\(\int (c+d x)^n (a-b x^2)^{3/2} \, dx\) [264]

Optimal result

Integrand size = 20, antiderivative size = 147 \[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\frac {(c+d x)^{1+n} \left (a-b x^2\right )^{3/2} \operatorname {AppellF1}\left (1+n,-\frac {3}{2},-\frac {3}{2},2+n,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d (1+n) \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2}} \] Output:

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

Mathematica [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx \] Input:

Integrate[(c + d*x)^n*(a - b*x^2)^(3/2),x]
 

Output:

Integrate[(c + d*x)^n*(a - b*x^2)^(3/2), x]
 

Rubi [A] (verified)

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.

\(\displaystyle \int \left (a-b x^2\right )^{3/2} (c+d x)^n \, dx\)

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

Int[(c + d*x)^n*(a - b*x^2)^(3/2),x]
 

Output:

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

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]
 

Maple [F]

\[\int \left (d x +c \right )^{n} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int \left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{n}\, dx \] Input:

integrate((d*x+c)**n*(-b*x**2+a)**(3/2),x)
 

Output:

Integral((a - b*x**2)**(3/2)*(c + d*x)**n, x)
 

Maxima [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int {\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^n \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=-\left (\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}\, x^{2}d x \right ) b +\left (\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}d x \right ) a \] Input:

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

Output:

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