Optimal. Leaf size=93 \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.0834996, antiderivative size = 93, 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 \[ \frac{2 b (d x)^{9/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 23.8377, size = 75, normalized size = 0.81 \[ \frac{8 a \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{45 d \left (a + b x^{2}\right )} + \frac{2 \left (d x\right )^{\frac{5}{2}} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{9 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(3/2)*((b*x**2+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0231847, size = 44, normalized size = 0.47 \[ \frac{2 x (d x)^{3/2} \sqrt{\left (a+b x^2\right )^2} \left (9 a+5 b x^2\right )}{45 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]
[Out]
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Maple [A] time = 0.004, size = 39, normalized size = 0.4 \[{\frac{2\, \left ( 5\,b{x}^{2}+9\,a \right ) x}{45\,b{x}^{2}+45\,a} \left ( dx \right ) ^{{\frac{3}{2}}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(3/2)*((b*x^2+a)^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.70954, size = 30, normalized size = 0.32 \[ \frac{2}{45} \,{\left (5 \, b d^{\frac{3}{2}} x^{3} + 9 \, a d^{\frac{3}{2}} x\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)*(d*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25838, size = 30, normalized size = 0.32 \[ \frac{2}{45} \,{\left (5 \, b d x^{4} + 9 \, a d x^{2}\right )} \sqrt{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^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*x**2+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265489, size = 55, normalized size = 0.59 \[ \frac{2}{9} \, \sqrt{d x} b d x^{4}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{5} \, \sqrt{d x} a d x^{2}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)^2)*(d*x)^(3/2),x, algorithm="giac")
[Out]