Optimal. Leaf size=195 \[ \frac{6 a b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.156643, antiderivative size = 195, 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 \[ \frac{6 a b^2 (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^5 \left (a+b x^2\right )}+\frac{6 a^2 b (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac{2 b^3 (d x)^{15/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 d^7 \left (a+b x^2\right )}+\frac{2 a^3 (d x)^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.1618, size = 156, normalized size = 0.8 \[ \frac{256 a^{3} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1155 d \left (a + b x^{2}\right )} + \frac{64 a^{2} \left (d x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{385 d} + \frac{8 a \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{55 d} + \frac{2 \left (d x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{15 d} \]
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 = 0.0352154, size = 66, normalized size = 0.34 \[ \frac{2 \sqrt{d x} \sqrt{\left (a+b x^2\right )^2} \left (385 a^3 x+495 a^2 b x^3+315 a b^2 x^5+77 b^3 x^7\right )}{1155 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d*x]*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 61, normalized size = 0.3 \[{\frac{2\,x \left ( 77\,{b}^{3}{x}^{6}+315\,a{x}^{4}{b}^{2}+495\,{a}^{2}b{x}^{2}+385\,{a}^{3} \right ) }{1155\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}\sqrt{dx}} \]
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 = 0.709566, size = 112, normalized size = 0.57 \[ \frac{2}{165} \,{\left (11 \, b^{3} \sqrt{d} x^{3} + 15 \, a b^{2} \sqrt{d} x\right )} x^{\frac{9}{2}} + \frac{4}{77} \,{\left (7 \, a b^{2} \sqrt{d} x^{3} + 11 \, a^{2} b \sqrt{d} x\right )} x^{\frac{5}{2}} + \frac{2}{21} \,{\left (3 \, a^{2} b \sqrt{d} x^{3} + 7 \, a^{3} \sqrt{d} x\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*sqrt(d*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270706, size = 54, normalized size = 0.28 \[ \frac{2}{1155} \,{\left (77 \, b^{3} x^{7} + 315 \, a b^{2} x^{5} + 495 \, a^{2} b x^{3} + 385 \, a^{3} x\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*sqrt(d*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{d x} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]
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 = 0.266292, size = 127, normalized size = 0.65 \[ \frac{2 \,{\left (77 \, \sqrt{d x} b^{3} d x^{7}{\rm sign}\left (b x^{2} + a\right ) + 315 \, \sqrt{d x} a b^{2} d x^{5}{\rm sign}\left (b x^{2} + a\right ) + 495 \, \sqrt{d x} a^{2} b d x^{3}{\rm sign}\left (b x^{2} + a\right ) + 385 \, \sqrt{d x} a^{3} d x{\rm sign}\left (b x^{2} + a\right )\right )}}{1155 \, d} \]
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]