Integrand size = 23, antiderivative size = 84 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (1+m),\frac {1}{3} (1-2 m),-\frac {b}{a \sqrt {c x^3}}\right )}{(1+m) \sqrt {1+\frac {b}{a \sqrt {c x^3}}}} \]
x*(d*x)^m*hypergeom([-1/2, -2/3-2/3*m],[1/3-2/3*m],-b/a/(c*x^3)^(1/2))*(a+ b/(c*x^3)^(1/2))^(1/2)/(1+m)/(1+b/a/(c*x^3)^(1/2))^(1/2)
Time = 1.57 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.06 \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\frac {4 x (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{6}+\frac {2 m}{3},\frac {7}{6}+\frac {2 m}{3},-\frac {a \sqrt {c x^3}}{b}\right )}{(1+4 m) \sqrt {1+\frac {a \sqrt {c x^3}}{b}}} \]
(4*x*(d*x)^m*Sqrt[a + b/Sqrt[c*x^3]]*Hypergeometric2F1[-1/2, 1/6 + (2*m)/3 , 7/6 + (2*m)/3, -((a*Sqrt[c*x^3])/b)])/((1 + 4*m)*Sqrt[1 + (a*Sqrt[c*x^3] )/b])
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {893, 866, 864, 862, 889, 888}
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.
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx\) |
\(\Big \downarrow \) 893 |
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}}dx\) |
\(\Big \downarrow \) 866 |
\(\displaystyle x^{-m} (d x)^m \int \sqrt {a+\frac {b}{\sqrt {c} x^{3/2}}} x^mdx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 2 x^{-m} (d x)^m \int \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m+1} \sqrt {\frac {b c x^3}{\left (c x^3\right )^{3/2}}+a}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\) |
\(\Big \downarrow \) 862 |
\(\displaystyle -2 x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \int \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{-2 m-3} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c^2 x^3}+a}d\frac {\sqrt {c} x}{\sqrt {c x^3}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle -\frac {2 x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c^2 x^3}} \int \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{-2 m-3} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a c^2 x^3}+1}d\frac {\sqrt {c} x}{\sqrt {c x^3}}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{a c^2 x^3}+1}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m-2 (m+1)} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c^2 x^3}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{3} (m+1),\frac {1}{3} (1-2 m),-\frac {b \left (c x^3\right )^{3/2}}{a c^2 x^3}\right )}{(m+1) \sqrt {\frac {b \left (c x^3\right )^{3/2}}{a c^2 x^3}+1}}\) |
((d*x)^m*((Sqrt[c]*x)/Sqrt[c*x^3])^(2*m - 2*(1 + m))*(Sqrt[c*x^3]/(Sqrt[c] *x))^(2*m)*Sqrt[a + (b*(c*x^3)^(3/2))/(c^2*x^3)]*Hypergeometric2F1[-1/2, ( -2*(1 + m))/3, (1 - 2*m)/3, -((b*(c*x^3)^(3/2))/(a*c^2*x^3))])/((1 + m)*x^ m*Sqrt[1 + (b*(c*x^3)^(3/2))/(a*c^2*x^3)])
3.30.80.3.1 Defintions of rubi rules used
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi nator[n]}, Simp[k Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x ^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]
Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^Int Part[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*x^n)^p, x], x] / ; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[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]
\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\sqrt {c \,x^{3}}}}d x\]
Exception generated. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: algl ogextint: unimplemented
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}}\, dx \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\sqrt {c x^{3}}}} \,d x } \]
Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\sqrt {c x^3}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{\sqrt {c\,x^3}}} \,d x \]