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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]