Integrand size = 20, antiderivative size = 69 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {2^p \left (\frac {c-d x}{c}\right )^{-2-p} \left (c^2-d^2 x^2\right )^{2+p} \operatorname {Hypergeometric2F1}\left (-p,2+p,3+p,\frac {c+d x}{2 c}\right )}{c^2 d (2+p)} \] Output:
2^p*((-d*x+c)/c)^(-2-p)*(-d^2*x^2+c^2)^(2+p)*hypergeom([-p, 2+p],[3+p],1/2 *(d*x+c)/c)/c^2/d/(2+p)
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=-\frac {\left (c^2-d^2 x^2\right )^{1+p}}{2 d (1+p)}+c x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right ) \] Input:
Integrate[(c + d*x)*(c^2 - d^2*x^2)^p,x]
Output:
-1/2*(c^2 - d^2*x^2)^(1 + p)/(d*(1 + p)) + (c*x*(c^2 - d^2*x^2)^p*Hypergeo metric2F1[1/2, -p, 3/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.20, 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.
\(\displaystyle \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx\) |
\(\Big \downarrow \) 455 |
\(\displaystyle c \int \left (c^2-d^2 x^2\right )^pdx-\frac {\left (c^2-d^2 x^2\right )^{p+1}}{2 d (p+1)}\) |
\(\Big \downarrow \) 238 |
\(\displaystyle c \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \int \left (1-\frac {d^2 x^2}{c^2}\right )^pdx-\frac {\left (c^2-d^2 x^2\right )^{p+1}}{2 d (p+1)}\) |
\(\Big \downarrow \) 237 |
\(\displaystyle c x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )-\frac {\left (c^2-d^2 x^2\right )^{p+1}}{2 d (p+1)}\) |
Input:
Int[(c + d*x)*(c^2 - d^2*x^2)^p,x]
Output:
-1/2*(c^2 - d^2*x^2)^(1 + p)/(d*(1 + p)) + (c*x*(c^2 - d^2*x^2)^p*Hypergeo metric2F1[1/2, -p, 3/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p
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]
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]
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]
\[\int \left (d x +c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{p}d x\]
Input:
int((d*x+c)*(-d^2*x^2+c^2)^p,x)
Output:
int((d*x+c)*(-d^2*x^2+c^2)^p,x)
\[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:
integrate((d*x+c)*(-d^2*x^2+c^2)^p,x, algorithm="fricas")
Output:
integral((d*x + c)*(-d^2*x^2 + c^2)^p, x)
Time = 1.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=c c^{2 p} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )} + d \left (\begin {cases} \frac {x^{2} \left (c^{2}\right )^{p}}{2} & \text {for}\: d^{2} = 0 \\- \frac {\begin {cases} \frac {\left (c^{2} - d^{2} x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (c^{2} - d^{2} x^{2} \right )} & \text {otherwise} \end {cases}}{2 d^{2}} & \text {otherwise} \end {cases}\right ) \] Input:
integrate((d*x+c)*(-d**2*x**2+c**2)**p,x)
Output:
c*c**(2*p)*x*hyper((1/2, -p), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + d*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))
\[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:
integrate((d*x+c)*(-d^2*x^2+c^2)^p,x, algorithm="maxima")
Output:
integrate((d*x + c)*(-d^2*x^2 + c^2)^p, x)
\[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\int { {\left (d x + c\right )} {\left (-d^{2} x^{2} + c^{2}\right )}^{p} \,d x } \] Input:
integrate((d*x+c)*(-d^2*x^2+c^2)^p,x, algorithm="giac")
Output:
integrate((d*x + c)*(-d^2*x^2 + c^2)^p, x)
Time = 8.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {c\,x\,{\left (c^2-d^2\,x^2\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},-p;\ \frac {3}{2};\ \frac {d^2\,x^2}{c^2}\right )}{{\left (1-\frac {d^2\,x^2}{c^2}\right )}^p}-\frac {{\left (c^2-d^2\,x^2\right )}^{p+1}}{2\,d\,\left (p+1\right )} \] Input:
int((c^2 - d^2*x^2)^p*(c + d*x),x)
Output:
(c*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 - (c^2 - d^2*x^2)^(p + 1)/(2*d*(p + 1))
\[ \int (c+d x) \left (c^2-d^2 x^2\right )^p \, dx=\frac {-2 \left (-d^{2} x^{2}+c^{2}\right )^{p} c^{2} p -\left (-d^{2} x^{2}+c^{2}\right )^{p} c^{2}+2 \left (-d^{2} x^{2}+c^{2}\right )^{p} c d p x +2 \left (-d^{2} x^{2}+c^{2}\right )^{p} c d x +2 \left (-d^{2} x^{2}+c^{2}\right )^{p} d^{2} p \,x^{2}+\left (-d^{2} x^{2}+c^{2}\right )^{p} d^{2} x^{2}+8 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) c^{3} d \,p^{3}+12 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) c^{3} d \,p^{2}+4 \left (\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{-2 d^{2} p \,x^{2}-d^{2} x^{2}+2 c^{2} p +c^{2}}d x \right ) c^{3} d p}{2 d \left (2 p^{2}+3 p +1\right )} \] Input:
int((d*x+c)*(-d^2*x^2+c^2)^p,x)
Output:
( - 2*(c**2 - d**2*x**2)**p*c**2*p - (c**2 - d**2*x**2)**p*c**2 + 2*(c**2 - d**2*x**2)**p*c*d*p*x + 2*(c**2 - d**2*x**2)**p*c*d*x + 2*(c**2 - d**2*x **2)**p*d**2*p*x**2 + (c**2 - d**2*x**2)**p*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)*c**3*d*p**3 + 1 2*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)*c**3*d*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)*c**3*d*p)/(2*d*(2*p**2 + 3*p + 1))