3.267 \(\int \frac{\left (a+b x^3+c x^6\right )^p}{x^4} \, dx\)

Optimal. Leaf size=164 \[ -\frac{4^p \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3} \]

[Out]

-(4^p*(a + b*x^3 + c*x^6)^p*AppellF1[1 - 2*p, -p, -p, 2*(1 - p), -(b - Sqrt[b^2
- 4*a*c])/(2*c*x^3), -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*(1 - 2*p)*x^3*((b -
Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x
^3))^p)

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Rubi [A]  time = 0.323034, antiderivative size = 164, 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 \[ -\frac{4^p \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (1-2 p;-p,-p;2 (1-p);-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 (1-2 p) x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^p/x^4,x]

[Out]

-(4^p*(a + b*x^3 + c*x^6)^p*AppellF1[1 - 2*p, -p, -p, 2*(1 - p), -(b - Sqrt[b^2
- 4*a*c])/(2*c*x^3), -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*(1 - 2*p)*x^3*((b -
Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x
^3))^p)

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Rubi in Sympy [A]  time = 28.9353, size = 151, normalized size = 0.92 \[ - \frac{\left (\frac{b + 2 c x^{3} - \sqrt{- 4 a c + b^{2}}}{2 c x^{3}}\right )^{- p} \left (\frac{b + 2 c x^{3} + \sqrt{- 4 a c + b^{2}}}{2 c x^{3}}\right )^{- p} \left (a + b x^{3} + c x^{6}\right )^{p} \left (\frac{1}{x^{3}}\right )^{2 p} \left (\frac{1}{x^{3}}\right )^{- 2 p + 1} \operatorname{appellf_{1}}{\left (- 2 p + 1,- p,- p,- 2 p + 2,- \frac{b - \sqrt{- 4 a c + b^{2}}}{2 c x^{3}},- \frac{b + \sqrt{- 4 a c + b^{2}}}{2 c x^{3}} \right )}}{3 \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**p/x**4,x)

[Out]

-((b + 2*c*x**3 - sqrt(-4*a*c + b**2))/(2*c*x**3))**(-p)*((b + 2*c*x**3 + sqrt(-
4*a*c + b**2))/(2*c*x**3))**(-p)*(a + b*x**3 + c*x**6)**p*(x**(-3))**(2*p)*(x**(
-3))**(-2*p + 1)*appellf1(-2*p + 1, -p, -p, -2*p + 2, -(b - sqrt(-4*a*c + b**2))
/(2*c*x**3), -(b + sqrt(-4*a*c + b**2))/(2*c*x**3))/(3*(-2*p + 1))

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Mathematica [B]  time = 3.31729, size = 510, normalized size = 3.11 \[ \frac{(p-1) \left (\sqrt{b^2-4 a c}-b-2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\frac{2 \left (b-\sqrt{b^2-4 a c}\right )}{c x^3}+4\right )^{-p} \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^p \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^p \left (a+b x^3+c x^6\right )^{p-1} F_1\left (1-2 p;-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )}{3 (2 p-1) \left (-4 c (p-1) x^3 F_1\left (1-2 p;-p,-p;2-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )+p \left (\sqrt{b^2-4 a c}+b\right ) F_1\left (2-2 p;1-p,-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )+p \left (b-\sqrt{b^2-4 a c}\right ) F_1\left (2-2 p;-p,1-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^3 + c*x^6)^p/x^4,x]

[Out]

((-1 + p)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x^3)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/
c)^p*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(c*x^3))^p*(b + Sqrt[b^2 - 4*a*c] + 2*c*
x^3)*(a + b*x^3 + c*x^6)^(-1 + p)*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -(b + Sqrt[
b^2 - 4*a*c])/(2*c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)])/(3*(-1 + 2*p)*(4 +
(2*(b - Sqrt[b^2 - 4*a*c]))/(c*x^3))^p*((b - Sqrt[b^2 - 4*a*c])/(2*c) + x^3)^p*
(-4*c*(-1 + p)*x^3*AppellF1[1 - 2*p, -p, -p, 2 - 2*p, -(b + Sqrt[b^2 - 4*a*c])/(
2*c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)] + (b + Sqrt[b^2 - 4*a*c])*p*Appell
F1[2 - 2*p, 1 - p, -p, 3 - 2*p, -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^3), (-b + Sqrt[b
^2 - 4*a*c])/(2*c*x^3)] + (b - Sqrt[b^2 - 4*a*c])*p*AppellF1[2 - 2*p, -p, 1 - p,
3 - 2*p, -(b + Sqrt[b^2 - 4*a*c])/(2*c*x^3), (-b + Sqrt[b^2 - 4*a*c])/(2*c*x^3)
]))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}}{{x}^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^p/x^4,x)

[Out]

int((c*x^6+b*x^3+a)^p/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p/x^4,x, algorithm="maxima")

[Out]

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p/x^4,x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)^p/x^4, 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**6+b*x**3+a)**p/x**4,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p/x^4,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^p/x^4, x)