Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]
[Out]
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Rubi [A] time = 0.0875, 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 \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2))/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.1622, size = 61, normalized size = 1. \[ - \frac{2 a^{2} c}{\sqrt{x}} + \frac{2 a x^{\frac{3}{2}} \left (a d + 2 b c\right )}{3} + \frac{2 b^{2} d x^{\frac{11}{2}}}{11} + \frac{2 b x^{\frac{7}{2}} \left (2 a d + b c\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0357795, size = 53, normalized size = 0.87 \[ \frac{2 \left (-231 a^2 c+33 b x^4 (2 a d+b c)+77 a x^2 (a d+2 b c)+21 b^2 d x^6\right )}{231 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2))/x^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 56, normalized size = 0.9 \[ -{\frac{-42\,{b}^{2}d{x}^{6}-132\,{x}^{4}abd-66\,{b}^{2}c{x}^{4}-154\,{x}^{2}{a}^{2}d-308\,abc{x}^{2}+462\,{a}^{2}c}{231}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)/x^(3/2),x)
[Out]
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Maxima [A] time = 1.37755, size = 69, normalized size = 1.13 \[ \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{7}{2}} - \frac{2 \, a^{2} c}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214484, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (21 \, b^{2} d x^{6} + 33 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 231 \, a^{2} c + 77 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{231 \, \sqrt{x}} \]
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 = 8.5719, size = 78, normalized size = 1.28 \[ - \frac{2 a^{2} c}{\sqrt{x}} + \frac{2 a^{2} d x^{\frac{3}{2}}}{3} + \frac{4 a b c x^{\frac{3}{2}}}{3} + \frac{4 a b d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d x^{\frac{11}{2}}}{11} \]
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)
[Out]
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GIAC/XCAS [A] time = 0.229528, size = 72, normalized size = 1.18 \[ \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{7}{2}} + \frac{4}{7} \, a b d x^{\frac{7}{2}} + \frac{4}{3} \, a b c x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d x^{\frac{3}{2}} - \frac{2 \, a^{2} c}{\sqrt{x}} \]
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="giac")
[Out]