\(\int (d x)^m \sqrt {a+b \sqrt {c x^2}} \, dx\) [50]

Optimal result

Integrand size = 23, antiderivative size = 78 \[ \int (d x)^m \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {(d x)^{1+m} \sqrt {a+b \sqrt {c x^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,-\frac {b \sqrt {c x^2}}{a}\right )}{d (1+m) \sqrt {1+\frac {b \sqrt {c x^2}}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int (d x)^m \sqrt {a+b \sqrt {c x^2}} \, dx=\frac {x (d x)^m \sqrt {a+b \sqrt {c x^2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+m,2+m,-\frac {b \sqrt {c x^2}}{a}\right )}{(1+m) \sqrt {1+\frac {b \sqrt {c x^2}}{a}}} \] Input:

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

Output:

(x*(d*x)^m*Sqrt[a + b*Sqrt[c*x^2]]*Hypergeometric2F1[-1/2, 1 + m, 2 + m, - 
((b*Sqrt[c*x^2])/a)])/((1 + m)*Sqrt[1 + (b*Sqrt[c*x^2])/a])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.28, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {892, 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.

\(\displaystyle \int (d x)^m \sqrt {a+b \sqrt {c x^2}} \, dx\)

\(\Big \downarrow \) 892

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

\(\Big \downarrow \) 77

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

\(\Big \downarrow \) 75

\(\displaystyle \frac {2 \left (c x^2\right )^{\frac {1}{2} (-m-1)+\frac {m}{2}} (d x)^{m+1} \left (a+b \sqrt {c x^2}\right )^{3/2} \left (-\frac {b \sqrt {c x^2}}{a}\right )^{-m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-m,\frac {5}{2},\frac {\sqrt {c x^2} b}{a}+1\right )}{3 b d}\)

Input:

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

Output:

(2*(d*x)^(1 + m)*(c*x^2)^((-1 - m)/2 + m/2)*(a + b*Sqrt[c*x^2])^(3/2)*Hype 
rgeometric2F1[3/2, -m, 5/2, 1 + (b*Sqrt[c*x^2])/a])/(3*b*d*(-((b*Sqrt[c*x^ 
2])/a))^m)
 

Defintions of rubi rules used

rule 75

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])
 

rule 77

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]
 

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)]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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