Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]
[Out]
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Rubi [A] time = 0.0382421, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{2 a^2 (d x)^{5/2}}{5 d}+\frac{4 a b (d x)^{9/2}}{9 d^3}+\frac{2 b^2 (d x)^{13/2}}{13 d^5} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]
[Out]
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Rubi in Sympy [A] time = 15.5899, size = 48, normalized size = 0.94 \[ \frac{2 a^{2} \left (d x\right )^{\frac{5}{2}}}{5 d} + \frac{4 a b \left (d x\right )^{\frac{9}{2}}}{9 d^{3}} + \frac{2 b^{2} \left (d x\right )^{\frac{13}{2}}}{13 d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2),x)
[Out]
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Mathematica [A] time = 0.0182458, size = 33, normalized size = 0.65 \[ \frac{2}{585} x (d x)^{3/2} \left (117 a^2+130 a b x^2+45 b^2 x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4),x]
[Out]
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Maple [A] time = 0.01, size = 30, normalized size = 0.6 \[{\frac{2\,x \left ( 45\,{b}^{2}{x}^{4}+130\,ab{x}^{2}+117\,{a}^{2} \right ) }{585} \left ( dx \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(3/2)*(b^2*x^4+2*a*b*x^2+a^2),x)
[Out]
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Maxima [A] time = 0.681239, size = 55, normalized size = 1.08 \[ \frac{2 \,{\left (45 \, \left (d x\right )^{\frac{13}{2}} b^{2} + 130 \, \left (d x\right )^{\frac{9}{2}} a b d^{2} + 117 \, \left (d x\right )^{\frac{5}{2}} a^{2} d^{4}\right )}}{585 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.256475, size = 46, normalized size = 0.9 \[ \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 130 \, a b d x^{4} + 117 \, a^{2} d x^{2}\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.81795, size = 49, normalized size = 0.96 \[ \frac{2 a^{2} d^{\frac{3}{2}} x^{\frac{5}{2}}}{5} + \frac{4 a b d^{\frac{3}{2}} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{\frac{3}{2}} x^{\frac{13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)**(3/2)*(b**2*x**4+2*a*b*x**2+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.26309, size = 57, normalized size = 1.12 \[ \frac{2}{13} \, \sqrt{d x} b^{2} d x^{6} + \frac{4}{9} \, \sqrt{d x} a b d x^{4} + \frac{2}{5} \, \sqrt{d x} a^{2} d x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(3/2),x, algorithm="giac")
[Out]