Optimal. Leaf size=113 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A b-2 a B}{2 a^3 \sqrt{a+b x^2}}-\frac{5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.240657, antiderivative size = 113, 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 \[ \frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{5 A b-2 a B}{2 a^3 \sqrt{a+b x^2}}-\frac{5 A b-2 a B}{6 a^2 \left (a+b x^2\right )^{3/2}}-\frac{A}{2 a x^2 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 21.3942, size = 99, normalized size = 0.88 \[ - \frac{A}{2 a x^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\frac{5 A b}{2} - B a}{3 a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{\frac{5 A b}{2} - B a}{a^{3} \sqrt{a + b x^{2}}} + \frac{\left (\frac{5 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**3/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [A] time = 0.318537, size = 114, normalized size = 1.01 \[ \frac{\frac{\sqrt{a} \left (-3 a^2 A+8 a^2 B x^2-20 a A b x^2+6 a b B x^4-15 A b^2 x^4\right )}{x^2 \left (a+b x^2\right )^{3/2}}+3 (5 A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\log (x) (6 a B-15 A b)}{6 a^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^3*(a + b*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.013, size = 140, normalized size = 1.2 \[ -{\frac{A}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{6\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{5\,Ab}{2\,{a}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{5\,Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}+{\frac{B}{3\,a} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{B}{{a}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^3/(b*x^2+a)^(5/2),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)^(5/2)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251177, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a} - 3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right )}{12 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{a}}, \frac{{\left (3 \,{\left (2 \, B a b - 5 \, A b^{2}\right )} x^{4} - 3 \, A a^{2} + 4 \,{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a} - 3 \,{\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 2 \,{\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right )}{6 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )} \sqrt{-a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 119.557, size = 1608, normalized size = 14.23 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**3/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.235652, size = 136, normalized size = 1.2 \[ \frac{{\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a^{3}} + \frac{3 \,{\left (b x^{2} + a\right )} B a + B a^{2} - 6 \,{\left (b x^{2} + a\right )} A b - A a b}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}} - \frac{\sqrt{b x^{2} + a} A}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*x^3),x, algorithm="giac")
[Out]