Integrand size = 19, antiderivative size = 143 \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\frac {a^3 c x^6 \sqrt {c \left (a+b x^2\right )^2}}{6 \left (a+b x^2\right )}+\frac {3 a^2 b c x^8 \sqrt {c \left (a+b x^2\right )^2}}{8 \left (a+b x^2\right )}+\frac {3 a b^2 c x^{10} \sqrt {c \left (a+b x^2\right )^2}}{10 \left (a+b x^2\right )}+\frac {b^3 c x^{12} \sqrt {c \left (a+b x^2\right )^2}}{12 \left (a+b x^2\right )} \]
1/6*a^3*c*x^6*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)+3/8*a^2*b*c*x^8*(c*(b*x^2+a) ^2)^(1/2)/(b*x^2+a)+3/10*a*b^2*c*x^10*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)+1/12 *b^3*c*x^12*(c*(b*x^2+a)^2)^(1/2)/(b*x^2+a)
Time = 0.83 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.44 \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\frac {x^6 \left (c \left (a+b x^2\right )^2\right )^{3/2} \left (20 a^3+45 a^2 b x^2+36 a b^2 x^4+10 b^3 x^6\right )}{120 \left (a+b x^2\right )^3} \]
(x^6*(c*(a + b*x^2)^2)^(3/2)*(20*a^3 + 45*a^2*b*x^2 + 36*a*b^2*x^4 + 10*b^ 3*x^6))/(120*(a + b*x^2)^3)
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2045, 27, 243, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2045 |
\(\displaystyle \frac {a^3 c \sqrt {c \left (a+b x^2\right )^2} \int \frac {x^5 \left (b x^2+a\right )^3}{a^3}dx}{a+b x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c \sqrt {c \left (a+b x^2\right )^2} \int x^5 \left (b x^2+a\right )^3dx}{a+b x^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {c \sqrt {c \left (a+b x^2\right )^2} \int x^4 \left (b x^2+a\right )^3dx^2}{2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {c \sqrt {c \left (a+b x^2\right )^2} \int \left (b^3 x^{10}+3 a b^2 x^8+3 a^2 b x^6+a^3 x^4\right )dx^2}{2 \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c \left (\frac {a^3 x^6}{3}+\frac {3}{4} a^2 b x^8+\frac {3}{5} a b^2 x^{10}+\frac {b^3 x^{12}}{6}\right ) \sqrt {c \left (a+b x^2\right )^2}}{2 \left (a+b x^2\right )}\) |
(c*Sqrt[c*(a + b*x^2)^2]*((a^3*x^6)/3 + (3*a^2*b*x^8)/4 + (3*a*b^2*x^10)/5 + (b^3*x^12)/6))/(2*(a + b*x^2))
3.3.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_.)*((c_.)*((a_) + (b_.)*(x_)^(n_.))^(q_))^(p_), x_Symbol] :> Simp[Si mp[(c*(a + b*x^n)^q)^p/(1 + b*(x^n/a))^(p*q)] Int[u*(1 + b*(x^n/a))^(p*q) , x], x] /; FreeQ[{a, b, c, n, p, q}, x] && !GeQ[a, 0]
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {x^{6} \left (10 b^{3} x^{6}+36 b^{2} x^{4} a +45 a^{2} b \,x^{2}+20 a^{3}\right ) {\left (c \left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{120 \left (b \,x^{2}+a \right )^{3}}\) | \(60\) |
default | \(\frac {x^{6} \left (10 b^{3} x^{6}+36 b^{2} x^{4} a +45 a^{2} b \,x^{2}+20 a^{3}\right ) {\left (c \left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}{120 \left (b \,x^{2}+a \right )^{3}}\) | \(60\) |
pseudoelliptic | \(\frac {x^{6} \left (10 b^{3} x^{6}+36 b^{2} x^{4} a +45 a^{2} b \,x^{2}+20 a^{3}\right ) c \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{120 b \,x^{2}+120 a}\) | \(63\) |
trager | \(\frac {c \,x^{6} \left (10 b^{3} x^{6}+36 b^{2} x^{4} a +45 a^{2} b \,x^{2}+20 a^{3}\right ) \sqrt {b^{2} c \,x^{4}+2 a b c \,x^{2}+c \,a^{2}}}{120 b \,x^{2}+120 a}\) | \(72\) |
risch | \(\frac {a^{3} c \,x^{6} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{6 b \,x^{2}+6 a}+\frac {3 a^{2} b c \,x^{8} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{8 \left (b \,x^{2}+a \right )}+\frac {3 a \,b^{2} c \,x^{10} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{10 \left (b \,x^{2}+a \right )}+\frac {b^{3} c \,x^{12} \sqrt {c \left (b \,x^{2}+a \right )^{2}}}{12 b \,x^{2}+12 a}\) | \(128\) |
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.52 \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\frac {{\left (10 \, b^{3} c x^{12} + 36 \, a b^{2} c x^{10} + 45 \, a^{2} b c x^{8} + 20 \, a^{3} c x^{6}\right )} \sqrt {b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c}}{120 \, {\left (b x^{2} + a\right )}} \]
1/120*(10*b^3*c*x^12 + 36*a*b^2*c*x^10 + 45*a^2*b*c*x^8 + 20*a^3*c*x^6)*sq rt(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)/(b*x^2 + a)
\[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\int x^{5} \left (c \left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Time = 0.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.95 \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{2} x^{2}}{8 \, b^{2}} + \frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {3}{2}} a^{3}}{8 \, b^{3}} + \frac {{\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {5}{2}} x^{2}}{12 \, b^{2} c} - \frac {7 \, {\left (b^{2} c x^{4} + 2 \, a b c x^{2} + a^{2} c\right )}^{\frac {5}{2}} a}{60 \, b^{3} c} \]
1/8*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)^(3/2)*a^2*x^2/b^2 + 1/8*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)^(3/2)*a^3/b^3 + 1/12*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2* c)^(5/2)*x^2/(b^2*c) - 7/60*(b^2*c*x^4 + 2*a*b*c*x^2 + a^2*c)^(5/2)*a/(b^3 *c)
Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.50 \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\frac {1}{120} \, {\left (10 \, b^{3} x^{12} \mathrm {sgn}\left (b x^{2} + a\right ) + 36 \, a b^{2} x^{10} \mathrm {sgn}\left (b x^{2} + a\right ) + 45 \, a^{2} b x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + 20 \, a^{3} x^{6} \mathrm {sgn}\left (b x^{2} + a\right )\right )} c^{\frac {3}{2}} \]
1/120*(10*b^3*x^12*sgn(b*x^2 + a) + 36*a*b^2*x^10*sgn(b*x^2 + a) + 45*a^2* b*x^8*sgn(b*x^2 + a) + 20*a^3*x^6*sgn(b*x^2 + a))*c^(3/2)
Timed out. \[ \int x^5 \left (c \left (a+b x^2\right )^2\right )^{3/2} \, dx=\int x^5\,{\left (c\,{\left (b\,x^2+a\right )}^2\right )}^{3/2} \,d x \]