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

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac{\left (a+b x^2\right )^{7/2}}{7 b^3}-\frac{2 a \left (a+b x^2\right )^{5/2}}{5 b^3} \]

[Out]

(a^2*(a + b*x^2)^(3/2))/(3*b^3) - (2*a*(a + b*x^2)^(5/2))/(5*b^3) + (a + b*x^2)^
(7/2)/(7*b^3)

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Rubi [A]  time = 0.0954688, antiderivative size = 59, 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 \[ \frac{a^2 \left (a+b x^2\right )^{3/2}}{3 b^3}+\frac{\left (a+b x^2\right )^{7/2}}{7 b^3}-\frac{2 a \left (a+b x^2\right )^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*x^2],x]

[Out]

(a^2*(a + b*x^2)^(3/2))/(3*b^3) - (2*a*(a + b*x^2)^(5/2))/(5*b^3) + (a + b*x^2)^
(7/2)/(7*b^3)

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Rubi in Sympy [A]  time = 11.843, size = 51, normalized size = 0.86 \[ \frac{a^{2} \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{3}} - \frac{2 a \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{3}} \]

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*(a + b*x**2)**(3/2)/(3*b**3) - 2*a*(a + b*x**2)**(5/2)/(5*b**3) + (a + b*x*
*2)**(7/2)/(7*b**3)

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Mathematica [A]  time = 0.0222199, size = 50, normalized size = 0.85 \[ \frac{\sqrt{a+b x^2} \left (8 a^3-4 a^2 b x^2+3 a b^2 x^4+15 b^3 x^6\right )}{105 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(8*a^3 - 4*a^2*b*x^2 + 3*a*b^2*x^4 + 15*b^3*x^6))/(105*b^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{15\,{b}^{2}{x}^{4}-12\,ab{x}^{2}+8\,{a}^{2}}{105\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(1/2),x)

[Out]

1/105*(b*x^2+a)^(3/2)*(15*b^2*x^4-12*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(sqrt(b*x^2 + a)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22806, size = 62, normalized size = 1.05 \[ \frac{{\left (15 \, b^{3} x^{6} + 3 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} + 8 \, a^{3}\right )} \sqrt{b x^{2} + a}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*x^5,x, algorithm="fricas")

[Out]

1/105*(15*b^3*x^6 + 3*a*b^2*x^4 - 4*a^2*b*x^2 + 8*a^3)*sqrt(b*x^2 + a)/b^3

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Sympy [A]  time = 1.87953, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases} \]

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**3*sqrt(a + b*x**2)/(105*b**3) - 4*a**2*x**2*sqrt(a + b*x**2)/(10
5*b**2) + a*x**4*sqrt(a + b*x**2)/(35*b) + x**6*sqrt(a + b*x**2)/7, Ne(b, 0)), (
sqrt(a)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.207517, size = 58, normalized size = 0.98 \[ \frac{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}}{105 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*x^5,x, algorithm="giac")

[Out]

1/105*(15*(b*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)
/b^3