Optimal. Leaf size=82 \[ b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.0885073, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 \sqrt{a+b x^2}}{x}-\frac{\left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(5/2)/x^6,x]
[Out]
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Rubi in Sympy [A] time = 10.9482, size = 70, normalized size = 0.85 \[ b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{b^{2} \sqrt{a + b x^{2}}}{x} - \frac{b \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(5/2)/x**6,x)
[Out]
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Mathematica [A] time = 0.0561669, size = 68, normalized size = 0.83 \[ b^{5/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (3 a^2+11 a b x^2+23 b^2 x^4\right )}{15 x^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(5/2)/x^6,x]
[Out]
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Maple [A] time = 0.012, size = 130, normalized size = 1.6 \[ -{\frac{1}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{15\,{a}^{2}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{b}^{2}}{15\,{a}^{3}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{b}^{3}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{3}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}x}{a}\sqrt{b{x}^{2}+a}}+{b}^{{\frac{5}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(5/2)/x^6,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((b*x^2 + a)^(5/2)/x^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247394, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{\frac{5}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{30 \, x^{5}}, \frac{15 \, \sqrt{-b} b^{2} x^{5} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left (23 \, b^{2} x^{4} + 11 \, a b x^{2} + 3 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)/x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.9194, size = 105, normalized size = 1.28 \[ - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{11 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 x^{2}} - \frac{23 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15} - \frac{b^{\frac{5}{2}} \log{\left (\frac{a}{b x^{2}} \right )}}{2} + b^{\frac{5}{2}} \log{\left (\sqrt{\frac{a}{b x^{2}} + 1} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(5/2)/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.226094, size = 227, normalized size = 2.77 \[ -\frac{1}{2} \, b^{\frac{5}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (45 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{2} b^{\frac{5}{2}} + 140 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{3} b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{4} b^{\frac{5}{2}} + 23 \, a^{5} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(5/2)/x^6,x, algorithm="giac")
[Out]