Optimal. Leaf size=115 \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]
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Rubi [A] time = 0.23321, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\left (a+b x^2\right )^{3/2} (4 a B+A b)}{8 a x^2}+\frac{3 b \sqrt{a+b x^2} (4 a B+A b)}{8 a}-\frac{3 b (4 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{A \left (a+b x^2\right )^{5/2}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]
[Out]
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Rubi in Sympy [A] time = 19.2761, size = 104, normalized size = 0.9 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{4 a x^{4}} + \frac{3 b \sqrt{a + b x^{2}} \left (A b + 4 B a\right )}{8 a} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b + 4 B a\right )}{8 a x^{2}} - \frac{3 b \left (A b + 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**5,x)
[Out]
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Mathematica [A] time = 0.222913, size = 100, normalized size = 0.87 \[ -\frac{3 b (4 a B+A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{8 \sqrt{a}}+\sqrt{a+b x^2} \left (\frac{-4 a B-5 A b}{8 x^2}-\frac{a A}{4 x^4}+b B\right )+\frac{3 b \log (x) (4 a B+A b)}{8 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^5,x]
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Maple [A] time = 0.014, size = 184, normalized size = 1.6 \[ -{\frac{A}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Ab}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}A}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,{b}^{2}A}{8\,a}\sqrt{b{x}^{2}+a}}-{\frac{B}{2\,a{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{Bb}{2\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,Bb}{2}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{3\,Bb}{2}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)*(B*x^2+A)/x^5,x)
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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 = 0.22874, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{4} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (8 \, B b x^{4} -{\left (4 \, B a + 5 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{a}}{16 \, \sqrt{a} x^{4}}, -\frac{3 \,{\left (4 \, B a b + A b^{2}\right )} x^{4} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, B b x^{4} -{\left (4 \, B a + 5 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{8 \, \sqrt{-a} x^{4}}\right ] \]
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 = 89.0397, size = 216, normalized size = 1.88 \[ - \frac{A a^{2}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A a \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{A b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 \sqrt{a}} - \frac{3 B \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} + \frac{B a \sqrt{b}}{x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{3}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**5,x)
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GIAC/XCAS [A] time = 0.230297, size = 177, normalized size = 1.54 \[ \frac{8 \, \sqrt{b x^{2} + a} B b^{2} + \frac{3 \,{\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{4 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{2} + a} B a^{2} b^{2} + 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A b^{3} - 3 \, \sqrt{b x^{2} + a} A a b^{3}}{b^{2} x^{4}}}{8 \, b} \]
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")
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