Optimal. Leaf size=288 \[ \frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{\sqrt [3]{-1} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(-1)^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )} \]
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Rubi [A] time = 0.685589, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{\sqrt [3]{-1} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac{(-1)^{2/3} (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(e + f*x)^n)/(a + b*x^3),x]
[Out]
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Rubi in Sympy [A] time = 96.6077, size = 252, normalized size = 0.88 \[ \frac{\sqrt [3]{-1} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e}} \right )}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (n + 1\right ) \left (\sqrt [3]{a} f - \left (-1\right )^{\frac{2}{3}} \sqrt [3]{b} e\right )} - \frac{\left (-1\right )^{\frac{2}{3}} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{-1} \sqrt [3]{b} \left (e + f x\right )}{\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e}} \right )}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (n + 1\right ) \left (\sqrt [3]{a} f + \sqrt [3]{-1} \sqrt [3]{b} e\right )} - \frac{\left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt [3]{b} \left (e + f x\right )}{- \sqrt [3]{a} f + \sqrt [3]{b} e}} \right )}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (n + 1\right ) \left (\sqrt [3]{a} f - \sqrt [3]{b} e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(f*x+e)**n/(b*x**3+a),x)
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Mathematica [C] time = 0.084591, size = 229, normalized size = 0.8 \[ -\frac{f (e+f x)^n \left (e \text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]-\text{RootSum}\left [-\text{$\#$1}^3 b+3 \text{$\#$1}^2 b e-3 \text{$\#$1} b e^2-a f^3+b e^3\&,\frac{\text{$\#$1} \left (\frac{e+f x}{-\text{$\#$1}+e+f x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\text{$\#$1}}{e+f x-\text{$\#$1}}\right )}{\text{$\#$1}^2-2 \text{$\#$1} e+e^2}\&\right ]\right )}{3 b n} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x*(e + f*x)^n)/(a + b*x^3),x]
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Maple [F] time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{x \left ( fx+e \right ) ^{n}}{b{x}^{3}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(f*x+e)^n/(b*x^3+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/(b*x^3 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x}{b x^{3} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/(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(x*(f*x+e)**n/(b*x**3+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x}{b x^{3} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/(b*x^3 + a),x, algorithm="giac")
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