\(\int (d+e x)^{1-2 p} (d^2-e^2 x^2)^p \, dx\) [385]

Optimal result

Integrand size = 26, antiderivative size = 77 \[ \int (d+e x)^{1-2 p} \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2^{1-p} (d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^{-1+p} \left (d^2-e^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-1+p,1+p,2+p,\frac {d-e x}{2 d}\right )}{e (1+p)} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.79 \[ \int (d+e x)^{1-2 p} \left (d^2-e^2 x^2\right )^p \, dx=\frac {(d+e x)^{-2 p} \left (1+\frac {e x}{d}\right )^p \left (e^2 x^2 (d-e x)^p (d+e x)^p \left (1-\frac {e x}{d}\right )^{-p} \operatorname {AppellF1}\left (2,-p,p,3,\frac {e x}{d},-\frac {e x}{d}\right )-\frac {2^{1-p} d (d-e x) \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (p,1+p,2+p,\frac {d-e x}{2 d}\right )}{1+p}\right )}{2 e} \] Input:

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

Output:

((1 + (e*x)/d)^p*((e^2*x^2*(d - e*x)^p*(d + e*x)^p*AppellF1[2, -p, p, 3, ( 
e*x)/d, -((e*x)/d)])/(1 - (e*x)/d)^p - (2^(1 - p)*d*(d - e*x)*(d^2 - e^2*x 
^2)^p*Hypergeometric2F1[p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(1 + p)))/(2*e* 
(d + e*x)^(2*p))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {474, 473, 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.

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

\(\Big \downarrow \) 474

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

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

-((2^(1 - p)*(1 + (e*x)/d)^(-1 + p)*(d^2 - e^2*x^2)^(1 + p)*Hypergeometric 
2F1[-1 + p, 1 + p, 2 + p, (d - e*x)/(2*d)])/(e*(1 + p)*(d + e*x)^(2*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 473

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

rule 474

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((-e^2*x^2 + d^2)^p*(e*x + d)^(-2*p + 1), x)
 

Sympy [F]

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

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

Output:

Integral((-(-d + e*x)*(d + e*x))**p*(d + e*x)**(1 - 2*p), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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