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

Optimal. Leaf size=155 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{2 e p} \]

[Out]

-((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])])/(2*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)

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Rubi [A]  time = 0.229199, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{2 e p} \]

Antiderivative was successfully verified.

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

[Out]

-((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])])/(2*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)

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Rubi in Sympy [A]  time = 31.7925, size = 144, normalized size = 0.93 \[ - \frac{\left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1\right )^{- p} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1\right )^{- p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{2 e p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + c*x**2)**p*(d + e*x)**(-2*p)*(sqrt(c)*(-d - e*x)/(sqrt(c)*d - e*sqrt(-a))
+ 1)**(-p)*(sqrt(c)*(-d - e*x)/(sqrt(c)*d + e*sqrt(-a)) + 1)**(-p)*appellf1(-2*p
, -p, -p, -2*p + 1, sqrt(c)*(d + e*x)/(sqrt(c)*d - e*sqrt(-a)), sqrt(c)*(d + e*x
)/(sqrt(c)*d + e*sqrt(-a)))/(2*e*p)

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Mathematica [A]  time = 0.180651, size = 160, normalized size = 1.03 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{-2 p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}\right )^{-p} F_1\left (-2 p;-p,-p;1-2 p;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{2 e p} \]

Warning: Unable to verify antiderivative.

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

[Out]

-((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)])/(2*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))

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{-1-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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