\(\int (d+e x)^{-2-2 p} (a+c x^2)^p \, dx\) [741]

Optimal result

Integrand size = 21, antiderivative size = 209 \[ \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx=-\frac {\left (\sqrt {-a}-\sqrt {c} x\right ) \left (-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) \left (\sqrt {-a}+\sqrt {c} x\right )}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+2 p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {741} \[ \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx=-\frac {\left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]

[In]

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

[Out]

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

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(Rt[(-a)*c, 2] - c*x)*(d + e*x)^(m
+ 1)*((a + c*x^2)^p/((m + 1)*(c*d + e*Rt[(-a)*c, 2])*((c*d + e*Rt[(-a)*c, 2])*((Rt[(-a)*c, 2] + c*x)/((c*d - e
*Rt[(-a)*c, 2])*(-Rt[(-a)*c, 2] + c*x))))^p))*Hypergeometric2F1[m + 1, -p, m + 2, 2*c*Rt[(-a)*c, 2]*((d + e*x)
/((c*d - e*Rt[(-a)*c, 2])*(Rt[(-a)*c, 2] - c*x)))], x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

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

Mathematica [A] (warning: unable to verify)

Time = 0.23 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx=\frac {\left (-\sqrt {-a}+\sqrt {c} x\right ) (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \left (1-\frac {c (d+e x)}{c d-\sqrt {-a} \sqrt {c} e}\right )^{-p} \left (1-\frac {c (d+e x)}{c d+\sqrt {-a} \sqrt {c} e}\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{\left (\sqrt {c} d+\sqrt {-a} e\right ) (1+2 p)} \]

[In]

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

[Out]

((-Sqrt[-a] + Sqrt[c]*x)*(d + e*x)^(-1 - 2*p)*(a + c*x^2)^p*(1 - (c*(d + e*x))/(c*d + Sqrt[-a]*Sqrt[c]*e))^p*H
ypergeometric2F1[-1 - 2*p, -p, -2*p, (2*Sqrt[-a]*Sqrt[c]*(d + e*x))/((Sqrt[c]*d - Sqrt[-a]*e)*(Sqrt[-a] - Sqrt
[c]*x))])/((Sqrt[c]*d + Sqrt[-a]*e)*(1 + 2*p)*(1 - (c*(d + e*x))/(c*d - Sqrt[-a]*Sqrt[c]*e))^p)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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