\(\int x (c+d x)^n \sqrt {a-b x^2} \, dx\) [257]

Optimal result

Integrand size = 21, antiderivative size = 168 \[ \int x (c+d x)^n \sqrt {a-b x^2} \, dx=-\frac {(c+d x)^n \left (a-b x^2\right )^{3/2}}{3 b}+\frac {(c+d x)^n \left (a-b x^2\right )^{3/2} \operatorname {AppellF1}\left (n,-\frac {3}{2},-\frac {3}{2},1+n,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{3 b \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2}} \] Output:

-1/3*(d*x+c)^n*(-b*x^2+a)^(3/2)/b+1/3*(d*x+c)^n*(-b*x^2+a)^(3/2)*AppellF1( 
n,-3/2,-3/2,1+n,(d*x+c)/(c-a^(1/2)*d/b^(1/2)),(d*x+c)/(c+a^(1/2)*d/b^(1/2) 
))/b/(1-(d*x+c)/(c-a^(1/2)*d/b^(1/2)))^(3/2)/(1-(d*x+c)/(c+a^(1/2)*d/b^(1/ 
2)))^(3/2)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.36 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {624, 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.

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

\(\Big \downarrow \) 624

\(\displaystyle \frac {\int (c+d x)^{n+1} \sqrt {a-b x^2}dx}{d}-\frac {c \int (c+d x)^n \sqrt {a-b x^2}dx}{d}\)

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\sqrt {a-b x^2} (c+d x)^{n+2} \operatorname {AppellF1}\left (n+2,-\frac {1}{2},-\frac {1}{2},n+3,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d^2 (n+2) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}}}-\frac {c \sqrt {a-b x^2} (c+d x)^{n+1} \operatorname {AppellF1}\left (n+1,-\frac {1}{2},-\frac {1}{2},n+2,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d^2 (n+1) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}}}\)

Input:

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

Output:

-((c*(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^2*(1 + n)*Sqrt[1 - (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b])]*Sqrt[1 - (c + 
d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])) + ((c + d*x)^(2 + n)*Sqrt[a - b*x^2]*App 
ellF1[2 + n, -1/2, -1/2, 3 + n, (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]), (c + 
d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])/(d^2*(2 + n)*Sqrt[1 - (c + d*x)/(c - (Sqr 
t[a]*d)/Sqrt[b])]*Sqrt[1 - (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])
 

Defintions of rubi rules used

rule 150

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])
 

rule 514

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]
 

rule 624

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[1/d   Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si 
mp[c/d   Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, 
 d, n, p}, x] && IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(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, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((c + d*x)**n*sqrt(a - b*x**2)*x,x)