Optimal. Leaf size=160 \[ \frac{b 2^{p-1} \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}}+\frac{\left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+1)} \]
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Rubi [A] time = 0.255272, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b 2^{p-1} \left (a+b x^2+c x^4\right )^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^2+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{c (p+1) \sqrt{b^2-4 a c}}+\frac{\left (a+b x^2+c x^4\right )^{p+1}}{4 c (p+1)} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^2 + c*x^4)^p,x]
[Out]
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Rubi in Sympy [A] time = 19.3745, size = 136, normalized size = 0.85 \[ \frac{b \left (\frac{- \frac{b}{2} - c x^{2} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}\right )^{- p - 1} \left (a + b x^{2} + c x^{4}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{b}{2} + c x^{2} + \frac{\sqrt{- 4 a c + b^{2}}}{2}}{\sqrt{- 4 a c + b^{2}}}} \right )}}{4 c \left (p + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{p + 1}}{4 c \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(c*x**4+b*x**2+a)**p,x)
[Out]
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Mathematica [C] time = 3.7756, size = 440, normalized size = 2.75 \[ \frac{3 c 2^{-p-3} x^4 \left (\sqrt{b^2-4 a c}+b\right ) \left (x^2 \left (b-\sqrt{b^2-4 a c}\right )+2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^2\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p+1} \left (a+x^2 \left (b+c x^2\right )\right )^{p-1} F_1\left (2;-p,-p;3;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (p x^2 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (3;1-p,-p;4;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (3;-p,1-p;4;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-6 a F_1\left (2;-p,-p;3;-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[x^3*(a + b*x^2 + c*x^4)^p,x]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(c*x^4+b*x^2+a)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^p*x^3,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^p*x^3,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(x**3*(c*x**4+b*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{p} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^p*x^3,x, algorithm="giac")
[Out]