Integrand size = 23, antiderivative size = 87 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=-\frac {2 b (d x)^{1+m} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},2+m,\frac {5}{2},1+\frac {b}{a \sqrt {c x^2}}\right )}{3 a^2 d \sqrt {c x^2}} \]
-2/3*b*(d*x)^(1+m)*hypergeom([3/2, 2+m],[5/2],1+b/a/(c*x^2)^(1/2))*(a+b/(c *x^2)^(1/2))^(3/2)*(-b/a/(c*x^2)^(1/2))^m/a^2/d/(c*x^2)^(1/2)
Time = 1.40 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\frac {2 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}+m,\frac {3}{2}+m,-\frac {a \sqrt {c x^2}}{b}\right )}{(1+2 m) \sqrt {1+\frac {a \sqrt {c x^2}}{b}}} \]
(2*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x^2]]*Hypergeometric2F1[-1/2, 1/2 + m, 3/2 + m, -((a*Sqrt[c*x^2])/b)])/((1 + 2*m)*Sqrt[1 + (a*Sqrt[c*x^2])/b])
Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {892, 862, 77, 75}
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.
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx\) |
\(\Big \downarrow \) 892 |
\(\displaystyle \frac {\left (c x^2\right )^{\frac {1}{2} (-m-1)} (d x)^{m+1} \int \left (c x^2\right )^{m/2} \sqrt {a+\frac {b}{\sqrt {c x^2}}}d\sqrt {c x^2}}{d}\) |
\(\Big \downarrow \) 862 |
\(\displaystyle -\frac {\left (c x^2\right )^{\frac {1}{2} (-m-1)} (d x)^{m+1} \int \left (c x^2\right )^{\frac {1}{2} (-m-2)} \sqrt {a+\frac {b}{\sqrt {c x^2}}}d\frac {1}{\sqrt {c x^2}}}{d}\) |
\(\Big \downarrow \) 77 |
\(\displaystyle -\frac {b^2 \left (c x^2\right )^{\frac {1}{2} (-m-1)-\frac {m}{2}} (d x)^{m+1} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \int \left (-\frac {b}{a \sqrt {c x^2}}\right )^{-m-2} \sqrt {a+\frac {b}{\sqrt {c x^2}}}d\frac {1}{\sqrt {c x^2}}}{a^2 d}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {2 b \left (c x^2\right )^{\frac {1}{2} (-m-1)-\frac {m}{2}} (d x)^{m+1} \left (a+\frac {b}{\sqrt {c x^2}}\right )^{3/2} \left (-\frac {b}{a \sqrt {c x^2}}\right )^m \operatorname {Hypergeometric2F1}\left (\frac {3}{2},m+2,\frac {5}{2},\frac {b}{a \sqrt {c x^2}}+1\right )}{3 a^2 d}\) |
(-2*b*(d*x)^(1 + m)*(c*x^2)^((-1 - m)/2 - m/2)*(-(b/(a*Sqrt[c*x^2])))^m*(a + b/Sqrt[c*x^2])^(3/2)*Hypergeometric2F1[3/2, 2 + m, 5/2, 1 + b/(a*Sqrt[c *x^2])])/(3*a^2*d)
3.30.57.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((-b)*(c/ d))^IntPart[m]*((b*x)^FracPart[m]/((-d)*(x/c))^FracPart[m]) Int[((-d)*(x/ c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-c^ (-1))*(c*x)^(m + 1)*(1/x)^(m + 1) Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] && !RationalQ[m]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(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)]
\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\sqrt {c \,x^{2}}}}d x\]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}}\, dx \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{2}}}} \,d x } \]
Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^2}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^2}}} \,d x \]