Optimal. Leaf size=274 \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]
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Rubi [A] time = 0.511617, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{3} e x^3 \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right )+c x \left (\frac{2 b x^2}{c-\sqrt{c^2-4 a b}}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \left (\frac{2 b x^2}{\sqrt{c^2-4 a b}+c}+1\right )^{-p} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c-\sqrt{c^2-4 a b}},-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]
[Out]
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Rubi in Sympy [A] time = 67.9417, size = 233, normalized size = 0.85 \[ c x \left (\frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (\frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (a + b x^{4} + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},- p,- p,\frac{3}{2},- \frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}},- \frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} \right )} + \frac{e x^{3} \left (\frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (\frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} + 1\right )^{- p} \left (a + b x^{4} + c x^{2}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{3}{2},- p,- p,\frac{5}{2},- \frac{2 b x^{2}}{c - \sqrt{- 4 a b + c^{2}}},- \frac{2 b x^{2}}{c + \sqrt{- 4 a b + c^{2}}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+c)*(b*x**4+c*x**2+a)**p,x)
[Out]
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Mathematica [B] time = 1.25916, size = 706, normalized size = 2.58 \[ \frac{1}{3} 2^{-p-3} x \left (\sqrt{c^2-4 a b}+c\right ) \left (x^2 \left (\sqrt{c^2-4 a b}-c\right )-2 a\right ) \left (\frac{c-\sqrt{c^2-4 a b}}{2 b}+x^2\right )^{-p} \left (\frac{-\sqrt{c^2-4 a b}+2 b x^2+c}{b}\right )^{p+1} \left (a+b x^4+c x^2\right )^{p-1} \left (\frac{5 e x^2 F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}{p x^2 \left (\left (\sqrt{c^2-4 a b}-c\right ) F_1\left (\frac{5}{2};1-p,-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )-\left (\sqrt{c^2-4 a b}+c\right ) F_1\left (\frac{5}{2};-p,1-p;\frac{7}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )\right )-5 a F_1\left (\frac{3}{2};-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}-\frac{9 c F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}{p x^2 \left (\left (c-\sqrt{c^2-4 a b}\right ) F_1\left (\frac{3}{2};1-p,-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )+\left (\sqrt{c^2-4 a b}+c\right ) F_1\left (\frac{3}{2};-p,1-p;\frac{5}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )\right )+3 a F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 b x^2}{c+\sqrt{c^2-4 a b}},\frac{2 b x^2}{\sqrt{c^2-4 a b}-c}\right )}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + e*x^2)*(a + c*x^2 + b*x^4)^p,x]
[Out]
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Maple [F] time = 0.034, size = 0, normalized size = 0. \[ \int \left ( e{x}^{2}+c \right ) \left ( b{x}^{4}+c{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+c)*(b*x^4+c*x^2+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + c\right )}{\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)*(b*x^4 + c*x^2 + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{2} + c\right )}{\left (b x^{4} + c x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)*(b*x^4 + c*x^2 + a)^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((e*x**2+c)*(b*x**4+c*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x^{2} + c\right )}{\left (b x^{4} + c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + c)*(b*x^4 + c*x^2 + a)^p,x, algorithm="giac")
[Out]