Integrand size = 20, antiderivative size = 147 \[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\frac {(c+d x)^{1+n} \sqrt {a-b x^2} \operatorname {AppellF1}\left (1+n,-\frac {1}{2},-\frac {1}{2},2+n,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d (1+n) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}}} \] Output:
(d*x+c)^(1+n)*(-b*x^2+a)^(1/2)*AppellF1(1+n,-1/2,-1/2,2+n,(d*x+c)/(c-a^(1/ 2)*d/b^(1/2)),(d*x+c)/(c+a^(1/2)*d/b^(1/2)))/d/(1+n)/(1-(d*x+c)/(c-a^(1/2) *d/b^(1/2)))^(1/2)/(1-(d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)
Time = 0.01 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.05 \[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\frac {(c+d x)^{1+n} \sqrt {a-b x^2} \operatorname {AppellF1}\left (1+n,-\frac {1}{2},-\frac {1}{2},2+n,\frac {c+d x}{c-\sqrt {\frac {a}{b}} d},\frac {c+d x}{c+\sqrt {\frac {a}{b}} d}\right )}{d (1+n) \sqrt {\frac {d \left (\sqrt {\frac {a}{b}}-x\right )}{c+\sqrt {\frac {a}{b}} d}} \sqrt {\frac {d \left (\sqrt {\frac {a}{b}}+x\right )}{-c+\sqrt {\frac {a}{b}} d}}} \] Input:
Integrate[(c + d*x)^n*Sqrt[a - b*x^2],x]
Output:
((c + d*x)^(1 + n)*Sqrt[a - b*x^2]*AppellF1[1 + n, -1/2, -1/2, 2 + n, (c + d*x)/(c - Sqrt[a/b]*d), (c + d*x)/(c + Sqrt[a/b]*d)])/(d*(1 + n)*Sqrt[(d* (Sqrt[a/b] - x))/(c + Sqrt[a/b]*d)]*Sqrt[(d*(Sqrt[a/b] + x))/(-c + Sqrt[a/ b]*d)])
Time = 0.24 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {514, 150}
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.
\(\displaystyle \int \sqrt {a-b x^2} (c+d x)^n \, dx\) |
\(\Big \downarrow \) 514 |
\(\displaystyle \frac {\sqrt {a-b x^2} \int (c+d x)^n \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}}d(c+d x)}{d \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\sqrt {a-b x^2} (c+d x)^{n+1} \operatorname {AppellF1}\left (n+1,-\frac {1}{2},-\frac {1}{2},n+2,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d (n+1) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}}}\) |
Input:
Int[(c + d*x)^n*Sqrt[a - b*x^2],x]
Output:
((c + d*x)^(1 + n)*Sqrt[a - b*x^2]*AppellF1[1 + n, -1/2, -1/2, 2 + n, (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])/(d* (1 + n)*Sqrt[1 - (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b])]*Sqrt[1 - (c + d*x)/( c + (Sqrt[a]*d)/Sqrt[b])])
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( c + d*x)/(c + d*q))^p) Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c^2 + a*d^2, 0]
\[\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}d x\]
Input:
int((d*x+c)^n*(-b*x^2+a)^(1/2),x)
Output:
int((d*x+c)^n*(-b*x^2+a)^(1/2),x)
\[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((d*x+c)^n*(-b*x^2+a)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(-b*x^2 + a)*(d*x + c)^n, x)
\[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int \sqrt {a - b x^{2}} \left (c + d x\right )^{n}\, dx \] Input:
integrate((d*x+c)**n*(-b*x**2+a)**(1/2),x)
Output:
Integral(sqrt(a - b*x**2)*(c + d*x)**n, x)
\[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((d*x+c)^n*(-b*x^2+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^n, x)
\[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int { \sqrt {-b x^{2} + a} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((d*x+c)^n*(-b*x^2+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-b*x^2 + a)*(d*x + c)^n, x)
Timed out. \[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int \sqrt {a-b\,x^2}\,{\left (c+d\,x\right )}^n \,d x \] Input:
int((a - b*x^2)^(1/2)*(c + d*x)^n,x)
Output:
int((a - b*x^2)^(1/2)*(c + d*x)^n, x)
\[ \int (c+d x)^n \sqrt {a-b x^2} \, dx=\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}d x \] Input:
int((d*x+c)^n*(-b*x^2+a)^(1/2),x)
Output:
int((c + d*x)**n*sqrt(a - b*x**2),x)