Optimal. Leaf size=173 \[ \frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.483241, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (d x)^{m+1} \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m/(a + b*x^3 + c*x^6),x]
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Rubi in Sympy [A] time = 31.7794, size = 148, normalized size = 0.86 \[ - \frac{2 c \left (d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{d \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} + \frac{2 c \left (d x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{d \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m/(c*x**6+b*x**3+a),x)
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Mathematica [C] time = 0.0940049, size = 84, normalized size = 0.49 \[ \frac{(d x)^m \text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right ]}{3 m} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(d*x)^m/(a + b*x^3 + c*x^6),x]
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Maple [F] time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{c{x}^{6}+b{x}^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m/(c*x^6+b*x^3+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{c x^{6} + b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^6 + b*x^3 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{c x^{6} + b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^6 + b*x^3 + 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((d*x)**m/(c*x**6+b*x**3+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d x\right )^{m}}{c x^{6} + b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^m/(c*x^6 + b*x^3 + a),x, algorithm="giac")
[Out]