3.2953 \(\int x^5 \sqrt{a+b \sqrt{c x^3}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{4 a^3 \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac{4 a^2 \left (a+b \sqrt{c x^3}\right )^{5/2}}{5 b^4 c^2}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{9/2}}{27 b^4 c^2}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

(-4*a^3*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^4*c^2) + (4*a^2*(a + b*Sqrt[c*x^3])^(5/2
))/(5*b^4*c^2) - (4*a*(a + b*Sqrt[c*x^3])^(7/2))/(7*b^4*c^2) + (4*(a + b*Sqrt[c*
x^3])^(9/2))/(27*b^4*c^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.196311, antiderivative size = 116, 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 \[ -\frac{4 a^3 \left (a+b \sqrt{c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac{4 a^2 \left (a+b \sqrt{c x^3}\right )^{5/2}}{5 b^4 c^2}+\frac{4 \left (a+b \sqrt{c x^3}\right )^{9/2}}{27 b^4 c^2}-\frac{4 a \left (a+b \sqrt{c x^3}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(-4*a^3*(a + b*Sqrt[c*x^3])^(3/2))/(9*b^4*c^2) + (4*a^2*(a + b*Sqrt[c*x^3])^(5/2
))/(5*b^4*c^2) - (4*a*(a + b*Sqrt[c*x^3])^(7/2))/(7*b^4*c^2) + (4*(a + b*Sqrt[c*
x^3])^(9/2))/(27*b^4*c^2)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \sqrt{a + b \sqrt{c x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(a+b*(c*x**3)**(1/2))**(1/2),x)

[Out]

Integral(x**5*sqrt(a + b*sqrt(c*x**3)), x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0540071, size = 86, normalized size = 0.74 \[ \frac{4 \sqrt{a+b \sqrt{c x^3}} \left (-16 a^4+8 a^3 b \sqrt{c x^3}-6 a^2 b^2 c x^3+5 a b^3 \left (c x^3\right )^{3/2}+35 b^4 c^2 x^6\right )}{945 b^4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a + b*Sqrt[c*x^3]],x]

[Out]

(4*Sqrt[a + b*Sqrt[c*x^3]]*(-16*a^4 - 6*a^2*b^2*c*x^3 + 35*b^4*c^2*x^6 + 8*a^3*b
*Sqrt[c*x^3] + 5*a*b^3*(c*x^3)^(3/2)))/(945*b^4*c^2)

_______________________________________________________________________________________

Maple [A]  time = 0.192, size = 103, normalized size = 0.9 \[{\frac{4}{945\,{c}^{2}{b}^{4}}\sqrt{a+b\sqrt{c{x}^{3}}} \left ( 35\,{c}^{2}{x}^{6}{b}^{4} \left ( c{x}^{3} \right ) ^{3/2}+5\,a{x}^{9}{c}^{3}{b}^{3}+8\,{a}^{3}{x}^{6}{c}^{2}b-6\,{a}^{2}c{x}^{3}{b}^{2} \left ( c{x}^{3} \right ) ^{3/2}-16\,{a}^{4} \left ( c{x}^{3} \right ) ^{3/2} \right ) \left ( c{x}^{3} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(a+b*(c*x^3)^(1/2))^(1/2),x)

[Out]

4/945/c^2*(a+b*(c*x^3)^(1/2))^(1/2)*(35*c^2*x^6*b^4*(c*x^3)^(3/2)+5*a*x^9*c^3*b^
3+8*a^3*x^6*c^2*b-6*a^2*c*x^3*b^2*(c*x^3)^(3/2)-16*a^4*(c*x^3)^(3/2))/b^4/(c*x^3
)^(3/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.36675, size = 115, normalized size = 0.99 \[ \frac{4 \,{\left (\frac{35 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (\sqrt{c x^{3}} b + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{945 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^3)*b + a)*x^5,x, algorithm="maxima")

[Out]

4/945*(35*(sqrt(c*x^3)*b + a)^(9/2)/b^4 - 135*(sqrt(c*x^3)*b + a)^(7/2)*a/b^4 +
189*(sqrt(c*x^3)*b + a)^(5/2)*a^2/b^4 - 105*(sqrt(c*x^3)*b + a)^(3/2)*a^3/b^4)/c
^2

_______________________________________________________________________________________

Fricas [A]  time = 0.347878, size = 101, normalized size = 0.87 \[ \frac{4 \,{\left (35 \, b^{4} c^{2} x^{6} - 6 \, a^{2} b^{2} c x^{3} - 16 \, a^{4} +{\left (5 \, a b^{3} c x^{3} + 8 \, a^{3} b\right )} \sqrt{c x^{3}}\right )} \sqrt{\sqrt{c x^{3}} b + a}}{945 \, b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^3)*b + a)*x^5,x, algorithm="fricas")

[Out]

4/945*(35*b^4*c^2*x^6 - 6*a^2*b^2*c*x^3 - 16*a^4 + (5*a*b^3*c*x^3 + 8*a^3*b)*sqr
t(c*x^3))*sqrt(sqrt(c*x^3)*b + a)/(b^4*c^2)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \sqrt{a + b \sqrt{c x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(a+b*(c*x**3)**(1/2))**(1/2),x)

[Out]

Integral(x**5*sqrt(a + b*sqrt(c*x**3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222413, size = 155, normalized size = 1.34 \[ \frac{4 \,{\left (\frac{16 \, \sqrt{a c} a^{4}}{b^{4} c} - \frac{105 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}} a^{3} c^{3} - 189 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{5}{2}} a^{2} c^{2} + 135 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{7}{2}} a c - 35 \,{\left (\sqrt{c x} b c x + a c\right )}^{\frac{9}{2}}}{b^{4} c^{5}}\right )}{\left | c \right |}}{945 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sqrt(c*x^3)*b + a)*x^5,x, algorithm="giac")

[Out]

4/945*(16*sqrt(a*c)*a^4/(b^4*c) - (105*(sqrt(c*x)*b*c*x + a*c)^(3/2)*a^3*c^3 - 1
89*(sqrt(c*x)*b*c*x + a*c)^(5/2)*a^2*c^2 + 135*(sqrt(c*x)*b*c*x + a*c)^(7/2)*a*c
- 35*(sqrt(c*x)*b*c*x + a*c)^(9/2))/(b^4*c^5))*abs(c)/c^(5/2)