3.30.58 \(\int (d x)^m \sqrt {a+\frac {b}{(c x^2)^{3/2}}} \, dx\) [2958]

3.30.58.1 Optimal result

Integrand size = 23, antiderivative size = 90 \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\frac {(d x)^{1+m} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3} (-1-m),\frac {2-m}{3},-\frac {b}{a \left (c x^2\right )^{3/2}}\right )}{d (1+m) \sqrt {1+\frac {b}{a \left (c x^2\right )^{3/2}}}} \]

output

(d*x)^(1+m)*hypergeom([-1/2, -1/3-1/3*m],[2/3-1/3*m],-b/a/(c*x^2)^(3/2))*( 
a+b/(c*x^2)^(3/2))^(1/2)/d/(1+m)/(1+b/a/(c*x^2)^(3/2))^(1/2)
 

3.30.58.2 Mathematica [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx \]

input

Integrate[(d*x)^m*Sqrt[a + b/(c*x^2)^(3/2)],x]
 

output

Integrate[(d*x)^m*Sqrt[a + b/(c*x^2)^(3/2)], x]
 

3.30.58.3 Rubi [A] (warning: unable to verify)

Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {892, 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.

\(\displaystyle \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx\)

\(\Big \downarrow \) 892

\(\displaystyle \frac {\left (c x^2\right )^{\frac {1}{2} (-m-1)} (d x)^{m+1} \int \left (c x^2\right )^{m/2} \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}}d\sqrt {c x^2}}{d}\)

\(\Big \downarrow \) 862

\(\displaystyle -\frac {\left (c x^2\right )^{\frac {1}{2} (-m-1)} (d x)^{m+1} \int \left (c x^2\right )^{\frac {1}{2} (-m-2)} \sqrt {b \left (c x^2\right )^{3/2}+a}d\frac {1}{\sqrt {c x^2}}}{d}\)

\(\Big \downarrow \) 889

\(\displaystyle -\frac {\left (c x^2\right )^{\frac {1}{2} (-m-1)} (d x)^{m+1} \sqrt {a+b \left (c x^2\right )^{3/2}} \int \left (c x^2\right )^{\frac {1}{2} (-m-2)} \sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}d\frac {1}{\sqrt {c x^2}}}{d \sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\left (c x^2\right )^{-m-1} (d x)^{m+1} \sqrt {a+b \left (c x^2\right )^{3/2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{3} (-m-1),\frac {2-m}{3},-\frac {b \left (c x^2\right )^{3/2}}{a}\right )}{d (m+1) \sqrt {\frac {b \left (c x^2\right )^{3/2}}{a}+1}}\)

input

Int[(d*x)^m*Sqrt[a + b/(c*x^2)^(3/2)],x]
 

output

((d*x)^(1 + m)*(c*x^2)^(-1 - m)*Sqrt[a + b*(c*x^2)^(3/2)]*Hypergeometric2F 
1[-1/2, (-1 - m)/3, (2 - m)/3, -((b*(c*x^2)^(3/2))/a)])/(d*(1 + m)*Sqrt[1 
+ (b*(c*x^2)^(3/2))/a])
 

3.30.58.3.1 Defintions of rubi rules used

rule 862

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]
 

rule 888

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])
 

rule 889

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])
 

rule 892

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)]
 

3.30.58.4 Maple [F]

\[\int \left (d x \right )^{m} \sqrt {a +\frac {b}{\left (c \,x^{2}\right )^{\frac {3}{2}}}}d x\]

input

int((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x)
 

output

int((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x)
 

3.30.58.5 Fricas [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{2}\right )^{\frac {3}{2}}}} \,d x } \]

input

integrate((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x, algorithm="fricas")
 

output

integral((d*x)^m*sqrt((a*c^2*x^4 + sqrt(c*x^2)*b)/(c^2*x^4)), x)
 

3.30.58.6 Sympy [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\int \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{2}\right )^{\frac {3}{2}}}}\, dx \]

input

integrate((d*x)**m*(a+b/(c*x**2)**(3/2))**(1/2),x)
 

output

Integral((d*x)**m*sqrt(a + b/(c*x**2)**(3/2)), x)
 

3.30.58.7 Maxima [F]

\[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\int { \left (d x\right )^{m} \sqrt {a + \frac {b}{\left (c x^{2}\right )^{\frac {3}{2}}}} \,d x } \]

input

integrate((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x, algorithm="maxima")
 

output

integrate((d*x)^m*sqrt(a + b/(c*x^2)^(3/2)), x)
 

3.30.58.8 Giac [F(-2)]

Exception generated. \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*x)^m*(a+b/(c*x^2)^(3/2))^(1/2),x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(con 
st gen &
 

3.30.58.9 Mupad [F(-1)]

Timed out. \[ \int (d x)^m \sqrt {a+\frac {b}{\left (c x^2\right )^{3/2}}} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a+\frac {b}{{\left (c\,x^2\right )}^{3/2}}} \,d x \]

input

int((d*x)^m*(a + b/(c*x^2)^(3/2))^(1/2),x)
 

output

int((d*x)^m*(a + b/(c*x^2)^(3/2))^(1/2), x)