Optimal. Leaf size=124 \[ \frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)}-\frac{c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)} \]
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Rubi [A] time = 0.153532, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{a (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{(n+1) (b c-a d) (b e-a f)}-\frac{c (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{(n+1) (b c-a d) (d e-c f)} \]
Antiderivative was successfully verified.
[In] Int[(x*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
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Rubi in Sympy [A] time = 23.4041, size = 94, normalized size = 0.76 \[ \frac{a \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{\left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} - \frac{c \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{\left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(f*x+e)**n/(b*x+a)/(d*x+c),x)
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Mathematica [A] time = 0.135411, size = 116, normalized size = 0.94 \[ \frac{(e+f x)^{n+1} \left (a (c f-d e) \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )+c (b e-a f) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )\right )}{(n+1) (b c-a d) (b e-a f) (c f-d e)} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
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Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{x \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(f*x+e)^n/(b*x+a)/(d*x+c),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (f x + e\right )}^{n} x}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/((b*x + a)*(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/((b*x + a)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(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. \[ \int \frac{{\left (f x + e\right )}^{n} x}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x/((b*x + a)*(d*x + c)),x, algorithm="giac")
[Out]