3.353 \(\int x^7 \sqrt{a+b x^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^4}-\frac{3 a \left (a+b x^2\right )^{7/2}}{7 b^4} \]

[Out]

-(a^3*(a + b*x^2)^(3/2))/(3*b^4) + (3*a^2*(a + b*x^2)^(5/2))/(5*b^4) - (3*a*(a +
b*x^2)^(7/2))/(7*b^4) + (a + b*x^2)^(9/2)/(9*b^4)

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Rubi [A]  time = 0.122207, antiderivative size = 80, 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 \[ -\frac{a^3 \left (a+b x^2\right )^{3/2}}{3 b^4}+\frac{3 a^2 \left (a+b x^2\right )^{5/2}}{5 b^4}+\frac{\left (a+b x^2\right )^{9/2}}{9 b^4}-\frac{3 a \left (a+b x^2\right )^{7/2}}{7 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7*Sqrt[a + b*x^2],x]

[Out]

-(a^3*(a + b*x^2)^(3/2))/(3*b^4) + (3*a^2*(a + b*x^2)^(5/2))/(5*b^4) - (3*a*(a +
b*x^2)^(7/2))/(7*b^4) + (a + b*x^2)^(9/2)/(9*b^4)

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Rubi in Sympy [A]  time = 15.6719, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{4}} + \frac{3 a^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{4}} - \frac{3 a \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b^{4}} + \frac{\left (a + b x^{2}\right )^{\frac{9}{2}}}{9 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**2+a)**(1/2),x)

[Out]

-a**3*(a + b*x**2)**(3/2)/(3*b**4) + 3*a**2*(a + b*x**2)**(5/2)/(5*b**4) - 3*a*(
a + b*x**2)**(7/2)/(7*b**4) + (a + b*x**2)**(9/2)/(9*b**4)

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Mathematica [A]  time = 0.029153, size = 61, normalized size = 0.76 \[ \frac{\sqrt{a+b x^2} \left (-16 a^4+8 a^3 b x^2-6 a^2 b^2 x^4+5 a b^3 x^6+35 b^4 x^8\right )}{315 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*Sqrt[a + b*x^2],x]

[Out]

(Sqrt[a + b*x^2]*(-16*a^4 + 8*a^3*b*x^2 - 6*a^2*b^2*x^4 + 5*a*b^3*x^6 + 35*b^4*x
^8))/(315*b^4)

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Maple [A]  time = 0.008, size = 47, normalized size = 0.6 \[ -{\frac{-35\,{b}^{3}{x}^{6}+30\,a{b}^{2}{x}^{4}-24\,{a}^{2}b{x}^{2}+16\,{a}^{3}}{315\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^2+a)^(1/2),x)

[Out]

-1/315*(b*x^2+a)^(3/2)*(-35*b^3*x^6+30*a*b^2*x^4-24*a^2*b*x^2+16*a^3)/b^4

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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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23268, size = 77, normalized size = 0.96 \[ \frac{{\left (35 \, b^{4} x^{8} + 5 \, a b^{3} x^{6} - 6 \, a^{2} b^{2} x^{4} + 8 \, a^{3} b x^{2} - 16 \, a^{4}\right )} \sqrt{b x^{2} + a}}{315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*x^7,x, algorithm="fricas")

[Out]

1/315*(35*b^4*x^8 + 5*a*b^3*x^6 - 6*a^2*b^2*x^4 + 8*a^3*b*x^2 - 16*a^4)*sqrt(b*x
^2 + a)/b^4

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Sympy [A]  time = 3.77464, size = 110, normalized size = 1.38 \[ \begin{cases} - \frac{16 a^{4} \sqrt{a + b x^{2}}}{315 b^{4}} + \frac{8 a^{3} x^{2} \sqrt{a + b x^{2}}}{315 b^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{6} \sqrt{a + b x^{2}}}{63 b} + \frac{x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**2+a)**(1/2),x)

[Out]

Piecewise((-16*a**4*sqrt(a + b*x**2)/(315*b**4) + 8*a**3*x**2*sqrt(a + b*x**2)/(
315*b**3) - 2*a**2*x**4*sqrt(a + b*x**2)/(105*b**2) + a*x**6*sqrt(a + b*x**2)/(6
3*b) + x**8*sqrt(a + b*x**2)/9, Ne(b, 0)), (sqrt(a)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.206812, size = 77, normalized size = 0.96 \[ \frac{35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}}{315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*x^7,x, algorithm="giac")

[Out]

1/315*(35*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^
2 - 105*(b*x^2 + a)^(3/2)*a^3)/b^4