Optimal. Leaf size=193 \[ \frac{2 a b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac{2 a^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.159151, antiderivative size = 193, 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 \[ \frac{2 a b^2 (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 d^7 \left (a+b x^2\right )}+\frac{2 a^3 \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/Sqrt[d*x],x]
[Out]
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Rubi in Sympy [A] time = 18.2877, size = 156, normalized size = 0.81 \[ \frac{256 a^{3} \sqrt{d x} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{195 d \left (a + b x^{2}\right )} + \frac{64 a^{2} \sqrt{d x} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{195 d} + \frac{8 a \sqrt{d x} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{39 d} + \frac{2 \sqrt{d x} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{13 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0322495, size = 66, normalized size = 0.34 \[ \frac{2 \sqrt{\left (a+b x^2\right )^2} \left (195 a^3 x+117 a^2 b x^3+65 a b^2 x^5+15 b^3 x^7\right )}{195 \sqrt{d x} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2)/Sqrt[d*x],x]
[Out]
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Maple [A] time = 0.009, size = 61, normalized size = 0.3 \[{\frac{2\, \left ( 15\,{b}^{3}{x}^{6}+65\,a{x}^{4}{b}^{2}+117\,{a}^{2}b{x}^{2}+195\,{a}^{3} \right ) x}{195\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{dx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^4+2*a*b*x^2+a^2)^(3/2)/(d*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.711173, size = 117, normalized size = 0.61 \[ \frac{2 \,{\left (5 \,{\left (9 \, b^{3} \sqrt{d} x^{3} + 13 \, a b^{2} \sqrt{d} x\right )} x^{\frac{7}{2}} + 26 \,{\left (5 \, a b^{2} \sqrt{d} x^{3} + 9 \, a^{2} b \sqrt{d} x\right )} x^{\frac{3}{2}} + \frac{117 \,{\left (a^{2} b \sqrt{d} x^{3} + 5 \, a^{3} \sqrt{d} x\right )}}{\sqrt{x}}\right )}}{585 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/sqrt(d*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271389, size = 57, normalized size = 0.3 \[ \frac{2 \,{\left (15 \, b^{3} x^{6} + 65 \, a b^{2} x^{4} + 117 \, a^{2} b x^{2} + 195 \, a^{3}\right )} \sqrt{d x}}{195 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/sqrt(d*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{\sqrt{d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**4+2*a*b*x**2+a**2)**(3/2)/(d*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.267712, size = 120, normalized size = 0.62 \[ \frac{2 \,{\left (15 \, \sqrt{d x} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 65 \, \sqrt{d x} a b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 117 \, \sqrt{d x} a^{2} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 195 \, \sqrt{d x} a^{3}{\rm sign}\left (b x^{2} + a\right )\right )}}{195 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)/sqrt(d*x),x, algorithm="giac")
[Out]