Optimal. Leaf size=63 \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]
[Out]
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Rubi [A] time = 0.0877595, 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 \[ \frac{2}{5} a^2 c x^{5/2}+\frac{2}{13} b x^{13/2} (2 a d+b c)+\frac{2}{9} a x^{9/2} (a d+2 b c)+\frac{2}{17} b^2 d x^{17/2} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.0836, size = 63, normalized size = 1. \[ \frac{2 a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 a x^{\frac{9}{2}} \left (a d + 2 b c\right )}{9} + \frac{2 b^{2} d x^{\frac{17}{2}}}{17} + \frac{2 b x^{\frac{13}{2}} \left (2 a d + b c\right )}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.0335752, size = 53, normalized size = 0.84 \[ \frac{2 x^{5/2} \left (1989 a^2 c+765 b x^4 (2 a d+b c)+1105 a x^2 (a d+2 b c)+585 b^2 d x^6\right )}{9945} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*(a + b*x^2)^2*(c + d*x^2),x]
[Out]
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Maple [A] time = 0.009, size = 56, normalized size = 0.9 \[{\frac{1170\,{b}^{2}d{x}^{6}+3060\,{x}^{4}abd+1530\,{b}^{2}c{x}^{4}+2210\,{x}^{2}{a}^{2}d+4420\,abc{x}^{2}+3978\,{a}^{2}c}{9945}{x}^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(b*x^2+a)^2*(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.35406, size = 69, normalized size = 1.1 \[ \frac{2}{17} \, b^{2} d x^{\frac{17}{2}} + \frac{2}{13} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{9}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20833, size = 76, normalized size = 1.21 \[ \frac{2}{9945} \,{\left (585 \, b^{2} d x^{8} + 765 \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + 1989 \, a^{2} c x^{2} + 1105 \,{\left (2 \, a b c + a^{2} d\right )} x^{4}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 20.4412, size = 80, normalized size = 1.27 \[ \frac{2 a^{2} c x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d x^{\frac{9}{2}}}{9} + \frac{4 a b c x^{\frac{9}{2}}}{9} + \frac{4 a b d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d x^{\frac{17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(b*x**2+a)**2*(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.228096, size = 72, normalized size = 1.14 \[ \frac{2}{17} \, b^{2} d x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c x^{\frac{13}{2}} + \frac{4}{13} \, a b d x^{\frac{13}{2}} + \frac{4}{9} \, a b c x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)*x^(3/2),x, algorithm="giac")
[Out]