3.485 \(\int \frac{x^5}{\sqrt{a+b x^2}} \, dx\)

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]

(a^2*Sqrt[a + b*x^2])/b^3 - (2*a*(a + b*x^2)^(3/2))/(3*b^3) + (a + b*x^2)^(5/2)/
(5*b^3)

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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]

(a^2*Sqrt[a + b*x^2])/b^3 - (2*a*(a + b*x^2)^(3/2))/(3*b^3) + (a + b*x^2)^(5/2)/
(5*b^3)

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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]

a**2*sqrt(a + b*x**2)/b**3 - 2*a*(a + b*x**2)**(3/2)/(3*b**3) + (a + b*x**2)**(5
/2)/(5*b**3)

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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]

(Sqrt[a + b*x^2]*(8*a^2 - 4*a*b*x^2 + 3*b^2*x^4))/(15*b^3)

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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]

1/15*(b*x^2+a)^(1/2)*(3*b^2*x^4-4*a*b*x^2+8*a^2)/b^3

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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]

Exception raised: ValueError

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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]

1/15*(3*b^2*x^4 - 4*a*b*x^2 + 8*a^2)*sqrt(b*x^2 + a)/b^3

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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]

Piecewise((8*a**2*sqrt(a + b*x**2)/(15*b**3) - 4*a*x**2*sqrt(a + b*x**2)/(15*b**
2) + x**4*sqrt(a + b*x**2)/(5*b), Ne(b, 0)), (x**6/(6*sqrt(a)), True))

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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]

1/15*(3*(b*x^2 + a)^(5/2) - 10*(b*x^2 + a)^(3/2)*a + 15*sqrt(b*x^2 + a)*a^2)/b^3