Optimal. Leaf size=138 \[ -\frac{8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt{b x^2+c x^4}}+\frac{2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt{b x^2+c x^4}}-\frac{7 b B-8 A c}{35 b^2 x^4 \sqrt{b x^2+c x^4}}-\frac{A}{7 b x^6 \sqrt{b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.507474, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{8 c^2 \left (b+2 c x^2\right ) (7 b B-8 A c)}{35 b^5 \sqrt{b x^2+c x^4}}+\frac{2 c (7 b B-8 A c)}{35 b^3 x^2 \sqrt{b x^2+c x^4}}-\frac{7 b B-8 A c}{35 b^2 x^4 \sqrt{b x^2+c x^4}}-\frac{A}{7 b x^6 \sqrt{b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 29.1504, size = 133, normalized size = 0.96 \[ - \frac{A}{7 b x^{6} \sqrt{b x^{2} + c x^{4}}} + \frac{8 A c - 7 B b}{35 b^{2} x^{4} \sqrt{b x^{2} + c x^{4}}} - \frac{2 c \left (8 A c - 7 B b\right )}{35 b^{3} x^{2} \sqrt{b x^{2} + c x^{4}}} + \frac{4 c^{2} \left (2 b + 4 c x^{2}\right ) \left (8 A c - 7 B b\right )}{35 b^{5} \sqrt{b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**5/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.116338, size = 108, normalized size = 0.78 \[ \frac{A \left (-5 b^4+8 b^3 c x^2-16 b^2 c^2 x^4+64 b c^3 x^6+128 c^4 x^8\right )-7 b B x^2 \left (b^3-2 b^2 c x^2+8 b c^2 x^4+16 c^3 x^6\right )}{35 b^5 x^6 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)),x]
[Out]
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Maple [A] time = 0.009, size = 118, normalized size = 0.9 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -128\,A{c}^{4}{x}^{8}+112\,Bb{c}^{3}{x}^{8}-64\,Ab{c}^{3}{x}^{6}+56\,B{b}^{2}{c}^{2}{x}^{6}+16\,A{b}^{2}{c}^{2}{x}^{4}-14\,B{b}^{3}c{x}^{4}-8\,A{b}^{3}c{x}^{2}+7\,B{b}^{4}{x}^{2}+5\,A{b}^{4} \right ) }{35\,{x}^{4}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^5/(c*x^4+b*x^2)^(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)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303843, size = 163, normalized size = 1.18 \[ -\frac{{\left (16 \,{\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} x^{8} + 8 \,{\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x^{6} + 5 \, A b^{4} - 2 \,{\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} x^{4} +{\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{35 \,{\left (b^{5} c x^{10} + b^{6} x^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{5} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**5/(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^(3/2)*x^5),x, algorithm="giac")
[Out]