Integrand size = 24, antiderivative size = 81 \[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {2^{\frac {1}{2}+n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-\frac {1}{2}-n} \sqrt {c^2-d^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {c-d x}{2 c}\right )}{c d} \] Output:
-2^(1/2+n)*(d*x+c)^n*(1+d*x/c)^(-1/2-n)*(-d^2*x^2+c^2)^(1/2)*hypergeom([1/ 2, 1/2-n],[3/2],1/2*(-d*x+c)/c)/c/d
Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=-\frac {2^{\frac {1}{2}+n} (c-d x) (c+d x)^n \left (1+\frac {d x}{c}\right )^{\frac {1}{2}-n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {c-d x}{2 c}\right )}{d \sqrt {c^2-d^2 x^2}} \] Input:
Integrate[(c + d*x)^n/Sqrt[c^2 - d^2*x^2],x]
Output:
-((2^(1/2 + n)*(c - d*x)*(c + d*x)^n*(1 + (d*x)/c)^(1/2 - n)*Hypergeometri c2F1[1/2, 1/2 - n, 3/2, (c - d*x)/(2*c)])/(d*Sqrt[c^2 - d^2*x^2]))
Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {474, 473, 79}
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.
\(\displaystyle \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 474 |
\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \int \frac {\left (\frac {d x}{c}+1\right )^n}{\sqrt {c^2-d^2 x^2}}dx\) |
\(\Big \downarrow \) 473 |
\(\displaystyle \frac {\sqrt {c^2-d^2 x^2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n-\frac {1}{2}} \int \frac {\left (\frac {d x}{c}+1\right )^{n-\frac {1}{2}}}{\sqrt {c^2-c d x}}dx}{\sqrt {c^2-c d x}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{n+\frac {1}{2}} \sqrt {c^2-d^2 x^2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-n,\frac {3}{2},\frac {c-d x}{2 c}\right )}{c d}\) |
Input:
Int[(c + d*x)^n/Sqrt[c^2 - d^2*x^2],x]
Output:
-((2^(1/2 + n)*(c + d*x)^n*(1 + (d*x)/c)^(-1/2 - n)*Sqrt[c^2 - d^2*x^2]*Hy pergeometric2F1[1/2, 1/2 - n, 3/2, (c - d*x)/(2*c)])/(c*d))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 1))) Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && !Gt Q[a, 0] && !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(1 + d *(x/c))^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && !(IntegerQ[n] || GtQ[c, 0])
\[\int \frac {\left (d x +c \right )^{n}}{\sqrt {-d^{2} x^{2}+c^{2}}}d x\]
Input:
int((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x)
Output:
int((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x)
\[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:
integrate((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-d^2*x^2 + c^2)*(d*x + c)^n/(d^2*x^2 - c^2), x)
\[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\left (c + d x\right )^{n}}{\sqrt {- \left (- c + d x\right ) \left (c + d x\right )}}\, dx \] Input:
integrate((d*x+c)**n/(-d**2*x**2+c**2)**(1/2),x)
Output:
Integral((c + d*x)**n/sqrt(-(-c + d*x)*(c + d*x)), x)
\[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:
integrate((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)^n/sqrt(-d^2*x^2 + c^2), x)
\[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n}}{\sqrt {-d^{2} x^{2} + c^{2}}} \,d x } \] Input:
integrate((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)^n/sqrt(-d^2*x^2 + c^2), x)
Timed out. \[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int \frac {{\left (c+d\,x\right )}^n}{\sqrt {c^2-d^2\,x^2}} \,d x \] Input:
int((c + d*x)^n/(c^2 - d^2*x^2)^(1/2),x)
Output:
int((c + d*x)^n/(c^2 - d^2*x^2)^(1/2), x)
\[ \int \frac {(c+d x)^n}{\sqrt {c^2-d^2 x^2}} \, dx=\int \frac {\left (d x +c \right )^{n}}{\sqrt {-d^{2} x^{2}+c^{2}}}d x \] Input:
int((d*x+c)^n/(-d^2*x^2+c^2)^(1/2),x)
Output:
int((c + d*x)**n/sqrt(c**2 - d**2*x**2),x)