Optimal. Leaf size=51 \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0391867, 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 \[ \frac{2 a^2 (d x)^{3/2}}{3 d}+\frac{4 a b (d x)^{7/2}}{7 d^3}+\frac{2 b^2 (d x)^{11/2}}{11 d^5} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.5266, size = 48, normalized size = 0.94 \[ \frac{2 a^{2} \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{4 a b \left (d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{2 b^{2} \left (d x\right )^{\frac{11}{2}}}{11 d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)*(d*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0133564, size = 33, normalized size = 0.65 \[ \frac{2}{231} x \sqrt{d x} \left (77 a^2+66 a b x^2+21 b^2 x^4\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 30, normalized size = 0.6 \[{\frac{2\,x \left ( 21\,{b}^{2}{x}^{4}+66\,ab{x}^{2}+77\,{a}^{2} \right ) }{231}\sqrt{dx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^4+2*a*b*x^2+a^2)*(d*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.691685, size = 55, normalized size = 1.08 \[ \frac{2 \,{\left (21 \, \left (d x\right )^{\frac{11}{2}} b^{2} + 66 \, \left (d x\right )^{\frac{7}{2}} a b d^{2} + 77 \, \left (d x\right )^{\frac{3}{2}} a^{2} d^{4}\right )}}{231 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.256995, size = 39, normalized size = 0.76 \[ \frac{2}{231} \,{\left (21 \, b^{2} x^{5} + 66 \, a b x^{3} + 77 \, a^{2} x\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.25467, size = 49, normalized size = 0.96 \[ \frac{2 a^{2} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{4 a b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} \]
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)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.261993, size = 61, normalized size = 1.2 \[ \frac{2 \,{\left (21 \, \sqrt{d x} b^{2} d x^{5} + 66 \, \sqrt{d x} a b d x^{3} + 77 \, \sqrt{d x} a^{2} d x\right )}}{231 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x),x, algorithm="giac")
[Out]