Integrand size = 27, antiderivative size = 83 \[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=-\frac {2 \sqrt {2} \sqrt {1-d x} (e+f x)^n \left (\frac {d (e+f x)}{d e+f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-d x),\frac {f (1-d x)}{d e+f}\right )}{d} \] Output:
-2*2^(1/2)*(-d*x+1)^(1/2)*(f*x+e)^n*AppellF1(1/2,-n,-1/2,3/2,f*(-d*x+1)/(d *e+f),-1/2*d*x+1/2)/d/((d*(f*x+e)/(d*e+f))^n)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx \] Input:
Integrate[((1 + d*x)*(e + f*x)^n)/Sqrt[1 - d^2*x^2],x]
Output:
Integrate[((1 + d*x)*(e + f*x)^n)/Sqrt[1 - d^2*x^2], x]
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {717, 156, 155}
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.
\(\displaystyle \int \frac {(d x+1) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx\) |
\(\Big \downarrow \) 717 |
\(\displaystyle \int \frac {\sqrt {d x+1} (e+f x)^n}{\sqrt {1-d x}}dx\) |
\(\Big \downarrow \) 156 |
\(\displaystyle (e+f x)^n \left (\frac {d (e+f x)}{d e+f}\right )^{-n} \int \frac {\sqrt {d x+1} \left (\frac {d e}{d e+f}+\frac {d f x}{d e+f}\right )^n}{\sqrt {1-d x}}dx\) |
\(\Big \downarrow \) 155 |
\(\displaystyle -\frac {2 \sqrt {2} \sqrt {1-d x} (e+f x)^n \left (\frac {d (e+f x)}{d e+f}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-d x),\frac {f (1-d x)}{d e+f}\right )}{d}\) |
Input:
Int[((1 + d*x)*(e + f*x)^n)/Sqrt[1 - d^2*x^2],x]
Output:
(-2*Sqrt[2]*Sqrt[1 - d*x]*(e + f*x)^n*AppellF1[1/2, -1/2, -n, 3/2, (1 - d* x)/2, (f*(1 - d*x))/(d*e + f)])/(d*((d*(e + f*x))/(d*e + f))^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[a , 0] && GtQ[d, 0]
\[\int \frac {\left (d x +1\right ) \left (f x +e \right )^{n}}{\sqrt {-d^{2} x^{2}+1}}d x\]
Input:
int((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x)
Output:
int((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:
integrate((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-d^2*x^2 + 1)*(f*x + e)^n/(d*x - 1), x)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {\left (e + f x\right )^{n} \left (d x + 1\right )}{\sqrt {- \left (d x - 1\right ) \left (d x + 1\right )}}\, dx \] Input:
integrate((d*x+1)*(f*x+e)**n/(-d**2*x**2+1)**(1/2),x)
Output:
Integral((e + f*x)**n*(d*x + 1)/sqrt(-(d*x - 1)*(d*x + 1)), x)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:
integrate((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + 1)*(f*x + e)^n/sqrt(-d^2*x^2 + 1), x)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int { \frac {{\left (d x + 1\right )} {\left (f x + e\right )}^{n}}{\sqrt {-d^{2} x^{2} + 1}} \,d x } \] Input:
integrate((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + 1)*(f*x + e)^n/sqrt(-d^2*x^2 + 1), x)
Timed out. \[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {{\left (e+f\,x\right )}^n\,\left (d\,x+1\right )}{\sqrt {1-d^2\,x^2}} \,d x \] Input:
int(((e + f*x)^n*(d*x + 1))/(1 - d^2*x^2)^(1/2),x)
Output:
int(((e + f*x)^n*(d*x + 1))/(1 - d^2*x^2)^(1/2), x)
\[ \int \frac {(1+d x) (e+f x)^n}{\sqrt {1-d^2 x^2}} \, dx=\int \frac {\left (f x +e \right )^{n}}{\sqrt {-d^{2} x^{2}+1}}d x +\left (\int \frac {\left (f x +e \right )^{n} x}{\sqrt {-d^{2} x^{2}+1}}d x \right ) d \] Input:
int((d*x+1)*(f*x+e)^n/(-d^2*x^2+1)^(1/2),x)
Output:
int((e + f*x)**n/sqrt( - d**2*x**2 + 1),x) + int(((e + f*x)**n*x)/sqrt( - d**2*x**2 + 1),x)*d