Optimal. Leaf size=297 \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.226831, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b^5 (d x)^{25/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{25 d^{11} \left (a+b x^2\right )}+\frac{10 a b^4 (d x)^{21/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 d^9 \left (a+b x^2\right )}+\frac{20 a^2 b^3 (d x)^{17/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 d^7 \left (a+b x^2\right )}+\frac{2 a^5 (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac{10 a^4 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{20 a^3 b^2 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^5 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 29.1006, size = 238, normalized size = 0.8 \[ \frac{16384 a^{5} \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{348075 d \left (a + b x^{2}\right )} + \frac{4096 a^{4} \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{69615 d} + \frac{512 a^{3} \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{7735 d} + \frac{128 a^{2} \left (d x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{1785 d} + \frac{8 a \left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{105 d} + \frac{2 \left (d x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{25 d} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0463342, size = 88, normalized size = 0.3 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (69615 a^5+193375 a^4 b x^2+267750 a^3 b^2 x^4+204750 a^2 b^3 x^6+82875 a b^4 x^8+13923 b^5 x^{10}\right )}{348075 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Maple [A] time = 0.008, size = 83, normalized size = 0.3 \[{\frac{2\,x \left ( 13923\,{b}^{5}{x}^{10}+82875\,a{b}^{4}{x}^{8}+204750\,{a}^{2}{b}^{3}{x}^{6}+267750\,{a}^{3}{b}^{2}{x}^{4}+193375\,{a}^{4}b{x}^{2}+69615\,{a}^{5} \right ) }{348075\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
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)^(5/2),x)
[Out]
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Maxima [A] time = 0.721859, size = 198, normalized size = 0.67 \[ \frac{2}{525} \,{\left (21 \, b^{5} d^{\frac{3}{2}} x^{3} + 25 \, a b^{4} d^{\frac{3}{2}} x\right )} x^{\frac{19}{2}} + \frac{8}{357} \,{\left (17 \, a b^{4} d^{\frac{3}{2}} x^{3} + 21 \, a^{2} b^{3} d^{\frac{3}{2}} x\right )} x^{\frac{15}{2}} + \frac{12}{221} \,{\left (13 \, a^{2} b^{3} d^{\frac{3}{2}} x^{3} + 17 \, a^{3} b^{2} d^{\frac{3}{2}} x\right )} x^{\frac{11}{2}} + \frac{8}{117} \,{\left (9 \, a^{3} b^{2} d^{\frac{3}{2}} x^{3} + 13 \, a^{4} b d^{\frac{3}{2}} x\right )} x^{\frac{7}{2}} + \frac{2}{45} \,{\left (5 \, a^{4} b d^{\frac{3}{2}} x^{3} + 9 \, a^{5} d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272575, size = 95, normalized size = 0.32 \[ \frac{2}{348075} \,{\left (13923 \, b^{5} d x^{12} + 82875 \, a b^{4} d x^{10} + 204750 \, a^{2} b^{3} d x^{8} + 267750 \, a^{3} b^{2} d x^{6} + 193375 \, a^{4} b d x^{4} + 69615 \, a^{5} d x^{2}\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.269821, size = 190, normalized size = 0.64 \[ \frac{2}{25} \, \sqrt{d x} b^{5} d x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{21} \, \sqrt{d x} a b^{4} d x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{20}{17} \, \sqrt{d x} a^{2} b^{3} d x^{8}{\rm sign}\left (b x^{2} + a\right ) + \frac{20}{13} \, \sqrt{d x} a^{3} b^{2} d x^{6}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{9} \, \sqrt{d x} a^{4} b d x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a^{5} d x^{2}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2),x, algorithm="giac")
[Out]