Optimal. Leaf size=168 \[ -\frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (2 (1-p);-p,-p;3-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 (1-p) x^6} \]
[Out]
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Rubi [A] time = 0.324099, antiderivative size = 168, 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 \[ -\frac{2^{2 p-1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (2 (1-p);-p,-p;3-2 p;-\frac{b-\sqrt{b^2-4 a c}}{2 c x^3},-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3}\right )}{3 (1-p) x^6} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3 + c*x^6)^p/x^7,x]
[Out]
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Rubi in Sympy [A] time = 29.5427, size = 150, normalized size = 0.89 \[ - \frac{\left (\frac{b + 2 c x^{3} - \sqrt{- 4 a c + b^{2}}}{2 c x^{3}}\right )^{- p} \left (\frac{b + 2 c x^{3} + \sqrt{- 4 a c + b^{2}}}{2 c x^{3}}\right )^{- p} \left (a + b x^{3} + c x^{6}\right )^{p} \left (\frac{1}{x^{3}}\right )^{2 p} \left (\frac{1}{x^{3}}\right )^{- 2 p + 2} \operatorname{appellf_{1}}{\left (- 2 p + 2,- p,- p,- 2 p + 3,- \frac{b - \sqrt{- 4 a c + b^{2}}}{2 c x^{3}},- \frac{b + \sqrt{- 4 a c + b^{2}}}{2 c x^{3}} \right )}}{6 \left (- p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**6+b*x**3+a)**p/x**7,x)
[Out]
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Mathematica [B] time = 3.1528, size = 507, normalized size = 3.02 \[ \frac{c 4^{-p-1} (2 p-3) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\frac{b-\sqrt{b^2-4 a c}}{2 c x^3}+1\right )^{-p} \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p+1} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c x^3}\right )^p \left (a+b x^3+c x^6\right )^{p-1} F_1\left (2-2 p;-p,-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )}{3 (p-1) x^3 \left (2 c (2 p-3) x^3 F_1\left (2-2 p;-p,-p;3-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )-p \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (3-2 p;1-p,-p;4-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (3-2 p;-p,1-p;4-2 p;-\frac{b+\sqrt{b^2-4 a c}}{2 c x^3},\frac{\sqrt{b^2-4 a c}-b}{2 c x^3}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3 + c*x^6)^p/x^7,x]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}}{{x}^{7}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^6+b*x^3+a)^p/x^7,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)^p/x^7,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)^p/x^7,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((c*x**6+b*x**3+a)**p/x**7,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{p}}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^6 + b*x^3 + a)^p/x^7,x, algorithm="giac")
[Out]