\(\int (d+e x)^{-1-2 p} (a+c x^2)^p \, dx\) [740]

Optimal result

Integrand size = 21, antiderivative size = 155 \[ \int (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, dx=-\frac {(d+e x)^{-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 e p} \]

[Out]

-1/2*(c*x^2+a)^p*AppellF1(-2*p,-p,-p,1-2*p,(e*x+d)/(d-e*(-a)^(1/2)/c^(1/2)),(e*x+d)/(d+e*(-a)^(1/2)/c^(1/2)))/
e/p/((e*x+d)^(2*p))/((1+(-e*x-d)/(d-e*(-a)^(1/2)/c^(1/2)))^p)/((1+(-e*x-d)/(d+e*(-a)^(1/2)/c^(1/2)))^p)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {774, 138} \[ \int (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, dx=-\frac {\left (a+c x^2\right )^p (d+e x)^{-2 p} \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{\frac {\sqrt {-a} e}{\sqrt {c}}+d}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 e p} \]

[In]

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

[Out]

-1/2*((a + c*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]), (d + e*x)/(d + (Sqrt
[-a]*e)/Sqrt[c])])/(e*p*(d + e*x)^(2*p)*(1 - (d + e*x)/(d - (Sqrt[-a]*e)/Sqrt[c]))^p*(1 - (d + e*x)/(d + (Sqrt
[-a]*e)/Sqrt[c]))^p)

Rule 138

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[c^n*e^p*((b*x)^(m +
1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rule 774

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[(a + c*x^
2)^p/(e*(1 - (d + e*x)/(d + e*(q/c)))^p*(1 - (d + e*x)/(d - e*(q/c)))^p), Subst[Int[x^m*Simp[1 - x/(d + e*(q/c
)), x]^p*Simp[1 - x/(d - e*(q/c)), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a
*e^2, 0] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p}\right ) \text {Subst}\left (\int x^{-1-2 p} \left (1-\frac {x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^p \left (1-\frac {x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^p \, dx,x,d+e x\right )}{e} \\ & = -\frac {(d+e x)^{-2 p} \left (a+c x^2\right )^p \left (1-\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} \left (1-\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac {d+e x}{d-\frac {\sqrt {-a} e}{\sqrt {c}}},\frac {d+e x}{d+\frac {\sqrt {-a} e}{\sqrt {c}}}\right )}{2 e p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, dx=-\frac {\left (\frac {e \left (\sqrt {-\frac {a}{c}}-x\right )}{d+\sqrt {-\frac {a}{c}} e}\right )^{-p} \left (\frac {e \left (\sqrt {-\frac {a}{c}}+x\right )}{-d+\sqrt {-\frac {a}{c}} e}\right )^{-p} (d+e x)^{-2 p} \left (a+c x^2\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d+e x}{d-\sqrt {-\frac {a}{c}} e},\frac {d+e x}{d+\sqrt {-\frac {a}{c}} e}\right )}{2 e p} \]

[In]

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

[Out]

-1/2*((a + c*x^2)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (d + e*x)/(d - Sqrt[-(a/c)]*e), (d + e*x)/(d + Sqrt[-(a/c)
]*e)])/(e*p*((e*(Sqrt[-(a/c)] - x))/(d + Sqrt[-(a/c)]*e))^p*((e*(Sqrt[-(a/c)] + x))/(-d + Sqrt[-(a/c)]*e))^p*(
d + e*x)^(2*p))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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