3.1119 \(\int \frac{\left (a+b x^2+c x^4\right )^p}{x} \, dx\)

Optimal. Leaf size=152 \[ \frac{4^{p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right )}{p} \]

[Out]

(4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -(b - Sqrt[b^2
- 4*a*c])/(2*c*x^2), -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/(p*((b - Sqrt[b^2 - 4
*a*c] + 2*c*x^2)/(c*x^2))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p)

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Rubi [A]  time = 0.337592, antiderivative size = 152, 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 \[ \frac{4^{p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (-2 p;-p,-p;1-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^2},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2}\right )}{p} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^p/x,x]

[Out]

(4^(-1 + p)*(a + b*x^2 + c*x^4)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -(b - Sqrt[b^2
- 4*a*c])/(2*c*x^2), -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/(p*((b - Sqrt[b^2 - 4
*a*c] + 2*c*x^2)/(c*x^2))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p)

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Rubi in Sympy [A]  time = 24.0463, size = 128, normalized size = 0.84 \[ \frac{\left (\frac{b + 2 c x^{2} - \sqrt{- 4 a c + b^{2}}}{2 c x^{2}}\right )^{- p} \left (\frac{b + 2 c x^{2} + \sqrt{- 4 a c + b^{2}}}{2 c x^{2}}\right )^{- p} \left (a + b x^{2} + c x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (- 2 p,- p,- p,- 2 p + 1,- \frac{b - \sqrt{- 4 a c + b^{2}}}{2 c x^{2}},- \frac{b + \sqrt{- 4 a c + b^{2}}}{2 c x^{2}} \right )}}{4 p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**p/x,x)

[Out]

((b + 2*c*x**2 - sqrt(-4*a*c + b**2))/(2*c*x**2))**(-p)*((b + 2*c*x**2 + sqrt(-4
*a*c + b**2))/(2*c*x**2))**(-p)*(a + b*x**2 + c*x**4)**p*appellf1(-2*p, -p, -p,
-2*p + 1, -(b - sqrt(-4*a*c + b**2))/(2*c*x**2), -(b + sqrt(-4*a*c + b**2))/(2*c
*x**2))/(4*p)

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Mathematica [B]  time = 2.86055, size = 497, normalized size = 3.27 \[ \frac{c 2^{-2 p-3} (2 p-1) x^2 \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (\frac{b-\sqrt{b^2-4 a c}}{2 c x^2}+1\right )^{-p} \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^2\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p+1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c x^2}\right )^p \left (a+b x^2+c x^4\right )^{p-1} F_1\left (-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right )}{p \left (2 c (2 p-1) x^2 F_1\left (-2 p;-p,-p;1-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right )-p \left (\sqrt{b^2-4 a c}+b\right ) F_1\left (1-2 p;1-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right )+p \left (\sqrt{b^2-4 a c}-b\right ) F_1\left (1-2 p;-p,1-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^2},\frac{\sqrt{b^2-4 a c}-b}{2 c x^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^2 + c*x^4)^p/x,x]

[Out]

(2^(-3 - 2*p)*c*(-1 + 2*p)*x^2*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/c)^(1 + p)*((b
- Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(c*x^2))^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)*(a
+ b*x^2 + c*x^4)^(-1 + p)*AppellF1[-2*p, -p, -p, 1 - 2*p, -(b + Sqrt[b^2 - 4*a*c
])/(2*c*x^2), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)])/(p*(1 + (b - Sqrt[b^2 - 4*a*c
])/(2*c*x^2))^p*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^2)^p*(-((b + Sqrt[b^2 - 4*a*c
])*p*AppellF1[1 - 2*p, 1 - p, -p, 2 - 2*p, -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^2), (
-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)]) + (-b + Sqrt[b^2 - 4*a*c])*p*AppellF1[1 - 2*
p, -p, 1 - p, 2 - 2*p, -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^2), (-b + Sqrt[b^2 - 4*a*
c])/(2*c*x^2)] + 2*c*(-1 + 2*p)*x^2*AppellF1[-2*p, -p, -p, 1 - 2*p, -(b + Sqrt[b
^2 - 4*a*c])/(2*c*x^2), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^2)]))

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^p/x,x)

[Out]

int((c*x^4+b*x^2+a)^p/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p/x,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^p/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{4} + b x^{2} + a\right )}^{p}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p/x,x, algorithm="fricas")

[Out]

integral((c*x^4 + b*x^2 + a)^p/x, 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**4+b*x**2+a)**p/x,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{p}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^p/x,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^p/x, x)