3.2935 \(\int x^8 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac{2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac{4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]

[Out]

(2*a^2*x^9*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b^3*(c*x^2)^(9/2)) - (4*a*x^9*(a + b*
(c*x^2)^(3/2))^(5/2))/(15*b^3*(c*x^2)^(9/2)) + (2*x^9*(a + b*(c*x^2)^(3/2))^(7/2
))/(21*b^3*(c*x^2)^(9/2))

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Rubi [A]  time = 0.160875, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 a^2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{3/2}}{9 b^3 \left (c x^2\right )^{9/2}}+\frac{2 x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{7/2}}{21 b^3 \left (c x^2\right )^{9/2}}-\frac{4 a x^9 \left (a+b \left (c x^2\right )^{3/2}\right )^{5/2}}{15 b^3 \left (c x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^8*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

(2*a^2*x^9*(a + b*(c*x^2)^(3/2))^(3/2))/(9*b^3*(c*x^2)^(9/2)) - (4*a*x^9*(a + b*
(c*x^2)^(3/2))^(5/2))/(15*b^3*(c*x^2)^(9/2)) + (2*x^9*(a + b*(c*x^2)^(3/2))^(7/2
))/(21*b^3*(c*x^2)^(9/2))

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Rubi in Sympy [A]  time = 15.1896, size = 105, normalized size = 0.93 \[ \frac{2 a^{2} x^{9} \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{3}{2}}}{9 b^{3} \left (c x^{2}\right )^{\frac{9}{2}}} - \frac{4 a x^{9} \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{5}{2}}}{15 b^{3} \left (c x^{2}\right )^{\frac{9}{2}}} + \frac{2 x^{9} \left (a + b \left (c x^{2}\right )^{\frac{3}{2}}\right )^{\frac{7}{2}}}{21 b^{3} \left (c x^{2}\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

2*a**2*x**9*(a + b*(c*x**2)**(3/2))**(3/2)/(9*b**3*(c*x**2)**(9/2)) - 4*a*x**9*(
a + b*(c*x**2)**(3/2))**(5/2)/(15*b**3*(c*x**2)**(9/2)) + 2*x**9*(a + b*(c*x**2)
**(3/2))**(7/2)/(21*b**3*(c*x**2)**(9/2))

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Mathematica [A]  time = 0.0720967, size = 0, normalized size = 0. \[ \int x^8 \sqrt{a+b \left (c x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[x^8*Sqrt[a + b*(c*x^2)^(3/2)],x]

[Out]

Integrate[x^8*Sqrt[a + b*(c*x^2)^(3/2)], x]

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{x}^{8}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(a+b*(c*x^2)^(3/2))^(1/2),x)

[Out]

int(x^8*(a+b*(c*x^2)^(3/2))^(1/2),x)

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Maxima [A]  time = 1.44528, size = 1091, normalized size = 9.65 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^8,x, algorithm="maxima")

[Out]

1/3*((c^7 + 3*c^6 + 2*c^5)*b^3*x^9 + (c^5 + c^4)*a*b^2*sqrt(c)*x^6 - 2*a^2*b*c^3
*x^3 + 2*a^3*sqrt(c))*sqrt(b*c^(3/2)*x^3 + a)/((c^8 + 6*c^7 + 11*c^6 + 6*c^5)*b^
3) + 2/315*(b^2*c^(11/2)*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(
3/2)*x^3 + a)^(5/2)*a/(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9
/2))) - b^2*c^5*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(3/2)*x^3
+ a)^(5/2)*a/(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9/2))) + 3
*b^2*c^(9/2)*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(3/2)*x^3 + a
)^(5/2)*a/(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9/2))) - 3*b^
2*c^4*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(3/2)*x^3 + a)^(5/2)
*a/(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9/2))) + 2*b^2*c^(7/
2)*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(3/2)*x^3 + a)^(5/2)*a/
(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9/2))) - 14*a*b*c^3*(3*
(b*c^(3/2)*x^3 + a)^(5/2)/(b^2*c^3) - 5*(b*c^(3/2)*x^3 + a)^(3/2)*a/(b^2*c^3)) -
2*b^2*c^3*(15*(b*c^(3/2)*x^3 + a)^(7/2)/(b^3*c^(9/2)) - 42*(b*c^(3/2)*x^3 + a)^
(5/2)*a/(b^3*c^(9/2)) + 35*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b^3*c^(9/2))) + 14*a*b
*c^(5/2)*(3*(b*c^(3/2)*x^3 + a)^(5/2)/(b^2*c^3) - 5*(b*c^(3/2)*x^3 + a)^(3/2)*a/
(b^2*c^3)) - 14*a*b*c^2*(3*(b*c^(3/2)*x^3 + a)^(5/2)/(b^2*c^3) - 5*(b*c^(3/2)*x^
3 + a)^(3/2)*a/(b^2*c^3)) + 14*a*b*c^(3/2)*(3*(b*c^(3/2)*x^3 + a)^(5/2)/(b^2*c^3
) - 5*(b*c^(3/2)*x^3 + a)^(3/2)*a/(b^2*c^3)) + 70*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/
(b*c) - 70*(b*c^(3/2)*x^3 + a)^(3/2)*a^2/(b*c^(3/2)))/((c^5 + 6*c^4 + 11*c^3 + 6
*c^2)*b^2*sqrt(c))

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Fricas [A]  time = 0.211175, size = 105, normalized size = 0.93 \[ \frac{2 \,{\left (15 \, b^{3} c^{5} x^{10} - 4 \, a^{2} b c^{2} x^{4} +{\left (3 \, a b^{2} c^{3} x^{6} + 8 \, a^{3}\right )} \sqrt{c x^{2}}\right )} \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{315 \, b^{3} c^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^8,x, algorithm="fricas")

[Out]

2/315*(15*b^3*c^5*x^10 - 4*a^2*b*c^2*x^4 + (3*a*b^2*c^3*x^6 + 8*a^3)*sqrt(c*x^2)
)*sqrt(sqrt(c*x^2)*b*c*x^2 + a)/(b^3*c^5*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{8} \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(a+b*(c*x**2)**(3/2))**(1/2),x)

[Out]

Integral(x**8*sqrt(a + b*(c*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.218875, size = 74, normalized size = 0.65 \[ \frac{2 \,{\left (15 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b c^{\frac{3}{2}} x^{3} + a\right )}^{\frac{3}{2}} a^{2}\right )}}{315 \, b^{3} c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)*x^8,x, algorithm="giac")

[Out]

2/315*(15*(b*c^(3/2)*x^3 + a)^(7/2) - 42*(b*c^(3/2)*x^3 + a)^(5/2)*a + 35*(b*c^(
3/2)*x^3 + a)^(3/2)*a^2)/(b^3*c^(9/2))