\(\int \frac {(1-\frac {d^2 x^2}{c^2})^p}{c+d x} \, dx\) [351]

Optimal result

Integrand size = 23, antiderivative size = 41 \[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\frac {2^p \left (\frac {c+d x}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-p,p,1+p,\frac {c+d x}{2 c}\right )}{d p} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85 \[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=-\frac {2^{-1+p} (c-d x) \left (1+\frac {d x}{c}\right )^{-p} \left (1-\frac {d^2 x^2}{c^2}\right )^p \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d (1+p)} \] Input:

Integrate[(1 - (d^2*x^2)/c^2)^p/(c + d*x),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {472, 79}

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.

\(\displaystyle \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx\)

\(\Big \downarrow \) 472

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

\(\Big \downarrow \) 79

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

Input:

Int[(1 - (d^2*x^2)/c^2)^p/(c + d*x),x]
 

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 472

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^(p + 1)*c^(n - 1)*(((c - d*x)/c)^(p + 1)/(a/c + b*(x/d))^(p + 1))   Int[( 
1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) && GtQ[a, 0] &&  !( 
IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

Maple [F]

\[\int \frac {\left (1-\frac {x^{2} d^{2}}{c^{2}}\right )^{p}}{d x +c}d x\]

Input:

int((1-1/c^2*x^2*d^2)^p/(d*x+c),x)
 

Output:

int((1-1/c^2*x^2*d^2)^p/(d*x+c),x)
 

Fricas [F]

\[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\int { \frac {{\left (-\frac {d^{2} x^{2}}{c^{2}} + 1\right )}^{p}}{d x + c} \,d x } \] Input:

integrate((1-d^2*x^2/c^2)^p/(d*x+c),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.37 (sec) , antiderivative size = 318, normalized size of antiderivative = 7.76 \[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\begin {cases} \frac {0^{p} \log {\left (-1 + \frac {d^{2} x^{2}}{c^{2}} \right )}}{2 d} + \frac {0^{p} \operatorname {acoth}{\left (\frac {d x}{c} \right )}}{d} + \frac {c^{1 - 2 p} d^{2 p - 2} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} + \frac {d x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 c^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {d^{2} x^{2}}{c^{2}}}\right | > 1 \\\frac {0^{p} \log {\left (1 - \frac {d^{2} x^{2}}{c^{2}} \right )}}{2 d} + \frac {0^{p} \operatorname {atanh}{\left (\frac {d x}{c} \right )}}{d} + \frac {c^{1 - 2 p} d^{2 p - 2} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} + \frac {d x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 c^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \] Input:

integrate((1-d**2*x**2/c**2)**p/(d*x+c),x)
 

Output:

Piecewise((0**p*log(-1 + d**2*x**2/c**2)/(2*d) + 0**p*acoth(d*x/c)/d + c** 
(1 - 2*p)*d**(2*p - 2)*p*x**(2*p - 1)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)* 
hyper((1 - p, 1/2 - p), (3/2 - p,), c**2/(d**2*x**2))/(2*gamma(3/2 - p)*ga 
mma(p + 1)) + d*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), d* 
*2*x**2*exp_polar(2*I*pi)/c**2)/(2*c**2*gamma(-p)*gamma(p + 1)), Abs(d**2* 
x**2/c**2) > 1), (0**p*log(1 - d**2*x**2/c**2)/(2*d) + 0**p*atanh(d*x/c)/d 
 + c**(1 - 2*p)*d**(2*p - 2)*p*x**(2*p - 1)*exp(I*pi*p)*gamma(p)*gamma(1/2 
 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), c**2/(d**2*x**2))/(2*gamma(3/2 - 
 p)*gamma(p + 1)) + d*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 
2), d**2*x**2*exp_polar(2*I*pi)/c**2)/(2*c**2*gamma(-p)*gamma(p + 1)), Tru 
e))
 

Maxima [F]

\[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\int { \frac {{\left (-\frac {d^{2} x^{2}}{c^{2}} + 1\right )}^{p}}{d x + c} \,d x } \] Input:

integrate((1-d^2*x^2/c^2)^p/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((-d^2*x^2/c^2 + 1)^p/(d*x + c), x)
 

Giac [F]

\[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\int { \frac {{\left (-\frac {d^{2} x^{2}}{c^{2}} + 1\right )}^{p}}{d x + c} \,d x } \] Input:

integrate((1-d^2*x^2/c^2)^p/(d*x+c),x, algorithm="giac")
 

Output:

integrate((-d^2*x^2/c^2 + 1)^p/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\int \frac {{\left (1-\frac {d^2\,x^2}{c^2}\right )}^p}{c+d\,x} \,d x \] Input:

int((1 - (d^2*x^2)/c^2)^p/(c + d*x),x)
 

Output:

int((1 - (d^2*x^2)/c^2)^p/(c + d*x), x)
 

Reduce [F]

\[ \int \frac {\left (1-\frac {d^2 x^2}{c^2}\right )^p}{c+d x} \, dx=\frac {\int \frac {\left (-d^{2} x^{2}+c^{2}\right )^{p}}{d x +c}d x}{c^{2 p}} \] Input:

int((1-d^2*x^2/c^2)^p/(d*x+c),x)
 

Output:

int((c**2 - d**2*x**2)**p/(c + d*x),x)/c**(2*p)