3.2964 \(\int x^8 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]

[Out]

(-4*a*(a + b*(c*x^3)^(3/2))^(3/2))/(27*b^2*c^3) + (4*(a + b*(c*x^3)^(3/2))^(5/2)
)/(45*b^2*c^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.0977529, antiderivative size = 56, 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 \[ \frac{4 \left (a+b \left (c x^3\right )^{3/2}\right )^{5/2}}{45 b^2 c^3}-\frac{4 a \left (a+b \left (c x^3\right )^{3/2}\right )^{3/2}}{27 b^2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a*(a + b*(c*x^3)^(3/2))^(3/2))/(27*b^2*c^3) + (4*(a + b*(c*x^3)^(3/2))^(5/2)
)/(45*b^2*c^3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0888599, size = 0, normalized size = 0. \[ \int x^8 \sqrt{a+b \left (c x^3\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{x}^{8}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.39793, size = 58, normalized size = 1.04 \[ \frac{4 \,{\left (\frac{3 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (\left (c x^{3}\right )^{\frac{3}{2}} b + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{135 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/135*(3*((c*x^3)^(3/2)*b + a)^(5/2)/b^2 - 5*((c*x^3)^(3/2)*b + a)^(3/2)*a/b^2)/
c^3

_______________________________________________________________________________________

Fricas [A]  time = 1.25408, size = 76, normalized size = 1.36 \[ \frac{4 \,{\left (3 \, b^{2} c^{3} x^{9} + \sqrt{c x^{3}} a b c x^{3} - 2 \, a^{2}\right )} \sqrt{\sqrt{c x^{3}} b c x^{3} + a}}{135 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/135*(3*b^2*c^3*x^9 + sqrt(c*x^3)*a*b*c*x^3 - 2*a^2)*sqrt(sqrt(c*x^3)*b*c*x^3 +
a)/(b^2*c^3)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222156, size = 115, normalized size = 2.05 \[ \frac{4 \,{\left (\frac{2 \, \sqrt{a c^{3}} a^{2}}{b^{2} c^{2}} - \frac{5 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{3}{2}} a c^{3} - 3 \,{\left (\sqrt{c x} b c^{4} x^{4} + a c^{3}\right )}^{\frac{5}{2}}}{b^{2} c^{8}}\right )}{\left | c \right |}}{135 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/135*(2*sqrt(a*c^3)*a^2/(b^2*c^2) - (5*(sqrt(c*x)*b*c^4*x^4 + a*c^3)^(3/2)*a*c^
3 - 3*(sqrt(c*x)*b*c^4*x^4 + a*c^3)^(5/2))/(b^2*c^8))*abs(c)/c^(7/2)