3.3.73 \(\int \frac {(d^2-e^2 x^2)^p}{x^2 (d+e x)} \, dx\) [273]

3.3.73.1 Optimal result

Integrand size = 25, antiderivative size = 106 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)} \, dx=-\frac {\left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{d x}+\frac {e \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d^2 p} \]

output

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

3.3.73.2 Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.58 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)} \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {2 d^2 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},\frac {e^2 x^2}{d^2}\right )}{x}+\frac {2^p e (-d+e x) \left (1+\frac {e x}{d}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )}{1+p}-\frac {d e \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {d^2}{e^2 x^2}\right )}{p}\right )}{2 d^3} \]

input

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

output

((d^2 - e^2*x^2)^p*((-2*d^2*Hypergeometric2F1[-1/2, -p, 1/2, (e^2*x^2)/d^2 
])/(x*(1 - (e^2*x^2)/d^2)^p) + (2^p*e*(-d + e*x)*Hypergeometric2F1[1 - p, 
1 + p, 2 + p, (d - e*x)/(2*d)])/((1 + p)*(1 + (e*x)/d)^p) - (d*e*Hypergeom 
etric2F1[-p, -p, 1 - p, d^2/(e^2*x^2)])/(p*(1 - d^2/(e^2*x^2))^p)))/(2*d^3 
)
 

3.3.73.3 Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {583, 542, 243, 75, 279, 278}

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.

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

\(\Big \downarrow \) 583

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

input

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

output

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

3.3.73.3.1 Defintions of rubi rules used

rule 75

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 278

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

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 542

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

rule 583

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, 0]
 

3.3.73.4 Maple [F]

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

input

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

output

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

3.3.73.5 Fricas [F]

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

input

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

output

integral((-e^2*x^2 + d^2)^p/(e*x^3 + d*x^2), x)
 

3.3.73.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.91 (sec) , antiderivative size = 434, normalized size of antiderivative = 4.09 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x^2 (d+e x)} \, dx=\begin {cases} - \frac {0^{p} d d^{2 p - 2}}{x} - \frac {0^{p} d^{2 p - 2} e \log {\left (\frac {e^{2} x^{2}}{d^{2}} \right )}}{2} + \frac {0^{p} d^{2 p - 2} e \log {\left (-1 + \frac {e^{2} x^{2}}{d^{2}} \right )}}{2} + 0^{p} d^{2 p - 2} e \operatorname {acoth}{\left (\frac {e x}{d} \right )} + \frac {d e^{2 p - 2} p x^{2 p - 3} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p - 1} p x^{2 p - 2} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {0^{p} d d^{2 p - 2}}{x} - \frac {0^{p} d^{2 p - 2} e \log {\left (\frac {e^{2} x^{2}}{d^{2}} \right )}}{2} + \frac {0^{p} d^{2 p - 2} e \log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )}}{2} + 0^{p} d^{2 p - 2} e \operatorname {atanh}{\left (\frac {e x}{d} \right )} + \frac {d e^{2 p - 2} p x^{2 p - 3} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p - 1} p x^{2 p - 2} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]

input

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

output

Piecewise((-0**p*d*d**(2*p - 2)/x - 0**p*d**(2*p - 2)*e*log(e**2*x**2/d**2 
)/2 + 0**p*d**(2*p - 2)*e*log(-1 + e**2*x**2/d**2)/2 + 0**p*d**(2*p - 2)*e 
*acoth(e*x/d) + d*e**(2*p - 2)*p*x**(2*p - 3)*exp(I*pi*p)*gamma(p)*gamma(3 
/2 - p)*hyper((1 - p, 3/2 - p), (5/2 - p,), d**2/(e**2*x**2))/(2*gamma(5/2 
 - p)*gamma(p + 1)) - e**(2*p - 1)*p*x**(2*p - 2)*exp(I*pi*p)*gamma(p)*gam 
ma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), d**2/(e**2*x**2))/(2*gamma(2 - p 
)*gamma(p + 1)), Abs(e**2*x**2/d**2) > 1), (-0**p*d*d**(2*p - 2)/x - 0**p* 
d**(2*p - 2)*e*log(e**2*x**2/d**2)/2 + 0**p*d**(2*p - 2)*e*log(1 - e**2*x* 
*2/d**2)/2 + 0**p*d**(2*p - 2)*e*atanh(e*x/d) + d*e**(2*p - 2)*p*x**(2*p - 
 3)*exp(I*pi*p)*gamma(p)*gamma(3/2 - p)*hyper((1 - p, 3/2 - p), (5/2 - p,) 
, d**2/(e**2*x**2))/(2*gamma(5/2 - p)*gamma(p + 1)) - e**(2*p - 1)*p*x**(2 
*p - 2)*exp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), 
d**2/(e**2*x**2))/(2*gamma(2 - p)*gamma(p + 1)), True))
 

3.3.73.7 Maxima [F]

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

input

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

output

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

3.3.73.8 Giac [F]

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

input

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

output

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

3.3.73.9 Mupad [F(-1)]

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

input

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

output

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