Optimal. Leaf size=163 \[ -\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
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Rubi [A] time = 0.197806, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{c} d-\sqrt{-a} e\right )}-\frac{(d+e x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{2 \sqrt{c} (n+1) \left (\sqrt{-a} e+\sqrt{c} d\right )} \]
Antiderivative was successfully verified.
[In] Int[(x*(d + e*x)^n)/(a + c*x^2),x]
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Rubi in Sympy [A] time = 38.3642, size = 129, normalized size = 0.79 \[ - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d + e \sqrt{- a}}} \right )}}{2 \sqrt{c} \left (n + 1\right ) \left (\sqrt{c} d + e \sqrt{- a}\right )} - \frac{\left (d + e x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{\sqrt{c} \left (d + e x\right )}{\sqrt{c} d - e \sqrt{- a}}} \right )}}{2 \sqrt{c} \left (n + 1\right ) \left (\sqrt{c} d - e \sqrt{- a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(e*x+d)**n/(c*x**2+a),x)
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Mathematica [C] time = 0.148661, size = 200, normalized size = 1.23 \[ \frac{(d+e x)^n \left (\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x-i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac{\sqrt{c} d+i \sqrt{a} e}{i \sqrt{a} e-\sqrt{c} e x}\right )+\left (\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} x e+i \sqrt{a} e}\right )\right )}{2 c n} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(d + e*x)^n)/(a + c*x^2),x]
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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{x \left ( ex+d \right ) ^{n}}{c{x}^{2}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(e*x+d)^n/(c*x^2+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x/(c*x^2 + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{n} x}{c x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x/(c*x^2 + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (d + e x\right )^{n}}{a + c x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(e*x+d)**n/(c*x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{n} x}{c x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^n*x/(c*x^2 + a),x, algorithm="giac")
[Out]