\(\int (A+B x) (c^2-d^2 x^2)^p \, dx\) [117]

Optimal result

Integrand size = 20, antiderivative size = 84 \[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=-\frac {B \left (c^2-d^2 x^2\right )^{1+p}}{2 d^2 (1+p)}+A x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right ) \] Output:

-1/2*B*(-d^2*x^2+c^2)^(p+1)/d^2/(p+1)+A*x*(-d^2*x^2+c^2)^p*hypergeom([1/2, 
 -p],[3/2],d^2*x^2/c^2)/((1-d^2*x^2/c^2)^p)
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.43 \[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {\left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \left (B d^2 x^2 \left (1-\frac {d^2 x^2}{c^2}\right )^p-B c^2 \left (-1+\left (1-\frac {d^2 x^2}{c^2}\right )^p\right )+2 A d^2 (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )\right )}{2 d^2 (1+p)} \] Input:

Integrate[(A + B*x)*(c^2 - d^2*x^2)^p,x]
 

Output:

((c^2 - d^2*x^2)^p*(B*d^2*x^2*(1 - (d^2*x^2)/c^2)^p - B*c^2*(-1 + (1 - (d^ 
2*x^2)/c^2)^p) + 2*A*d^2*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (d^2*x^ 
2)/c^2]))/(2*d^2*(1 + p)*(1 - (d^2*x^2)/c^2)^p)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 238, 237}

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.

\(\displaystyle \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx\)

\(\Big \downarrow \) 455

\(\displaystyle A \int \left (c^2-d^2 x^2\right )^pdx-\frac {B \left (c^2-d^2 x^2\right )^{p+1}}{2 d^2 (p+1)}\)

\(\Big \downarrow \) 238

\(\displaystyle A \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \int \left (1-\frac {d^2 x^2}{c^2}\right )^pdx-\frac {B \left (c^2-d^2 x^2\right )^{p+1}}{2 d^2 (p+1)}\)

\(\Big \downarrow \) 237

\(\displaystyle A x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )-\frac {B \left (c^2-d^2 x^2\right )^{p+1}}{2 d^2 (p+1)}\)

Input:

Int[(A + B*x)*(c^2 - d^2*x^2)^p,x]
 

Output:

-1/2*(B*(c^2 - d^2*x^2)^(1 + p))/(d^2*(1 + p)) + (A*x*(c^2 - d^2*x^2)^p*Hy 
pergeometric2F1[1/2, -p, 3/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p
 

Defintions of rubi rules used

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

Maple [F]

\[\int \left (B x +A \right ) \left (-d^{2} x^{2}+c^{2}\right )^{p}d x\]

Input:

int((B*x+A)*(-d^2*x^2+c^2)^p,x)
 

Output:

int((B*x+A)*(-d^2*x^2+c^2)^p,x)
 

Fricas [F]

\[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^p,x, algorithm="fricas")
 

Output:

integral((B*x + A)*(-d^2*x^2 + c^2)^p, x)
 

Sympy [A] (verification not implemented)

Time = 1.15 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98 \[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=A c^{2 p} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + B \left (\begin {cases} \frac {x^{2} \left (c^{2}\right )^{p}}{2} & \text {for}\: d^{2} = 0 \\- \frac {\begin {cases} \frac {\left (c^{2} - d^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (c^{2} - d^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 d^{2}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((B*x+A)*(-d**2*x**2+c**2)**p,x)
 

Output:

A*c**(2*p)*x*hyper((1/2, -p), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 
B*Piecewise((x**2*(c**2)**p/2, Eq(d**2, 0)), (-Piecewise(((c**2 - d**2*x** 
2)**(p + 1)/(p + 1), Ne(p, -1)), (log(c**2 - d**2*x**2), True))/(2*d**2), 
True))
 

Maxima [F]

\[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^p,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(-d^2*x^2 + c^2)^p, x)
 

Giac [F]

\[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:

integrate((B*x+A)*(-d^2*x^2+c^2)^p,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(-d^2*x^2 + c^2)^p, x)
 

Mupad [B] (verification not implemented)

Time = 10.99 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.94 \[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {A\,x\,{\left (c^2-d^2\,x^2\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {d^2\,x^2}{c^2}\right )}{{\left (1-\frac {d^2\,x^2}{c^2}\right )}^p}-\frac {B\,{\left (c^2-d^2\,x^2\right )}^{p+1}}{2\,d^2\,\left (p+1\right )} \] Input:

int((c^2 - d^2*x^2)^p*(A + B*x),x)
 

Output:

(A*x*(c^2 - d^2*x^2)^p*hypergeom([1/2, -p], 3/2, (d^2*x^2)/c^2))/(1 - (d^2 
*x^2)/c^2)^p - (B*(c^2 - d^2*x^2)^(p + 1))/(2*d^2*(p + 1))
 

Reduce [F]

\[ \int (A+B x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {2 \left (-d^{2} x^{2}+c^{2}\right )^{p} a \,d^{2} p x +2 \left (-d^{2} x^{2}+c^{2}\right )^{p} a \,d^{2} x -2 \left (-d^{2} x^{2}+c^{2}\right )^{p} b \,c^{2} p -\left (-d^{2} x^{2}+c^{2}\right )^{p} b \,c^{2}+2 \left (-d^{2} x^{2}+c^{2}\right )^{p} b \,d^{2} p \,x^{2}+\left (-d^{2} x^{2}+c^{2}\right )^{p} b \,d^{2} x^{2}+8 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) a \,c^{2} d^{2} p^{3}+12 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) a \,c^{2} d^{2} p^{2}+4 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) a \,c^{2} d^{2} p}{2 d^{2} \left (2 p^{2}+3 p +1\right )} \] Input:

int((B*x+A)*(-d^2*x^2+c^2)^p,x)
 

Output:

(2*(c**2 - d**2*x**2)**p*a*d**2*p*x + 2*(c**2 - d**2*x**2)**p*a*d**2*x - 2 
*(c**2 - d**2*x**2)**p*b*c**2*p - (c**2 - d**2*x**2)**p*b*c**2 + 2*(c**2 - 
 d**2*x**2)**p*b*d**2*p*x**2 + (c**2 - d**2*x**2)**p*b*d**2*x**2 + 8*int(( 
c**2 - d**2*x**2)**p/(2*c**2*p + c**2 - 2*d**2*p*x**2 - d**2*x**2),x)*a*c* 
*2*d**2*p**3 + 12*int((c**2 - d**2*x**2)**p/(2*c**2*p + c**2 - 2*d**2*p*x* 
*2 - d**2*x**2),x)*a*c**2*d**2*p**2 + 4*int((c**2 - d**2*x**2)**p/(2*c**2* 
p + c**2 - 2*d**2*p*x**2 - d**2*x**2),x)*a*c**2*d**2*p)/(2*d**2*(2*p**2 + 
3*p + 1))