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

Optimal. Leaf size=160 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1-2 p)} \]

[Out]

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

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Rubi [A]  time = 0.242276, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (1-\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}}\right )^{-p} \left (1-\frac{d+e x}{\frac{\sqrt{-a} e}{\sqrt{c}}+d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\frac{\sqrt{-a} e}{\sqrt{c}}},\frac{d+e x}{d+\frac{\sqrt{-a} e}{\sqrt{c}}}\right )}{e (1-2 p)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.1387, size = 146, normalized size = 0.91 \[ \frac{\left (a + c x^{2}\right )^{p} \left (d + e x\right )^{- 2 p + 1} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d - e \sqrt{- a}} + 1\right )^{- p} \left (\frac{\sqrt{c} \left (- d - e x\right )}{\sqrt{c} d + e \sqrt{- a}} + 1\right )^{- p} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}},\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}} \right )}}{e \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.205051, size = 166, normalized size = 1.04 \[ -\frac{\left (a+c x^2\right )^p (d+e x)^{1-2 p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}-x\right )}{e \sqrt{-\frac{a}{c}}+d}\right )^{-p} \left (\frac{e \left (\sqrt{-\frac{a}{c}}+x\right )}{e \sqrt{-\frac{a}{c}}-d}\right )^{-p} F_1\left (1-2 p;-p,-p;2-2 p;\frac{d+e x}{d-\sqrt{-\frac{a}{c}} e},\frac{d+e x}{d+\sqrt{-\frac{a}{c}} e}\right )}{e (2 p-1)} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{2}+a \right ) ^{p}}{ \left ( ex+d \right ) ^{2\,p}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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