x^2)*Sqrt[5 + x^4])/4 - (45*ArcSi
nh[x^2/Sqrt[5]])/4

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Rubi in Sympy [A]  time = 11.2306, size = 51, normalized size = 0.88 \[ - \frac{x^{4} \left (30 x^{2} + 20\right )}{20 \sqrt{x^{4} + 5}} + \frac{\left (90 x^{2} + 80\right ) \sqrt{x^{4} + 5}}{40} - \frac{45 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4*(30*x**2 + 20)/(20*sqrt(x**4 + 5)) + (90*x**2 + 80)*sqrt(x**4 + 5)/40 - 45
*asinh(sqrt(5)*x**2/5)/4

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Mathematica [A]  time = 0.0589764, size = 44, normalized size = 0.76 \[ \frac{1}{4} \left (\frac{3 x^6+4 x^4+45 x^2+40}{\sqrt{x^4+5}}-45 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((40 + 45*x^2 + 4*x^4 + 3*x^6)/Sqrt[5 + x^4] - 45*ArcSinh[x^2/Sqrt[5]])/4

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Maple [A]  time = 0.026, size = 50, normalized size = 0.9 \[{({x}^{4}+10){\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3\,{x}^{6}}{4}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{45\,{x}^{2}}{4}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{45}{4}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(x^4+5)^(1/2)*(x^4+10)+3/4*x^6/(x^4+5)^(1/2)+45/4*x^2/(x^4+5)^(1/2)-45/4*arcsi
nh(1/5*5^(1/2)*x^2)

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Maxima [A]  time = 0.774079, size = 120, normalized size = 2.07 \[ \sqrt{x^{4} + 5} - \frac{15 \,{\left (\frac{3 \,{\left (x^{4} + 5\right )}}{x^{4}} - 2\right )}}{4 \,{\left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}} + \frac{5}{\sqrt{x^{4} + 5}} - \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^4 + 5) - 15/4*(3*(x^4 + 5)/x^4 - 2)/(sqrt(x^4 + 5)/x^2 - (x^4 + 5)^(3/2)/
x^6) + 5/sqrt(x^4 + 5) - 45/8*log(sqrt(x^4 + 5)/x^2 + 1) + 45/8*log(sqrt(x^4 + 5
)/x^2 - 1)

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Fricas [A]  time = 0.297073, size = 205, normalized size = 3.53 \[ -\frac{12 \, x^{12} + 16 \, x^{10} + 105 \, x^{8} + 220 \, x^{6} - 75 \, x^{4} + 600 \, x^{2} - 45 \,{\left (4 \, x^{8} + 25 \, x^{4} -{\left (4 \, x^{6} + 15 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (12 \, x^{10} + 16 \, x^{8} + 75 \, x^{6} + 180 \, x^{4} - 225 \, x^{2} + 200\right )} \sqrt{x^{4} + 5} - 750}{4 \,{\left (4 \, x^{8} + 25 \, x^{4} -{\left (4 \, x^{6} + 15 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(12*x^12 + 16*x^10 + 105*x^8 + 220*x^6 - 75*x^4 + 600*x^2 - 45*(4*x^8 + 25*
x^4 - (4*x^6 + 15*x^2)*sqrt(x^4 + 5) + 25)*log(-x^2 + sqrt(x^4 + 5)) - (12*x^10
+ 16*x^8 + 75*x^6 + 180*x^4 - 225*x^2 + 200)*sqrt(x^4 + 5) - 750)/(4*x^8 + 25*x^
4 - (4*x^6 + 15*x^2)*sqrt(x^4 + 5) + 25)

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Sympy [A]  time = 30.3557, size = 66, normalized size = 1.14 \[ \frac{3 x^{6}}{4 \sqrt{x^{4} + 5}} + \frac{x^{4}}{\sqrt{x^{4} + 5}} + \frac{45 x^{2}}{4 \sqrt{x^{4} + 5}} - \frac{45 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} + \frac{10}{\sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**6/(4*sqrt(x**4 + 5)) + x**4/sqrt(x**4 + 5) + 45*x**2/(4*sqrt(x**4 + 5)) - 4
5*asinh(sqrt(5)*x**2/5)/4 + 10/sqrt(x**4 + 5)

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GIAC/XCAS [A]  time = 0.268689, size = 61, normalized size = 1.05 \[ \frac{{\left ({\left (3 \, x^{2} + 4\right )} x^{2} + 45\right )} x^{2} + 40}{4 \, \sqrt{x^{4} + 5}} + \frac{45}{4} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(((3*x^2 + 4)*x^2 + 45)*x^2 + 40)/sqrt(x^4 + 5) + 45/4*ln(-x^2 + sqrt(x^4 +
5))