Integrand size = 21, antiderivative size = 76 \[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\frac {(d x)^{1+m} \sqrt {a+b \sqrt {c x}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2 (1+m),3+2 m,-\frac {b \sqrt {c x}}{a}\right )}{d (1+m) \sqrt {1+\frac {b \sqrt {c x}}{a}}} \] Output:
(d*x)^(1+m)*(a+b*(c*x)^(1/2))^(1/2)*hypergeom([-1/2, 2+2*m],[3+2*m],-b*(c* x)^(1/2)/a)/d/(1+m)/(1+b*(c*x)^(1/2)/a)^(1/2)
Time = 0.86 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\frac {4 (d x)^m \left (-\frac {b \sqrt {c x}}{a}\right )^{-2 m} \left (a+b \sqrt {c x}\right )^{3/2} \left (-5 a \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 m,\frac {5}{2},1+\frac {b \sqrt {c x}}{a}\right )+3 \left (a+b \sqrt {c x}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-2 m,\frac {7}{2},1+\frac {b \sqrt {c x}}{a}\right )\right )}{15 b^2 c} \] Input:
Integrate[(d*x)^m*Sqrt[a + b*Sqrt[c*x]],x]
Output:
(4*(d*x)^m*(a + b*Sqrt[c*x])^(3/2)*(-5*a*Hypergeometric2F1[3/2, -2*m, 5/2, 1 + (b*Sqrt[c*x])/a] + 3*(a + b*Sqrt[c*x])*Hypergeometric2F1[5/2, -2*m, 7 /2, 1 + (b*Sqrt[c*x])/a]))/(15*b^2*c*(-((b*Sqrt[c*x])/a))^(2*m))
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {891, 866, 864, 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.
\(\displaystyle \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int (d x)^m \sqrt {a+b \sqrt {c x}}d(c x)}{c}\) |
\(\Big \downarrow \) 866 |
\(\displaystyle \frac {(c x)^{-m} (d x)^m \int (c x)^m \sqrt {a+b \sqrt {c x}}d(c x)}{c}\) |
\(\Big \downarrow \) 864 |
\(\displaystyle \frac {2 (c x)^{-m} (d x)^m \int (c x)^{\frac {1}{2} (2 m+1)} \sqrt {a+b \sqrt {c x}}d\sqrt {c x}}{c}\) |
\(\Big \downarrow \) 77 |
\(\displaystyle -\frac {2 a (d x)^m \left (-\frac {b \sqrt {c x}}{a}\right )^{-2 m} \int \left (-\frac {b \sqrt {c x}}{a}\right )^{2 m+1} \sqrt {a+b \sqrt {c x}}d\sqrt {c x}}{b c}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {4 a (d x)^m \left (a+b \sqrt {c x}\right )^{3/2} \left (-\frac {b \sqrt {c x}}{a}\right )^{-2 m} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-2 m-1,\frac {5}{2},\frac {\sqrt {c x} b}{a}+1\right )}{3 b^2 c}\) |
Input:
Int[(d*x)^m*Sqrt[a + b*Sqrt[c*x]],x]
Output:
(-4*a*(d*x)^m*(a + b*Sqrt[c*x])^(3/2)*Hypergeometric2F1[3/2, -1 - 2*m, 5/2 , 1 + (b*Sqrt[c*x])/a])/(3*b^2*c*(-((b*Sqrt[c*x])/a))^(2*m))
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[(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[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[1/c Subst[Int[(d*(x/c))^m*(a + b*x^n)^p, x], x, c*x], x] /; FreeQ[{a , b, c, d, m, n, p}, x]
\[\int \left (d x \right )^{m} \sqrt {a +b \sqrt {c x}}d x\]
Input:
int((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x)
Output:
int((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x)
Exception generated. \[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: algl ogextint: unimplemented
\[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\int \left (d x\right )^{m} \sqrt {a + b \sqrt {c x}}\, dx \] Input:
integrate((d*x)**m*(a+b*(c*x)**(1/2))**(1/2),x)
Output:
Integral((d*x)**m*sqrt(a + b*sqrt(c*x)), x)
\[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\int { \sqrt {\sqrt {c x} b + a} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(sqrt(c*x)*b + a)*(d*x)^m, x)
\[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\int { \sqrt {\sqrt {c x} b + a} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sqrt(c*x)*b + a)*(d*x)^m, x)
Timed out. \[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\int \sqrt {a+b\,\sqrt {c\,x}}\,{\left (d\,x\right )}^m \,d x \] Input:
int((a + b*(c*x)^(1/2))^(1/2)*(d*x)^m,x)
Output:
int((a + b*(c*x)^(1/2))^(1/2)*(d*x)^m, x)
\[ \int (d x)^m \sqrt {a+b \sqrt {c x}} \, dx=\frac {4 d^{m} \left (x^{m +\frac {1}{2}} \sqrt {c}\, \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}\, a b -4 x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}\, a^{2} m -2 x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}\, a^{2}+4 x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}\, b^{2} c m x +3 x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}\, b^{2} c x +4 \left (\int \frac {x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}}{x}d x \right ) a^{2} m^{2}+2 \left (\int \frac {x^{m} \sqrt {\sqrt {x}\, \sqrt {c}\, b +a}}{x}d x \right ) a^{2} m \right )}{b^{2} c \left (16 m^{2}+32 m +15\right )} \] Input:
int((d*x)^m*(a+b*(c*x)^(1/2))^(1/2),x)
Output:
(4*d**m*(x**((2*m + 1)/2)*sqrt(c)*sqrt(sqrt(x)*sqrt(c)*b + a)*a*b - 4*x**m *sqrt(sqrt(x)*sqrt(c)*b + a)*a**2*m - 2*x**m*sqrt(sqrt(x)*sqrt(c)*b + a)*a **2 + 4*x**m*sqrt(sqrt(x)*sqrt(c)*b + a)*b**2*c*m*x + 3*x**m*sqrt(sqrt(x)* sqrt(c)*b + a)*b**2*c*x + 4*int((x**m*sqrt(sqrt(x)*sqrt(c)*b + a))/x,x)*a* *2*m**2 + 2*int((x**m*sqrt(sqrt(x)*sqrt(c)*b + a))/x,x)*a**2*m))/(b**2*c*( 16*m**2 + 32*m + 15))