Optimal. Leaf size=61 \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
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Rubi [A] time = 0.0878789, 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 \[ 2 a^2 A \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a B+A b)+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{13} b^2 B x^{13/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 12.6494, size = 61, normalized size = 1. \[ 2 A a^{2} \sqrt{x} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{2 a x^{\frac{5}{2}} \left (2 A b + B a\right )}{5} + \frac{2 b x^{\frac{9}{2}} \left (A b + 2 B a\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0328795, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 A+65 b x^4 (2 a B+A b)+117 a x^2 (a B+2 A b)+45 b^2 B x^6\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(A + B*x^2))/Sqrt[x],x]
[Out]
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Maple [A] time = 0.009, size = 56, normalized size = 0.9 \[{\frac{90\,{b}^{2}B{x}^{6}+130\,A{b}^{2}{x}^{4}+260\,{x}^{4}abB+468\,aAb{x}^{2}+234\,{x}^{2}{a}^{2}B+1170\,{a}^{2}A}{585}\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(B*x^2+A)/x^(1/2),x)
[Out]
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Maxima [A] time = 1.33669, size = 69, normalized size = 1.13 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{9}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217327, size = 72, normalized size = 1.18 \[ \frac{2}{585} \,{\left (45 \, B b^{2} x^{6} + 65 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 585 \, A a^{2} + 117 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.71726, size = 78, normalized size = 1.28 \[ 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{5}{2}}}{5} + \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5} + \frac{4 B a b x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(B*x**2+A)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209922, size = 72, normalized size = 1.18 \[ \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{9} \, B a b x^{\frac{9}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + 2 \, A a^{2} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^2/sqrt(x),x, algorithm="giac")
[Out]