3.1124 \(\int \left (a+b x^2+c x^4\right )^p \, dx\)

Optimal. Leaf size=133 \[ x \left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.178851, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ x \left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 28.8024, size = 112, normalized size = 0.84 \[ x \left (\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{2} + c x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2},- p,- p,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(2*c*x**2/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**2/(b + sqrt(-4*a*c + b*
*2)) + 1)**(-p)*(a + b*x**2 + c*x**4)**p*appellf1(1/2, -p, -p, 3/2, -2*c*x**2/(b
- sqrt(-4*a*c + b**2)), -2*c*x**2/(b + sqrt(-4*a*c + b**2)))

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Mathematica [B]  time = 3.59406, size = 487, normalized size = 3.66 \[ \frac{3\ 4^{-p-1} x \left (\sqrt{b^2-4 a c}+b\right ) \left (x^2 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^2\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b}{2 c}+x^2\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p+1} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p-1} \left (a+b x^2+c x^4\right )^{p-1} F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{\left (\sqrt{b^2-4 a c}-b\right ) \left (p x^2 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{3}{2};1-p,-p;\frac{5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{3}{2};-p,1-p;\frac{5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )-3 a F_1\left (\frac{1}{2};-p,-p;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*4^(-1 - p)*(b + Sqrt[b^2 - 4*a*c])*x*((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)^(-1 + p)*(-2*a + (-b + Sqrt[b^2 - 4*
a*c])*x^2)^2*(a + b*x^2 + c*x^4)^(-1 + p)*AppellF1[1/2, -p, -p, 3/2, (-2*c*x^2)/
(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((-b + Sqrt[b^2 -
4*a*c])*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^2)^p*((b + Sqrt[b^2 - 4*a*c])/(2*c) +
x^2)^p*(-3*a*AppellF1[1/2, -p, -p, 3/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*
c*x^2)/(-b + Sqrt[b^2 - 4*a*c])] + p*x^2*((-b + Sqrt[b^2 - 4*a*c])*AppellF1[3/2,
1 - p, -p, 5/2, (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 -
4*a*c])] - (b + Sqrt[b^2 - 4*a*c])*AppellF1[3/2, -p, 1 - p, 5/2, (-2*c*x^2)/(b +
Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2 + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + b*x^2 + a)^p, x)