3.1126 \(\int \frac{\left (a+b x^2+c x^4\right )^p}{x^4} \, dx\)

Optimal. Leaf size=138 \[ -\frac{\left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (-\frac{3}{2};-p,-p;-\frac{1}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.275932, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\left (\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p F_1\left (-\frac{3}{2};-p,-p;-\frac{1}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.1248, size = 121, normalized size = 0.88 \[ - \frac{\left (\frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{2} + c x^{4}\right )^{p} \operatorname{appellf_{1}}{\left (- \frac{3}{2},- p,- p,- \frac{1}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [B]  time = 3.18373, size = 456, normalized size = 3.3 \[ \frac{c 2^{-p-2} \left (\sqrt{b^2-4 a c}+b\right ) \left (x^2 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^2\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^2}{c}\right )^{p+1} \left (a+b x^2+c x^4\right )^{p-1} F_1\left (-\frac{3}{2};-p,-p;-\frac{1}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )}{3 x^3 \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (p x^2 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (-\frac{1}{2};1-p,-p;\frac{1}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (-\frac{1}{2};-p,1-p;\frac{1}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )+a F_1\left (-\frac{3}{2};-p,-p;-\frac{1}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{p}}{{x}^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^p/x^4,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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**4,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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