Optimal. Leaf size=56 \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.0927026, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{5/2}}{5 b^3}-\frac{2 a \left (a+b x^2\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^5/Sqrt[a + b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 11.7623, size = 49, normalized size = 0.88 \[ \frac{a^{2} \sqrt{a + b x^{2}}}{b^{3}} - \frac{2 a \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0268568, size = 39, normalized size = 0.7 \[ \frac{\sqrt{a+b x^2} \left (8 a^2-4 a b x^2+3 b^2 x^4\right )}{15 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/Sqrt[a + b*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 36, normalized size = 0.6 \[{\frac{3\,{b}^{2}{x}^{4}-4\,ab{x}^{2}+8\,{a}^{2}}{15\,{b}^{3}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234902, size = 47, normalized size = 0.84 \[ \frac{{\left (3 \, b^{2} x^{4} - 4 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.67692, size = 68, normalized size = 1.21 \[ \begin{cases} \frac{8 a^{2} \sqrt{a + b x^{2}}}{15 b^{3}} - \frac{4 a x^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{x^{4} \sqrt{a + b x^{2}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.202612, size = 58, normalized size = 1.04 \[ \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 10 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b x^{2} + a} a^{2}}{15 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]