Optimal. Leaf size=63 \[ \frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2} \]
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Rubi [A] time = 0.0850662, antiderivative size = 63, 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 \[ \frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.3509, size = 63, normalized size = 1. \[ \frac{2 a^{2} c x^{\frac{3}{2}}}{3} + \frac{2 a x^{\frac{7}{2}} \left (a d + 2 b c\right )}{7} + \frac{2 b^{2} d x^{\frac{15}{2}}}{15} + \frac{2 b x^{\frac{11}{2}} \left (2 a d + b c\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)*x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0318511, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 a^2 c+105 b x^4 (2 a d+b c)+165 a x^2 (a d+2 b c)+77 b^2 d x^6\right )}{1155} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Maple [A] time = 0.007, size = 56, normalized size = 0.9 \[{\frac{154\,{b}^{2}d{x}^{6}+420\,{x}^{4}abd+210\,{b}^{2}c{x}^{4}+330\,{x}^{2}{a}^{2}d+660\,abc{x}^{2}+770\,{a}^{2}c}{1155}{x}^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)*x^(1/2),x)
[Out]
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Maxima [A] time = 1.37178, size = 69, normalized size = 1.1 \[ \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225149, size = 73, normalized size = 1.16 \[ \frac{2}{1155} \,{\left (77 \, b^{2} d x^{7} + 105 \,{\left (b^{2} c + 2 \, a b d\right )} x^{5} + 385 \, a^{2} c x + 165 \,{\left (2 \, a b c + a^{2} d\right )} x^{3}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.85478, size = 66, normalized size = 1.05 \[ \frac{2 a^{2} c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d x^{\frac{15}{2}}}{15} + \frac{2 x^{\frac{11}{2}} \left (2 a b d + b^{2} c\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (a^{2} d + 2 a b c\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)*x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.225723, size = 72, normalized size = 1.14 \[ \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c x^{\frac{11}{2}} + \frac{4}{11} \, a b d x^{\frac{11}{2}} + \frac{4}{7} \, a b c x^{\frac{7}{2}} + \frac{2}{7} \, a^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*sqrt(x),x, algorithm="giac")
[Out]