Optimal. Leaf size=158 \[ -\frac{a^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b (n+1) (b c-a d) (b e-a f)}+\frac{c^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d (n+1) (b c-a d) (d e-c f)}+\frac{(e+f x)^{n+1}}{b d f (n+1)} \]
[Out]
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Rubi [A] time = 0.31919, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b (n+1) (b c-a d) (b e-a f)}+\frac{c^2 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d (n+1) (b c-a d) (d e-c f)}+\frac{(e+f x)^{n+1}}{b d f (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(x^2*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 48.1351, size = 117, normalized size = 0.74 \[ - \frac{a^{2} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{b \left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} + \frac{c^{2} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{d \left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} + \frac{\left (e + f x\right )^{n + 1}}{b d f \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(f*x+e)**n/(b*x+a)/(d*x+c),x)
[Out]
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Mathematica [A] time = 0.272, size = 153, normalized size = 0.97 \[ \frac{(e+f x)^{n+1} \left (a^2 d f (c f-d e) \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )+(b e-a f) \left (b c^2 f \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )-(b c-a d) (c f-d e)\right )\right )}{b d f (n+1) (b c-a d) (b e-a f) (d e-c f)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^2*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
[Out]
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Maple [F] time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2} \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(f*x+e)^n/(b*x+a)/(d*x+c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^2/((b*x + a)*(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{2}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^2/((b*x + a)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(f*x+e)**n/(b*x+a)/(d*x+c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x^{2}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^2/((b*x + a)*(d*x + c)),x, algorithm="giac")
[Out]