Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
[Out]
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Rubi [A] time = 0.0890327, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 a^2 A}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a B+A b)+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{11} b^2 B x^{11/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(A + B*x^2))/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.7595, size = 61, normalized size = 1. \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{2 a x^{\frac{3}{2}} \left (2 A b + B a\right )}{3} + \frac{2 b x^{\frac{7}{2}} \left (A b + 2 B a\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0336501, size = 53, normalized size = 0.87 \[ \frac{2 \left (-231 a^2 A+33 b x^4 (2 a B+A b)+77 a x^2 (a B+2 A b)+21 b^2 B x^6\right )}{231 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(A + B*x^2))/x^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 56, normalized size = 0.9 \[ -{\frac{-42\,{b}^{2}B{x}^{6}-66\,A{b}^{2}{x}^{4}-132\,{x}^{4}abB-308\,aAb{x}^{2}-154\,{x}^{2}{a}^{2}B+462\,{a}^{2}A}{231}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(B*x^2+A)/x^(3/2),x)
[Out]
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Maxima [A] time = 1.34332, size = 69, normalized size = 1.13 \[ \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209474, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (21 \, B b^{2} x^{6} + 33 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 231 \, A a^{2} + 77 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{231 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.57423, size = 78, normalized size = 1.28 \[ - \frac{2 A a^{2}}{\sqrt{x}} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(B*x**2+A)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221667, size = 72, normalized size = 1.18 \[ \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{2} x^{\frac{3}{2}} + \frac{4}{3} \, A a b x^{\frac{3}{2}} - \frac{2 \, A a^{2}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/x^(3/2),x, algorithm="giac")
[Out]