Optimal. Leaf size=339 \[ \frac{x \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{b x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c^2}+\frac{x^3 \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{3}{2},-q;\frac{5}{2};-\frac{e x^2}{d}\right )}{3 c} \]
[Out]
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Rubi [A] time = 1.41638, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{x \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (b-\sqrt{b^2-4 a c}\right )}+\frac{x \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} F_1\left (\frac{1}{2};1,-q;\frac{3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},-\frac{e x^2}{d}\right )}{c^2 \left (\sqrt{b^2-4 a c}+b\right )}-\frac{b x \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{1}{2},-q;\frac{3}{2};-\frac{e x^2}{d}\right )}{c^2}+\frac{x^3 \left (d+e x^2\right )^q \left (\frac{e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac{3}{2},-q;\frac{5}{2};-\frac{e x^2}{d}\right )}{3 c} \]
Antiderivative was successfully verified.
[In] Int[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]
[Out]
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Rubi in Sympy [A] time = 163.142, size = 308, normalized size = 0.91 \[ - \frac{b x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{e x^{2}}{d}} \right )}}{c^{2}} + \frac{x^{3} \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q}{{}_{2}F_{1}\left (\begin{matrix} - q, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{e x^{2}}{d}} \right )}}{3 c} + \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b \left (- 3 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b + \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c^{2} \left (b + \sqrt{- 4 a c + b^{2}}\right ) \sqrt{- 4 a c + b^{2}}} - \frac{x \left (1 + \frac{e x^{2}}{d}\right )^{- q} \left (d + e x^{2}\right )^{q} \left (b \left (- 3 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (- a c + b^{2}\right )\right ) \operatorname{appellf_{1}}{\left (\frac{1}{2},1,- q,\frac{3}{2},- \frac{2 c x^{2}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{e x^{2}}{d} \right )}}{c^{2} \left (b - \sqrt{- 4 a c + b^{2}}\right ) \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(e*x**2+d)**q/(c*x**4+b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.31375, size = 0, normalized size = 0. \[ \int \frac{x^6 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(x^6*(d + e*x^2)^q)/(a + b*x^2 + c*x^4),x]
[Out]
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Maple [F] time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{{x}^{6} \left ( e{x}^{2}+d \right ) ^{q}}{c{x}^{4}+b{x}^{2}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6*(e*x^2+d)^q/(c*x^4+b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),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(x**6*(e*x**2+d)**q/(c*x**4+b*x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x^2 + d)^q*x^6/(c*x^4 + b*x^2 + a),x, algorithm="giac")
[Out]