Optimal. Leaf size=63 \[ \frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} b x^{15/2} (2 a d+b c)+\frac{2}{11} a x^{11/2} (a d+2 b c)+\frac{2}{19} b^2 d x^{19/2} \]
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Rubi [A] time = 0.0926184, 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 \[ \frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} b x^{15/2} (2 a d+b c)+\frac{2}{11} a x^{11/2} (a d+2 b c)+\frac{2}{19} b^2 d x^{19/2} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.0796, size = 63, normalized size = 1. \[ \frac{2 a^{2} c x^{\frac{7}{2}}}{7} + \frac{2 a x^{\frac{11}{2}} \left (a d + 2 b c\right )}{11} + \frac{2 b^{2} d x^{\frac{19}{2}}}{19} + \frac{2 b x^{\frac{15}{2}} \left (2 a d + b c\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0298307, size = 63, normalized size = 1. \[ \frac{2}{7} a^2 c x^{7/2}+\frac{2}{15} b x^{15/2} (2 a d+b c)+\frac{2}{11} a x^{11/2} (a d+2 b c)+\frac{2}{19} b^2 d x^{19/2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Maple [A] time = 0.008, size = 56, normalized size = 0.9 \[{\frac{2310\,{b}^{2}d{x}^{6}+5852\,{x}^{4}abd+2926\,{b}^{2}c{x}^{4}+3990\,{x}^{2}{a}^{2}d+7980\,abc{x}^{2}+6270\,{a}^{2}c}{21945}{x}^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(b*x^2+a)^2*(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.35049, size = 69, normalized size = 1.1 \[ \frac{2}{19} \, b^{2} d x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} c x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{11}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217673, size = 76, normalized size = 1.21 \[ \frac{2}{21945} \,{\left (1155 \, b^{2} d x^{9} + 1463 \,{\left (b^{2} c + 2 \, a b d\right )} x^{7} + 3135 \, a^{2} c x^{3} + 1995 \,{\left (2 \, a b c + a^{2} d\right )} x^{5}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 37.752, size = 80, normalized size = 1.27 \[ \frac{2 a^{2} c x^{\frac{7}{2}}}{7} + \frac{2 a^{2} d x^{\frac{11}{2}}}{11} + \frac{4 a b c x^{\frac{11}{2}}}{11} + \frac{4 a b d x^{\frac{15}{2}}}{15} + \frac{2 b^{2} c x^{\frac{15}{2}}}{15} + \frac{2 b^{2} d x^{\frac{19}{2}}}{19} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.234895, size = 72, normalized size = 1.14 \[ \frac{2}{19} \, b^{2} d x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} c x^{\frac{15}{2}} + \frac{4}{15} \, a b d x^{\frac{15}{2}} + \frac{4}{11} \, a b c x^{\frac{11}{2}} + \frac{2}{11} \, a^{2} d x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} c x^{\frac{7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(5/2),x, algorithm="giac")
[Out]