Integrand size = 22, antiderivative size = 123 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\frac {d (2 b c-a d) \sqrt {c+\frac {d}{x^2}} x^2}{16 c^2}+\frac {(2 b c-a d) \sqrt {c+\frac {d}{x^2}} x^4}{8 c}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^6}{6 c}-\frac {d^2 (2 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{16 c^{5/2}} \]
1/6*a*(c+d/x^2)^(3/2)*x^6/c-1/16*d^2*(-a*d+2*b*c)*arctanh((c+d/x^2)^(1/2)/ c^(1/2))/c^(5/2)+1/16*d*(-a*d+2*b*c)*x^2*(c+d/x^2)^(1/2)/c^2+1/8*(-a*d+2*b *c)*x^4*(c+d/x^2)^(1/2)/c
Time = 0.64 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\frac {\sqrt {c+\frac {d}{x^2}} x \left (\sqrt {c} x \left (6 b c \left (d+2 c x^2\right )+a \left (-3 d^2+2 c d x^2+8 c^2 x^4\right )\right )+\frac {6 d^2 (-2 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {d}+\sqrt {d+c x^2}}\right )}{\sqrt {d+c x^2}}\right )}{48 c^{5/2}} \]
(Sqrt[c + d/x^2]*x*(Sqrt[c]*x*(6*b*c*(d + 2*c*x^2) + a*(-3*d^2 + 2*c*d*x^2 + 8*c^2*x^4)) + (6*d^2*(-2*b*c + a*d)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[d] + Sqr t[d + c*x^2])])/Sqrt[d + c*x^2]))/(48*c^(5/2))
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {948, 87, 51, 52, 73, 221}
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.
| \(\displaystyle \int x^5 \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\frac {1}{2} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^8d\frac {1}{x^2}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-\frac {(2 b c-a d) \int \sqrt {c+\frac {d}{x^2}} x^6d\frac {1}{x^2}}{2 c}\right )\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} \left (\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-\frac {(2 b c-a d) \left (\frac {1}{4} d \int \frac {x^4}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}-\frac {1}{2} x^4 \sqrt {c+\frac {d}{x^2}}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {1}{2} \left (\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-\frac {(2 b c-a d) \left (\frac {1}{4} d \left (-\frac {d \int \frac {x^2}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}}{2 c}-\frac {x^2 \sqrt {c+\frac {d}{x^2}}}{c}\right )-\frac {1}{2} x^4 \sqrt {c+\frac {d}{x^2}}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-\frac {(2 b c-a d) \left (\frac {1}{4} d \left (-\frac {\int \frac {1}{\frac {1}{d x^4}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{c}-\frac {x^2 \sqrt {c+\frac {d}{x^2}}}{c}\right )-\frac {1}{2} x^4 \sqrt {c+\frac {d}{x^2}}\right )}{2 c}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {a x^6 \left (c+\frac {d}{x^2}\right )^{3/2}}{3 c}-\frac {(2 b c-a d) \left (\frac {1}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {x^2 \sqrt {c+\frac {d}{x^2}}}{c}\right )-\frac {1}{2} x^4 \sqrt {c+\frac {d}{x^2}}\right )}{2 c}\right )\) |
((a*(c + d/x^2)^(3/2)*x^6)/(3*c) - ((2*b*c - a*d)*(-1/2*(Sqrt[c + d/x^2]*x ^4) + (d*(-((Sqrt[c + d/x^2]*x^2)/c) + (d*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]] )/c^(3/2)))/4))/(2*c))/2
3.10.29.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {x^{2} \left (8 a \,x^{4} c^{2}+2 a c d \,x^{2}+12 b \,c^{2} x^{2}-3 a \,d^{2}+6 b c d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{48 c^{2}}+\frac {d^{2} \left (a d -2 b c \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}{16 c^{\frac {5}{2}} \sqrt {c \,x^{2}+d}}\) | \(116\) |
| default | \(\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \left (8 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {3}{2}} a \,x^{3}-6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c}\, a d x +12 \left (c \,x^{2}+d \right )^{\frac {3}{2}} c^{\frac {3}{2}} b x +3 \sqrt {c \,x^{2}+d}\, \sqrt {c}\, a \,d^{2} x -6 \sqrt {c \,x^{2}+d}\, c^{\frac {3}{2}} b d x +3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a \,d^{3}-6 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b c \,d^{2}\right )}{48 \sqrt {c \,x^{2}+d}\, c^{\frac {5}{2}}}\) | \(162\) |
1/48*x^2*(8*a*c^2*x^4+2*a*c*d*x^2+12*b*c^2*x^2-3*a*d^2+6*b*c*d)/c^2*((c*x^ 2+d)/x^2)^(1/2)+1/16*d^2*(a*d-2*b*c)/c^(5/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2)) *((c*x^2+d)/x^2)^(1/2)*x/(c*x^2+d)^(1/2)
Time = 0.43 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.97 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\left [-\frac {3 \, {\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \, {\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{96 \, c^{3}}, \frac {3 \, {\left (2 \, b c d^{2} - a d^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + {\left (8 \, a c^{3} x^{6} + 2 \, {\left (6 \, b c^{3} + a c^{2} d\right )} x^{4} + 3 \, {\left (2 \, b c^{2} d - a c d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, c^{3}}\right ] \]
[-1/96*(3*(2*b*c*d^2 - a*d^3)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c *x^2 + d)/x^2) - d) - 2*(8*a*c^3*x^6 + 2*(6*b*c^3 + a*c^2*d)*x^4 + 3*(2*b* c^2*d - a*c*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/c^3, 1/48*(3*(2*b*c*d^2 - a*d ^3)*sqrt(-c)*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + (8*a *c^3*x^6 + 2*(6*b*c^3 + a*c^2*d)*x^4 + 3*(2*b*c^2*d - a*c*d^2)*x^2)*sqrt(( c*x^2 + d)/x^2))/c^3]
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (107) = 214\).
Time = 24.90 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.84 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\frac {a c x^{7}}{6 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {5 a \sqrt {d} x^{5}}{24 \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{\frac {3}{2}} x^{3}}{48 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{\frac {5}{2}} x}{16 c^{2} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {a d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{16 c^{\frac {5}{2}}} + \frac {b c x^{5}}{4 \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {3 b \sqrt {d} x^{3}}{8 \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {b d^{\frac {3}{2}} x}{8 c \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{8 c^{\frac {3}{2}}} \]
a*c*x**7/(6*sqrt(d)*sqrt(c*x**2/d + 1)) + 5*a*sqrt(d)*x**5/(24*sqrt(c*x**2 /d + 1)) - a*d**(3/2)*x**3/(48*c*sqrt(c*x**2/d + 1)) - a*d**(5/2)*x/(16*c* *2*sqrt(c*x**2/d + 1)) + a*d**3*asinh(sqrt(c)*x/sqrt(d))/(16*c**(5/2)) + b *c*x**5/(4*sqrt(d)*sqrt(c*x**2/d + 1)) + 3*b*sqrt(d)*x**3/(8*sqrt(c*x**2/d + 1)) + b*d**(3/2)*x/(8*c*sqrt(c*x**2/d + 1)) - b*d**2*asinh(sqrt(c)*x/sq rt(d))/(8*c**(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (103) = 206\).
Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.98 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=-\frac {1}{96} \, {\left (\frac {3 \, d^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} - 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c d^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d^{3}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} c^{2} - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} c^{3} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} c^{4} - c^{5}}\right )} a + \frac {1}{16} \, {\left (\frac {d^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} + \sqrt {c + \frac {d}{x^{2}}} c d^{2}\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} c - 2 \, {\left (c + \frac {d}{x^{2}}\right )} c^{2} + c^{3}}\right )} b \]
-1/96*(3*d^3*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c)))/ c^(5/2) + 2*(3*(c + d/x^2)^(5/2)*d^3 - 8*(c + d/x^2)^(3/2)*c*d^3 - 3*sqrt( c + d/x^2)*c^2*d^3)/((c + d/x^2)^3*c^2 - 3*(c + d/x^2)^2*c^3 + 3*(c + d/x^ 2)*c^4 - c^5))*a + 1/16*(d^2*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x ^2) + sqrt(c)))/c^(3/2) + 2*((c + d/x^2)^(3/2)*d^2 + sqrt(c + d/x^2)*c*d^2 )/((c + d/x^2)^2*c - 2*(c + d/x^2)*c^2 + c^3))*b
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.16 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, a x^{2} \mathrm {sgn}\left (x\right ) + \frac {6 \, b c^{4} \mathrm {sgn}\left (x\right ) + a c^{3} d \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x^{2} + \frac {3 \, {\left (2 \, b c^{3} d \mathrm {sgn}\left (x\right ) - a c^{2} d^{2} \mathrm {sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt {c x^{2} + d} x + \frac {{\left (2 \, b c d^{2} \mathrm {sgn}\left (x\right ) - a d^{3} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right )}{16 \, c^{\frac {5}{2}}} - \frac {{\left (2 \, b c d^{2} \log \left ({\left | d \right |}\right ) - a d^{3} \log \left ({\left | d \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{32 \, c^{\frac {5}{2}}} \]
1/48*(2*(4*a*x^2*sgn(x) + (6*b*c^4*sgn(x) + a*c^3*d*sgn(x))/c^4)*x^2 + 3*( 2*b*c^3*d*sgn(x) - a*c^2*d^2*sgn(x))/c^4)*sqrt(c*x^2 + d)*x + 1/16*(2*b*c* d^2*sgn(x) - a*d^3*sgn(x))*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))/c^(5/2) - 1/32*(2*b*c*d^2*log(abs(d)) - a*d^3*log(abs(d)))*sgn(x)/c^(5/2)
Time = 10.01 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09 \[ \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^5 \, dx=\frac {a\,x^6\,\sqrt {c+\frac {d}{x^2}}}{16}+\frac {b\,x^4\,\sqrt {c+\frac {d}{x^2}}}{8}+\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{6\,c}-\frac {a\,x^6\,{\left (c+\frac {d}{x^2}\right )}^{5/2}}{16\,c^2}+\frac {b\,x^4\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{8\,c}-\frac {b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{8\,c^{3/2}}-\frac {a\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c+\frac {d}{x^2}}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{16\,c^{5/2}} \]