3.2960 \(\int x^5 \sqrt {a+b \sqrt {c x^3}} \, dx\)

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]

-4/9*a^3*(a+b*(c*x^3)^(1/2))^(3/2)/b^4/c^2+4/5*a^2*(a+b*(c*x^3)^(1/2))^(5/2)/b^4/c^2-4/7*a*(a+b*(c*x^3)^(1/2))
^(7/2)/b^4/c^2+4/27*(a+b*(c*x^3)^(1/2))^(9/2)/b^4/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac {4 a^2 \left (a+b \sqrt {c x^3}\right )^{5/2}}{5 b^4 c^2}-\frac {4 a^3 \left (a+b \sqrt {c x^3}\right )^{3/2}}{9 b^4 c^2}+\frac {4 \left (a+b \sqrt {c x^3}\right )^{9/2}}{27 b^4 c^2}-\frac {4 a \left (a+b \sqrt {c x^3}\right )^{7/2}}{7 b^4 c^2} \]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rubi steps

\begin {align*} \int x^5 \sqrt {a+b \sqrt {c x^3}} \, dx &=\operatorname {Subst}\left (\int x^5 \sqrt {a+b \sqrt {c} x^{3/2}} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{3} \operatorname {Subst}\left (\int x^3 \sqrt {a+b \sqrt {c} x} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3 \sqrt {a+b \sqrt {c} x}}{b^3 c^{3/2}}+\frac {3 a^2 \left (a+b \sqrt {c} x\right )^{3/2}}{b^3 c^{3/2}}-\frac {3 a \left (a+b \sqrt {c} x\right )^{5/2}}{b^3 c^{3/2}}+\frac {\left (a+b \sqrt {c} x\right )^{7/2}}{b^3 c^{3/2}}\right ) \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=-\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 a \left (a+b \sqrt {c x^3}\right )^{7/2}}{7 b^4 c^2}+\frac {4 \left (a+b \sqrt {c x^3}\right )^{9/2}}{27 b^4 c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 72, normalized size = 0.62 \[ \frac {4 \left (a+b \sqrt {c x^3}\right )^{3/2} \left (-16 a^3+24 a^2 b \sqrt {c x^3}-30 a b^2 c x^3+35 b^3 \left (c x^3\right )^{3/2}\right )}{945 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a + b*Sqrt[c*x^3])^(3/2)*(-16*a^3 - 30*a*b^2*c*x^3 + 24*a^2*b*Sqrt[c*x^3] + 35*b^3*(c*x^3)^(3/2)))/(945*b^
4*c^2)

________________________________________________________________________________________

fricas [A]  time = 1.30, size = 75, normalized size = 0.65 \[ \frac {4 \, {\left (35 \, b^{4} c^{2} x^{6} - 6 \, a^{2} b^{2} c x^{3} - 16 \, a^{4} + {\left (5 \, a b^{3} c x^{3} + 8 \, a^{3} b\right )} \sqrt {c x^{3}}\right )} \sqrt {\sqrt {c x^{3}} b + a}}{945 \, b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*(c*x^3)^(1/2))^(1/2),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)*sqrt(c*x^3))*sqrt(sqrt(c*x^3)*b +
a)/(b^4*c^2)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 111, normalized size = 0.96 \[ -\frac {4 \, {\left (105 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {3}{2}} a^{3} c^{6} - 189 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {5}{2}} a^{2} c^{4} + 135 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {7}{2}} a c^{2} - 35 \, {\left (\sqrt {c x} b c^{2} x + a c^{2}\right )}^{\frac {9}{2}}\right )} {\left | c \right |}}{945 \, b^{4} c^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*(c*x^3)^(1/2))^(1/2),x, algorithm="giac")

[Out]

-4/945*(105*(sqrt(c*x)*b*c^2*x + a*c^2)^(3/2)*a^3*c^6 - 189*(sqrt(c*x)*b*c^2*x + a*c^2)^(5/2)*a^2*c^4 + 135*(s
qrt(c*x)*b*c^2*x + a*c^2)^(7/2)*a*c^2 - 35*(sqrt(c*x)*b*c^2*x + a*c^2)^(9/2))*abs(c)/(b^4*c^12)

________________________________________________________________________________________

maple [A]  time = 0.43, size = 103, normalized size = 0.89 \[ \frac {4 \sqrt {a +\sqrt {c \,x^{3}}\, b}\, \left (5 a \,b^{3} c^{3} x^{9}+8 a^{3} b \,c^{2} x^{6}+35 \left (c \,x^{3}\right )^{\frac {3}{2}} b^{4} c^{2} x^{6}-6 \left (c \,x^{3}\right )^{\frac {3}{2}} a^{2} b^{2} c \,x^{3}-16 \left (c \,x^{3}\right )^{\frac {3}{2}} a^{4}\right )}{945 \left (c \,x^{3}\right )^{\frac {3}{2}} b^{4} c^{2}} \]

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 = 0.64, size = 85, normalized size = 0.73 \[ \frac {4 \, {\left (\frac {35 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {9}{2}}}{b^{4}} - \frac {135 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {7}{2}} a}{b^{4}} + \frac {189 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {5}{2}} a^{2}}{b^{4}} - \frac {105 \, {\left (\sqrt {c x^{3}} b + a\right )}^{\frac {3}{2}} a^{3}}{b^{4}}\right )}}{945 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*(c*x^3)^(1/2))^(1/2),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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\sqrt {a+b\,\sqrt {c\,x^3}} \,d x \]

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]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \sqrt {a + b \sqrt {c x^{3}}}\, dx \]

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)

________________________________________________________________________________________