Optimal. Leaf size=120 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{3 \sqrt{a+b x^2} (5 A b-4 a B)}{8 a^3 x^2}-\frac{5 A b-4 a B}{4 a^2 x^2 \sqrt{a+b x^2}}-\frac{A}{4 a x^4 \sqrt{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.242107, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{3 \sqrt{a+b x^2} (5 A b-4 a B)}{8 a^3 x^2}-\frac{5 A b-4 a B}{4 a^2 x^2 \sqrt{a+b x^2}}-\frac{A}{4 a x^4 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 20.4755, size = 114, normalized size = 0.95 \[ - \frac{A}{4 a x^{4} \sqrt{a + b x^{2}}} - \frac{5 A b - 4 B a}{4 a^{2} x^{2} \sqrt{a + b x^{2}}} + \frac{3 \sqrt{a + b x^{2}} \left (5 A b - 4 B a\right )}{8 a^{3} x^{2}} - \frac{3 b \left (5 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**5/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.275984, size = 115, normalized size = 0.96 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 \left (A+2 B x^2\right )+a b x^2 \left (5 A-12 B x^2\right )+15 A b^2 x^4\right )}{x^4 \sqrt{a+b x^2}}+3 b (4 a B-5 A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+3 b \log (x) (5 A b-4 a B)}{8 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^5*(a + b*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.015, size = 153, normalized size = 1.3 \[ -{\frac{A}{4\,a{x}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Ab}{8\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,{b}^{2}A}{8\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{B}{2\,a{x}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{3\,Bb}{2\,{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{3\,Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^5/(b*x^2+a)^(3/2),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)/((b*x^2 + a)^(3/2)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245352, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} + 3 \,{\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} +{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{16 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )} \sqrt{a}}, -\frac{{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left ({\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} +{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{8 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 56.0351, size = 180, normalized size = 1.5 \[ A \left (- \frac{1}{4 a \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{5 \sqrt{b}}{8 a^{2} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{15 b^{\frac{3}{2}}}{8 a^{3} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{7}{2}}}\right ) + B \left (- \frac{1}{2 a \sqrt{b} x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 \sqrt{b}}{2 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2 a^{\frac{5}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**5/(b*x**2+a)**(3/2),x)
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GIAC/XCAS [A] time = 0.233742, size = 185, normalized size = 1.54 \[ -\frac{3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{3}} - \frac{B a b - A b^{2}}{\sqrt{b x^{2} + a} a^{3}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x^{2} + a} B a^{2} b - 7 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{2} + 9 \, \sqrt{b x^{2} + a} A a b^{2}}{8 \, a^{3} b^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*x^5),x, algorithm="giac")
[Out]