Optimal. Leaf size=131 \[ \frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]
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Rubi [A] time = 0.151779, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{2 a^6 (d x)^{3/2}}{3 d}+\frac{12 a^5 b (d x)^{7/2}}{7 d^3}+\frac{30 a^4 b^2 (d x)^{11/2}}{11 d^5}+\frac{8 a^3 b^3 (d x)^{15/2}}{3 d^7}+\frac{30 a^2 b^4 (d x)^{19/2}}{19 d^9}+\frac{12 a b^5 (d x)^{23/2}}{23 d^{11}}+\frac{2 b^6 (d x)^{27/2}}{27 d^{13}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 36.0538, size = 129, normalized size = 0.98 \[ \frac{2 a^{6} \left (d x\right )^{\frac{3}{2}}}{3 d} + \frac{12 a^{5} b \left (d x\right )^{\frac{7}{2}}}{7 d^{3}} + \frac{30 a^{4} b^{2} \left (d x\right )^{\frac{11}{2}}}{11 d^{5}} + \frac{8 a^{3} b^{3} \left (d x\right )^{\frac{15}{2}}}{3 d^{7}} + \frac{30 a^{2} b^{4} \left (d x\right )^{\frac{19}{2}}}{19 d^{9}} + \frac{12 a b^{5} \left (d x\right )^{\frac{23}{2}}}{23 d^{11}} + \frac{2 b^{6} \left (d x\right )^{\frac{27}{2}}}{27 d^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**3*(d*x)**(1/2),x)
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Mathematica [A] time = 0.0235069, size = 77, normalized size = 0.59 \[ \frac{2 x \sqrt{d x} \left (302841 a^6+778734 a^5 b x^2+1238895 a^4 b^2 x^4+1211364 a^3 b^3 x^6+717255 a^2 b^4 x^8+237006 a b^5 x^{10}+33649 b^6 x^{12}\right )}{908523} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]
[Out]
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Maple [A] time = 0.011, size = 74, normalized size = 0.6 \[{\frac{2\,x \left ( 33649\,{b}^{6}{x}^{12}+237006\,a{b}^{5}{x}^{10}+717255\,{a}^{2}{b}^{4}{x}^{8}+1211364\,{a}^{3}{b}^{3}{x}^{6}+1238895\,{a}^{4}{b}^{2}{x}^{4}+778734\,{a}^{5}b{x}^{2}+302841\,{a}^{6} \right ) }{908523}\sqrt{dx}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^4+2*a*b*x^2+a^2)^3*(d*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.703632, size = 142, normalized size = 1.08 \[ \frac{2 \,{\left (33649 \, \left (d x\right )^{\frac{27}{2}} b^{6} + 237006 \, \left (d x\right )^{\frac{23}{2}} a b^{5} d^{2} + 717255 \, \left (d x\right )^{\frac{19}{2}} a^{2} b^{4} d^{4} + 1211364 \, \left (d x\right )^{\frac{15}{2}} a^{3} b^{3} d^{6} + 1238895 \, \left (d x\right )^{\frac{11}{2}} a^{4} b^{2} d^{8} + 778734 \, \left (d x\right )^{\frac{7}{2}} a^{5} b d^{10} + 302841 \, \left (d x\right )^{\frac{3}{2}} a^{6} d^{12}\right )}}{908523 \, d^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*sqrt(d*x),x, algorithm="maxima")
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Fricas [A] time = 0.257206, size = 99, normalized size = 0.76 \[ \frac{2}{908523} \,{\left (33649 \, b^{6} x^{13} + 237006 \, a b^{5} x^{11} + 717255 \, a^{2} b^{4} x^{9} + 1211364 \, a^{3} b^{3} x^{7} + 1238895 \, a^{4} b^{2} x^{5} + 778734 \, a^{5} b x^{3} + 302841 \, a^{6} x\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*sqrt(d*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.01914, size = 131, normalized size = 1. \[ \frac{2 a^{6} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{12 a^{5} b \sqrt{d} x^{\frac{7}{2}}}{7} + \frac{30 a^{4} b^{2} \sqrt{d} x^{\frac{11}{2}}}{11} + \frac{8 a^{3} b^{3} \sqrt{d} x^{\frac{15}{2}}}{3} + \frac{30 a^{2} b^{4} \sqrt{d} x^{\frac{19}{2}}}{19} + \frac{12 a b^{5} \sqrt{d} x^{\frac{23}{2}}}{23} + \frac{2 b^{6} \sqrt{d} x^{\frac{27}{2}}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**4+2*a*b*x**2+a**2)**3*(d*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.264919, size = 153, normalized size = 1.17 \[ \frac{2 \,{\left (33649 \, \sqrt{d x} b^{6} d x^{13} + 237006 \, \sqrt{d x} a b^{5} d x^{11} + 717255 \, \sqrt{d x} a^{2} b^{4} d x^{9} + 1211364 \, \sqrt{d x} a^{3} b^{3} d x^{7} + 1238895 \, \sqrt{d x} a^{4} b^{2} d x^{5} + 778734 \, \sqrt{d x} a^{5} b d x^{3} + 302841 \, \sqrt{d x} a^{6} d x\right )}}{908523 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^3*sqrt(d*x),x, algorithm="giac")
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