\(\int \frac {(1-\frac {e^2 x^2}{d^2})^p}{d+e x} \, dx\) [964]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.03 (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 = {690, 71} \[ \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{d+e x} \, dx=-\frac {2^{p-1} \left (\frac {d-e x}{d}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1-p,p+1,p+2,\frac {d-e x}{2 d}\right )}{e (p+1)} \]

[In]

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

[Out]

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

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 690

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[a^(p + 1)*d^(m - 1)*(((d - e*x)/d)^
(p + 1)/(a/d + c*(x/e))^(p + 1)), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, c, d, e, m}
, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) && GtQ[a, 0] &&  !(IGtQ[m, 0] &&
(IntegerQ[3*p] || IntegerQ[4*p]))

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85 \[ \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{d+e x} \, dx=-\frac {2^{-1+p} (d-e x) \left (1+\frac {e x}{d}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^p \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )}{d e (1+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.29 (sec) , antiderivative size = 318, normalized size of antiderivative = 7.76 \[ \int \frac {\left (1-\frac {e^2 x^2}{d^2}\right )^p}{d+e x} \, dx=\begin {cases} \frac {0^{p} \log {\left (-1 + \frac {e^{2} x^{2}}{d^{2}} \right )}}{2 e} + \frac {0^{p} \operatorname {acoth}{\left (\frac {e x}{d} \right )}}{e} + \frac {d^{1 - 2 p} e^{2 p - 2} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} + \frac {e x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {0^{p} \log {\left (1 - \frac {e^{2} x^{2}}{d^{2}} \right )}}{2 e} + \frac {0^{p} \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{e} + \frac {d^{1 - 2 p} e^{2 p - 2} p x^{2 p - 1} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} 1 - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} + \frac {e x^{2} \Gamma \left (p\right ) \Gamma \left (1 - p\right ) {{}_{3}F_{2}\left (\begin {matrix} 2, 1, 1 - p \\ 2, 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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