Optimal. Leaf size=95 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
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Rubi [A] time = 0.132047, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2)^2)/Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 22.323, size = 100, normalized size = 1.05 \[ 2 a^{2} c^{2} \sqrt{x} + \frac{4 a c x^{\frac{5}{2}} \left (a d + b c\right )}{5} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} + \frac{4 b d x^{\frac{13}{2}} \left (a d + b c\right )}{13} + x^{\frac{9}{2}} \left (\frac{2 a^{2} d^{2}}{9} + \frac{8 a b c d}{9} + \frac{2 b^{2} c^{2}}{9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.050901, size = 95, normalized size = 1. \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/Sqrt[x],x]
[Out]
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Maple [A] time = 0.009, size = 97, normalized size = 1. \[{\frac{1170\,{b}^{2}{d}^{2}{x}^{8}+3060\,{x}^{6}ab{d}^{2}+3060\,{x}^{6}{b}^{2}cd+2210\,{x}^{4}{a}^{2}{d}^{2}+8840\,{x}^{4}abcd+2210\,{x}^{4}{b}^{2}{c}^{2}+7956\,{x}^{2}{a}^{2}cd+7956\,a{c}^{2}b{x}^{2}+19890\,{a}^{2}{c}^{2}}{9945}\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^2/x^(1/2),x)
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Maxima [A] time = 1.34102, size = 115, normalized size = 1.21 \[ \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{9}{2}} + 2 \, a^{2} c^{2} \sqrt{x} + \frac{4}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.22173, size = 117, normalized size = 1.23 \[ \frac{2}{9945} \,{\left (585 \, b^{2} d^{2} x^{8} + 1530 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 1105 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + 9945 \, a^{2} c^{2} + 3978 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.235, size = 134, normalized size = 1.41 \[ 2 a^{2} c^{2} \sqrt{x} + \frac{4 a^{2} c d x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d^{2} x^{\frac{9}{2}}}{9} + \frac{4 a b c^{2} x^{\frac{5}{2}}}{5} + \frac{8 a b c d x^{\frac{9}{2}}}{9} + \frac{4 a b d^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 b^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23393, size = 127, normalized size = 1.34 \[ \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b^{2} c d x^{\frac{13}{2}} + \frac{4}{13} \, a b d^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c^{2} x^{\frac{9}{2}} + \frac{8}{9} \, a b c d x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b c^{2} x^{\frac{5}{2}} + \frac{4}{5} \, a^{2} c d x^{\frac{5}{2}} + 2 \, a^{2} c^{2} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^2/sqrt(x),x, algorithm="giac")
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