Optimal. Leaf size=57 \[ \frac{x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} F_1\left (\frac{1}{3};1,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a} \]
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Rubi [A] time = 0.0747915, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (c+d x^3\right )^q \left (\frac{d x^3}{c}+1\right )^{-q} F_1\left (\frac{1}{3};1,-q;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^q/(a + b*x^3),x]
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Rubi in Sympy [A] time = 21.6503, size = 42, normalized size = 0.74 \[ \frac{x \left (1 + \frac{d x^{3}}{c}\right )^{- q} \left (c + d x^{3}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{3},1,- q,\frac{4}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**q/(b*x**3+a),x)
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Mathematica [B] time = 0.284151, size = 162, normalized size = 2.84 \[ \frac{4 a c x \left (c+d x^3\right )^q F_1\left (\frac{1}{3};-q,1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (a d q F_1\left (\frac{4}{3};1-q,1;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-b c F_1\left (\frac{4}{3};-q,2;\frac{7}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )+4 a c F_1\left (\frac{1}{3};-q,1;\frac{4}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^q/(a + b*x^3),x]
[Out]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{ \left ( d{x}^{3}+c \right ) ^{q}}{b{x}^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^q/(b*x^3+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{q}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^q/(b*x^3 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{3} + c\right )}^{q}}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^q/(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**3+c)**q/(b*x**3+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{q}}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^q/(b*x^3 + a),x, algorithm="giac")
[Out]