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

Optimal. Leaf size=209 \[ -\frac {\left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]

[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]  time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {727} \[ -\frac {\left (\sqrt {-a}-\sqrt {c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac {\left (\sqrt {-a}+\sqrt {c} x\right ) \left (\sqrt {-a} e+\sqrt {c} d\right )}{\left (\sqrt {-a}-\sqrt {c} x\right ) \left (\sqrt {c} d-\sqrt {-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]

Antiderivative was successfully verified.

[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 727

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*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))])/((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), 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*} \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 \, _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]  time = 0.14, size = 200, normalized size = 0.96 \[ \frac {\left (\sqrt {c} x-\sqrt {-a}\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (1-\frac {c (d+e x)}{c d-\sqrt {-a} \sqrt {c} e}\right )^{-p} \left (1-\frac {c (d+e x)}{\sqrt {-a} \sqrt {c} e+c d}\right )^p \, _2F_1\left (-2 p-1,-p;-2 p;\frac {2 \sqrt {-a} \sqrt {c} (d+e x)}{\left (\sqrt {c} d-\sqrt {-a} e\right ) \left (\sqrt {-a}-\sqrt {c} x\right )}\right )}{(2 p+1) \left (\sqrt {-a} e+\sqrt {c} d\right )} \]

Warning: Unable to verify antiderivative.

[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)

________________________________________________________________________________________

fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [F]  time = 0.69, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+a \right )^{p} \left (e x +d \right )^{-2 p -2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (c x^{2} + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________