Integrand size = 23, antiderivative size = 102 \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\frac {x (d x)^m \sqrt {a+\frac {b c^3 x^9}{\left (c x^3\right )^{9/2}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{9} (1+m),\frac {1}{9} (7-2 m),-\frac {b c^3 x^9}{a \left (c x^3\right )^{9/2}}\right )}{(1+m) \sqrt {1+\frac {b c^3 x^9}{a \left (c x^3\right )^{9/2}}}} \]
x*(d*x)^m*hypergeom([-1/2, -2/9-2/9*m],[7/9-2/9*m],-b*c^3*x^9/a/(c*x^3)^(9 /2))*(a+b*c^3*x^9/(c*x^3)^(9/2))^(1/2)/(1+m)/(1+b*c^3*x^9/a/(c*x^3)^(9/2)) ^(1/2)
\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx \]
Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.51, 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.
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx\) |
\(\Big \downarrow \) 893 |
\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{c^{3/2} x^{9/2}}}dx\) |
\(\Big \downarrow \) 866 |
\(\displaystyle x^{-m} (d x)^m \int \sqrt {a+\frac {b}{c^{3/2} x^{9/2}}} x^mdx\) |
\(\Big \downarrow \) 864 |
\(\displaystyle 2 x^{-m} (d x)^m \int \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m+1} \sqrt {\frac {b c^3 x^9}{\left (c x^3\right )^{9/2}}+a}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\) |
\(\Big \downarrow \) 862 |
\(\displaystyle -2 x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \int \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{-2 m-3} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{c^6 x^9}+a}d\frac {\sqrt {c} x}{\sqrt {c x^3}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle -\frac {2 x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^6 x^9}} \int \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{-2 m-3} \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^6 x^9}+1}d\frac {\sqrt {c} x}{\sqrt {c x^3}}}{\sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^6 x^9}+1}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {x^{-m} \left (\frac {\sqrt {c} x}{\sqrt {c x^3}}\right )^{2 m-2 (m+1)} \left (\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )^{2 m} (d x)^m \sqrt {a+\frac {b \left (c x^3\right )^{9/2}}{c^6 x^9}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{9} (m+1),\frac {1}{9} (7-2 m),-\frac {b \left (c x^3\right )^{9/2}}{a c^6 x^9}\right )}{(m+1) \sqrt {\frac {b \left (c x^3\right )^{9/2}}{a c^6 x^9}+1}}\) |
((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)^(9/2))/(c^6*x^9)]*Hypergeometric2F1[-1/2, ( -2*(1 + m))/9, (7 - 2*m)/9, -((b*(c*x^3)^(9/2))/(a*c^6*x^9))])/((1 + m)*x^ m*Sqrt[1 + (b*(c*x^3)^(9/2))/(a*c^6*x^9)])
3.30.81.3.1 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]
\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\left (c \,x^{3}\right )^{\frac {3}{2}}}}d x\]
Exception generated. \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: algl ogextint: unimplemented
\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{3}\right )^{\frac {3}{2}}}}\, dx \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{3}\right )^{\frac {3}{2}}}} \,d x } \]
\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{3}\right )^{\frac {3}{2}}}} \,d x } \]
Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^3\right )^{3/2}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{{\left (c\,x^3\right )}^{3/2}}} \,d x \]