3.2930 \(\int x^5 \sqrt {a+b \sqrt {c x^2}} \, dx\)

Optimal. Leaf size=174 \[ -\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3} \]

[Out]

-2/3*a^5*(a+b*(c*x^2)^(1/2))^(3/2)/b^6/c^3+2*a^4*(a+b*(c*x^2)^(1/2))^(5/2)/b^6/c^3-20/7*a^3*(a+b*(c*x^2)^(1/2)
)^(7/2)/b^6/c^3+20/9*a^2*(a+b*(c*x^2)^(1/2))^(9/2)/b^6/c^3-10/11*a*(a+b*(c*x^2)^(1/2))^(11/2)/b^6/c^3+2/13*(a+
b*(c*x^2)^(1/2))^(13/2)/b^6/c^3

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {368, 43} \[ \frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(-2*a^5*(a + b*Sqrt[c*x^2])^(3/2))/(3*b^6*c^3) + (2*a^4*(a + b*Sqrt[c*x^2])^(5/2))/(b^6*c^3) - (20*a^3*(a + b*
Sqrt[c*x^2])^(7/2))/(7*b^6*c^3) + (20*a^2*(a + b*Sqrt[c*x^2])^(9/2))/(9*b^6*c^3) - (10*a*(a + b*Sqrt[c*x^2])^(
11/2))/(11*b^6*c^3) + (2*(a + b*Sqrt[c*x^2])^(13/2))/(13*b^6*c^3)

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 368

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*((c*x^q
)^(1/q))^(m + 1)), Subst[Int[x^m*(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q
}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]

Rubi steps

\begin {align*} \int x^5 \sqrt {a+b \sqrt {c x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int x^5 \sqrt {a+b x} \, dx,x,\sqrt {c x^2}\right )}{c^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^5 \sqrt {a+b x}}{b^5}+\frac {5 a^4 (a+b x)^{3/2}}{b^5}-\frac {10 a^3 (a+b x)^{5/2}}{b^5}+\frac {10 a^2 (a+b x)^{7/2}}{b^5}-\frac {5 a (a+b x)^{9/2}}{b^5}+\frac {(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt {c x^2}\right )}{c^3}\\ &=-\frac {2 a^5 \left (a+b \sqrt {c x^2}\right )^{3/2}}{3 b^6 c^3}+\frac {2 a^4 \left (a+b \sqrt {c x^2}\right )^{5/2}}{b^6 c^3}-\frac {20 a^3 \left (a+b \sqrt {c x^2}\right )^{7/2}}{7 b^6 c^3}+\frac {20 a^2 \left (a+b \sqrt {c x^2}\right )^{9/2}}{9 b^6 c^3}-\frac {10 a \left (a+b \sqrt {c x^2}\right )^{11/2}}{11 b^6 c^3}+\frac {2 \left (a+b \sqrt {c x^2}\right )^{13/2}}{13 b^6 c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 103, normalized size = 0.59 \[ \frac {2 \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {c x^2}-480 a^3 b^2 c x^2+560 a^2 b^3 \left (c x^2\right )^{3/2}-630 a b^4 c^2 x^4+693 b^5 \left (c x^2\right )^{5/2}\right )}{9009 b^6 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*Sqrt[a + b*Sqrt[c*x^2]],x]

[Out]

(2*(a + b*Sqrt[c*x^2])^(3/2)*(-256*a^5 - 480*a^3*b^2*c*x^2 - 630*a*b^4*c^2*x^4 + 384*a^4*b*Sqrt[c*x^2] + 560*a
^2*b^3*(c*x^2)^(3/2) + 693*b^5*(c*x^2)^(5/2)))/(9009*b^6*c^3)

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 103, normalized size = 0.59 \[ \frac {2 \, {\left (693 \, b^{6} c^{3} x^{6} - 70 \, a^{2} b^{4} c^{2} x^{4} - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} + {\left (63 \, a b^{5} c^{2} x^{4} + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{9009 \, b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(693*b^6*c^3*x^6 - 70*a^2*b^4*c^2*x^4 - 96*a^4*b^2*c*x^2 - 256*a^6 + (63*a*b^5*c^2*x^4 + 80*a^3*b^3*c*x
^2 + 128*a^5*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a)/(b^6*c^3)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 231, normalized size = 1.33 \[ \frac {2 \, {\left (\frac {13 \, {\left (63 \, {\left (b \sqrt {c} x + a\right )}^{\frac {11}{2}} \sqrt {c} - 385 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} a \sqrt {c} + 990 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a^{2} \sqrt {c} - 1386 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{3} \sqrt {c} + 1155 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{4} \sqrt {c} - 693 \, \sqrt {b \sqrt {c} x + a} a^{5} \sqrt {c}\right )} a}{b^{5} c^{3}} + \frac {3 \, {\left (231 \, {\left (b \sqrt {c} x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b \sqrt {c} x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b \sqrt {c} x + a} a^{6}\right )}}{b^{5} c^{\frac {5}{2}}}\right )}}{9009 \, b \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(13*(63*(b*sqrt(c)*x + a)^(11/2)*sqrt(c) - 385*(b*sqrt(c)*x + a)^(9/2)*a*sqrt(c) + 990*(b*sqrt(c)*x + a
)^(7/2)*a^2*sqrt(c) - 1386*(b*sqrt(c)*x + a)^(5/2)*a^3*sqrt(c) + 1155*(b*sqrt(c)*x + a)^(3/2)*a^4*sqrt(c) - 69
3*sqrt(b*sqrt(c)*x + a)*a^5*sqrt(c))*a/(b^5*c^3) + 3*(231*(b*sqrt(c)*x + a)^(13/2) - 1638*(b*sqrt(c)*x + a)^(1
1/2)*a + 5005*(b*sqrt(c)*x + a)^(9/2)*a^2 - 8580*(b*sqrt(c)*x + a)^(7/2)*a^3 + 9009*(b*sqrt(c)*x + a)^(5/2)*a^
4 - 6006*(b*sqrt(c)*x + a)^(3/2)*a^5 + 3003*sqrt(b*sqrt(c)*x + a)*a^6)/(b^5*c^(5/2)))/(b*sqrt(c))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 92, normalized size = 0.53 \[ \frac {2 \left (a +\sqrt {c \,x^{2}}\, b \right )^{\frac {3}{2}} \left (-630 a \,b^{4} c^{2} x^{4}-480 a^{3} b^{2} c \,x^{2}-256 a^{5}+384 \sqrt {c \,x^{2}}\, a^{4} b +560 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{2} b^{3}+693 \left (c \,x^{2}\right )^{\frac {5}{2}} b^{5}\right )}{9009 b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(a+b*(c*x^2)^(1/2))^(3/2)*(693*(c*x^2)^(5/2)*b^5-630*c^2*x^4*a*b^4+560*(c*x^2)^(3/2)*a^2*b^3-480*c*x^2*
a^3*b^2+384*(c*x^2)^(1/2)*a^4*b-256*a^5)/c^3/b^6

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 127, normalized size = 0.73 \[ \frac {2 \, {\left (\frac {693 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {13}{2}}}{b^{6}} - \frac {4095 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {11}{2}} a}{b^{6}} + \frac {10010 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {9}{2}} a^{2}}{b^{6}} - \frac {12870 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {7}{2}} a^{3}}{b^{6}} + \frac {9009 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {3003 \, {\left (\sqrt {c x^{2}} b + a\right )}^{\frac {3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9009*(693*(sqrt(c*x^2)*b + a)^(13/2)/b^6 - 4095*(sqrt(c*x^2)*b + a)^(11/2)*a/b^6 + 10010*(sqrt(c*x^2)*b + a)
^(9/2)*a^2/b^6 - 12870*(sqrt(c*x^2)*b + a)^(7/2)*a^3/b^6 + 9009*(sqrt(c*x^2)*b + a)^(5/2)*a^4/b^6 - 3003*(sqrt
(c*x^2)*b + a)^(3/2)*a^5/b^6)/c^3

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\sqrt {a+b\,\sqrt {c\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{5} \sqrt {a + b \sqrt {c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*sqrt(a + b*sqrt(c*x**2)), x)

________________________________________________________________________________________