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