\(\int \frac {(d^2-e^2 x^2)^p}{x (d+e x)} \, dx\) [272]

Optimal result

Integrand size = 25, antiderivative size = 104 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)} \, dx=-\frac {e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {e^2 x^2}{d^2}\right )}{2 d p} \]

[Out]

-e*x*(-e^2*x^2+d^2)^p*hypergeom([1/2, 1-p],[3/2],e^2*x^2/d^2)/d^2/((1-e^2*x^2/d^2)^p)-1/2*(-e^2*x^2+d^2)^p*hyp
ergeom([1, p],[p+1],1-e^2*x^2/d^2)/d/p

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {864, 778, 272, 67, 252, 251} \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)} \, dx=-\frac {e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,p,p+1,1-\frac {e^2 x^2}{d^2}\right )}{2 d p} \]

[In]

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

[Out]

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

Rule 67

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] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 251

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

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 778

Int[(x_)^(m_.)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[f, Int[x^m*(a + c*x^2)^p, x]
, x] + Dist[g, Int[x^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && IntegerQ[m] &&  !IntegerQ[2
*p]

Rule 864

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{-1+p}}{x} \, dx \\ & = d \int \frac {\left (d^2-e^2 x^2\right )^{-1+p}}{x} \, dx-e \int \left (d^2-e^2 x^2\right )^{-1+p} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {\left (d^2-e^2 x\right )^{-1+p}}{x} \, dx,x,x^2\right )-\frac {\left (e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int \left (1-\frac {e^2 x^2}{d^2}\right )^{-1+p} \, dx}{d^2} \\ & = -\frac {e x \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1}{2},1-p;\frac {3}{2};\frac {e^2 x^2}{d^2}\right )}{d^2}-\frac {\left (d^2-e^2 x^2\right )^p \, _2F_1\left (1,p;1+p;1-\frac {e^2 x^2}{d^2}\right )}{2 d p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.45 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)} \, dx=\frac {2^{-1+p} \left (1-\frac {d^2}{e^2 x^2}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (p \left (1-\frac {d^2}{e^2 x^2}\right )^p (d-e x) \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )+d (1+p) \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {d^2}{e^2 x^2}\right )\right )}{d^2 p (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.39 (sec) , antiderivative size = 348, normalized size of antiderivative = 3.35 \[ \int \frac {\left (d^2-e^2 x^2\right )^p}{x (d+e x)} \, dx=\begin {cases} - \frac {0^{p} d^{2 p - 1} \log {\left (\frac {d^{2}}{e^{2} x^{2}} - 1 \right )}}{2} - 0^{p} d^{2 p - 1} \operatorname {acoth}{\left (\frac {d}{e x} \right )} + \frac {d e^{2 p - 2} p x^{2 p - 2} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p - 1} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {0^{p} d^{2 p - 1} \log {\left (- \frac {d^{2}}{e^{2} x^{2}} + 1 \right )}}{2} - 0^{p} d^{2 p - 1} \operatorname {atanh}{\left (\frac {d}{e x} \right )} + \frac {d e^{2 p - 2} p x^{2 p - 2} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (2 - p\right ) \Gamma \left (p + 1\right )} - \frac {e^{2 p - 1} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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