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

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]

-(((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)*(Sqr
t[-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))

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Rubi [A]  time = 0.227345, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]

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)*(Sqr
t[-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))

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Rubi in Sympy [A]  time = 13.4932, size = 173, normalized size = 0.83 \[ - \frac{\left (\frac{\left (\sqrt{c} d + e \sqrt{- a}\right ) \left (\sqrt{c} x + \sqrt{- a}\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (\sqrt{c} x - \sqrt{- a}\right )}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p - 1} \left (- \sqrt{c} x + \sqrt{- a}\right ){{}_{2}F_{1}\left (\begin{matrix} - 2 p - 1, - p \\ - 2 p \end{matrix}\middle |{\frac{2 \sqrt{c} \sqrt{- a} \left (d + e x\right )}{\left (\sqrt{c} d - e \sqrt{- a}\right ) \left (- \sqrt{c} x + \sqrt{- a}\right )}} \right )}}{\left (2 p + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-2-2*p)*(c*x**2+a)**p,x)

[Out]

-((sqrt(c)*d + e*sqrt(-a))*(sqrt(c)*x + sqrt(-a))/((sqrt(c)*d - e*sqrt(-a))*(sqr
t(c)*x - sqrt(-a))))**(-p)*(a + c*x**2)**p*(d + e*x)**(-2*p - 1)*(-sqrt(c)*x + s
qrt(-a))*hyper((-2*p - 1, -p), (-2*p,), 2*sqrt(c)*sqrt(-a)*(d + e*x)/((sqrt(c)*d
- e*sqrt(-a))*(-sqrt(c)*x + sqrt(-a))))/((2*p + 1)*(sqrt(c)*d + e*sqrt(-a)))

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Mathematica [A]  time = 0.412015, size = 195, normalized size = 0.93 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{1-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}\right )^{-p} \left (1-\frac{d+e x}{e \sqrt{-\frac{a}{c}}+d}\right )^{2 p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-\frac{a}{c}} (d+e x)}{\left (d-\sqrt{-\frac{a}{c}} e\right ) \left (\sqrt{-\frac{a}{c}}-x\right )}\right )}{2 e p+e} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.122, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-2-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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