3.2072 \(\int \left (a+\frac{b}{x^4}\right )^{5/2} x^3 \, dx\)

Optimal. Leaf size=80 \[ \frac{5}{4} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )+\frac{1}{4} x^4 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5}{12} b \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{5}{4} a b \sqrt{a+\frac{b}{x^4}} \]

[Out]

(-5*a*b*Sqrt[a + b/x^4])/4 - (5*b*(a + b/x^4)^(3/2))/12 + ((a + b/x^4)^(5/2)*x^4
)/4 + (5*a^(3/2)*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/4

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Rubi [A]  time = 0.137096, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5}{4} a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )+\frac{1}{4} x^4 \left (a+\frac{b}{x^4}\right )^{5/2}-\frac{5}{12} b \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{5}{4} a b \sqrt{a+\frac{b}{x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(5/2)*x^3,x]

[Out]

(-5*a*b*Sqrt[a + b/x^4])/4 - (5*b*(a + b/x^4)^(3/2))/12 + ((a + b/x^4)^(5/2)*x^4
)/4 + (5*a^(3/2)*b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/4

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Rubi in Sympy [A]  time = 10.9803, size = 73, normalized size = 0.91 \[ \frac{5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4} - \frac{5 a b \sqrt{a + \frac{b}{x^{4}}}}{4} - \frac{5 b \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{12} + \frac{x^{4} \left (a + \frac{b}{x^{4}}\right )^{\frac{5}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*b*atanh(sqrt(a + b/x**4)/sqrt(a))/4 - 5*a*b*sqrt(a + b/x**4)/4 - 5*b*
(a + b/x**4)**(3/2)/12 + x**4*(a + b/x**4)**(5/2)/4

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Mathematica [A]  time = 0.118249, size = 95, normalized size = 1.19 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (15 a^{3/2} b x^6 \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )+\sqrt{a x^4+b} \left (3 a^2 x^8-14 a b x^4-2 b^2\right )\right )}{12 x^4 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*(Sqrt[b + a*x^4]*(-2*b^2 - 14*a*b*x^4 + 3*a^2*x^8) + 15*a^(3/2)
*b*x^6*ArcTanh[(Sqrt[a]*x^2)/Sqrt[b + a*x^4]]))/(12*x^4*Sqrt[b + a*x^4])

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Maple [A]  time = 0.029, size = 103, normalized size = 1.3 \[{\frac{{x}^{4}}{12} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{a}^{3/2}b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{6}+3\,{a}^{2}{x}^{8}\sqrt{a{x}^{4}+b}-14\,ab\sqrt{a{x}^{4}+b}{x}^{4}-2\,{b}^{2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(5/2)*x^3,x)

[Out]

1/12*((a*x^4+b)/x^4)^(5/2)*x^4*(15*a^(3/2)*b*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*x^6
+3*a^2*x^8*(a*x^4+b)^(1/2)-14*a*b*(a*x^4+b)^(1/2)*x^4-2*b^2*(a*x^4+b)^(1/2))/(a*
x^4+b)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261971, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{\frac{3}{2}} b x^{4} \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) + 2 \,{\left (3 \, a^{2} x^{8} - 14 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{24 \, x^{4}}, \frac{15 \, \sqrt{-a} a b x^{4} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (3 \, a^{2} x^{8} - 14 \, a b x^{4} - 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{12 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(15*a^(3/2)*b*x^4*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b)
+ 2*(3*a^2*x^8 - 14*a*b*x^4 - 2*b^2)*sqrt((a*x^4 + b)/x^4))/x^4, 1/12*(15*sqrt(-
a)*a*b*x^4*arctan(a/(sqrt(-a)*sqrt((a*x^4 + b)/x^4))) + (3*a^2*x^8 - 14*a*b*x^4
- 2*b^2)*sqrt((a*x^4 + b)/x^4))/x^4]

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Sympy [A]  time = 26.1637, size = 112, normalized size = 1.4 \[ \frac{a^{\frac{5}{2}} x^{4} \sqrt{1 + \frac{b}{a x^{4}}}}{4} - \frac{7 a^{\frac{3}{2}} b \sqrt{1 + \frac{b}{a x^{4}}}}{6} - \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b}{a x^{4}} \right )}}{8} + \frac{5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4} - \frac{\sqrt{a} b^{2} \sqrt{1 + \frac{b}{a x^{4}}}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*x**4*sqrt(1 + b/(a*x**4))/4 - 7*a**(3/2)*b*sqrt(1 + b/(a*x**4))/6 - 5*a
**(3/2)*b*log(b/(a*x**4))/8 + 5*a**(3/2)*b*log(sqrt(1 + b/(a*x**4)) + 1)/4 - sqr
t(a)*b**2*sqrt(1 + b/(a*x**4))/(6*x**4)

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GIAC/XCAS [A]  time = 0.243232, size = 192, normalized size = 2.4 \[ \frac{1}{4} \, \sqrt{a x^{4} + b} a^{2} x^{2} - \frac{5}{8} \, a^{\frac{3}{2}} b{\rm ln}\left ({\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{2}\right ) + \frac{9 \,{\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{4} a^{\frac{3}{2}} b^{2} - 12 \,{\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{2} a^{\frac{3}{2}} b^{3} + 7 \, a^{\frac{3}{2}} b^{4}}{3 \,{\left ({\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{2} - b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(a*x^4 + b)*a^2*x^2 - 5/8*a^(3/2)*b*ln((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2
) + 1/3*(9*(sqrt(a)*x^2 - sqrt(a*x^4 + b))^4*a^(3/2)*b^2 - 12*(sqrt(a)*x^2 - sqr
t(a*x^4 + b))^2*a^(3/2)*b^3 + 7*a^(3/2)*b^4)/((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2
- b)^3