Optimal. Leaf size=86 \[ -\frac{A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{3/2}}{3 x^3} \]
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Rubi [A] time = 0.110976, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{A \left (a+b x^2\right )^{5/2}}{5 a x^5}+b^{3/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]
[Out]
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Rubi in Sympy [A] time = 13.8667, size = 75, normalized size = 0.87 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} + B b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{B b \sqrt{a + b x^{2}}}{x} - \frac{B \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**6,x)
[Out]
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Mathematica [A] time = 0.10902, size = 88, normalized size = 1.02 \[ \sqrt{a+b x^2} \left (\frac{-5 a B-6 A b}{15 x^3}-\frac{b (20 a B+3 A b)}{15 a x}-\frac{a A}{5 x^5}\right )+b^{3/2} B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/x^6,x]
[Out]
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Maple [A] time = 0.015, size = 115, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B}{3\,a{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Bb}{3\,{a}^{2}x} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{b}^{2}Bx}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}Bx}{a}\sqrt{b{x}^{2}+a}}+B{b}^{{\frac{3}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^(3/2)*(B*x^2+A)/x^6,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^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231343, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a b^{\frac{3}{2}} x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{30 \, a x^{5}}, \frac{15 \, B a \sqrt{-b} b x^{5} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left ({\left (20 \, B a b + 3 \, A b^{2}\right )} x^{4} + 3 \, A a^{2} +{\left (5 \, B a^{2} + 6 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, a x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.7144, size = 184, normalized size = 2.14 \[ - \frac{A a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{2 A b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{2}} - \frac{A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 a} - \frac{B \sqrt{a} b}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{3 x^{2}} - \frac{B b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{3} + B b^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{B b^{2} x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(3/2)*(B*x**2+A)/x**6,x)
[Out]
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GIAC/XCAS [A] time = 0.250725, size = 319, normalized size = 3.71 \[ -\frac{1}{2} \, B b^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} + 15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} + 110 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} + 30 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 70 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} + 20 \, B a^{5} b^{\frac{3}{2}} + 3 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/x^6,x, algorithm="giac")
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