Optimal. Leaf size=319 \[ -\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]
[Out]
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Rubi [A] time = 0.719059, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
[Out]
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Rubi in Sympy [A] time = 97.9068, size = 267, normalized size = 0.84 \[ - \frac{a^{4} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{b \left (- e - f x\right )}{a f - b e}} \right )}}{b^{3} \left (n + 1\right ) \left (a d - b c\right ) \left (a f - b e\right )} + \frac{c^{4} \left (e + f x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{\frac{d \left (- e - f x\right )}{c f - d e}} \right )}}{d^{3} \left (n + 1\right ) \left (a d - b c\right ) \left (c f - d e\right )} + \frac{e^{2} \left (e + f x\right )^{n + 1}}{b d f^{3} \left (n + 1\right )} - \frac{2 e \left (e + f x\right )^{n + 2}}{b d f^{3} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 3}}{b d f^{3} \left (n + 3\right )} + \frac{e \left (e + f x\right )^{n + 1} \left (a d + b c\right )}{b^{2} d^{2} f^{2} \left (n + 1\right )} - \frac{\left (e + f x\right )^{n + 2} \left (a d + b c\right )}{b^{2} d^{2} f^{2} \left (n + 2\right )} + \frac{\left (e + f x\right )^{n + 1} \left (a^{2} d^{2} + a b c d + b^{2} c^{2}\right )}{b^{3} d^{3} f \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(f*x+e)**n/(b*x+a)/(d*x+c),x)
[Out]
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Mathematica [C] time = 0.884739, size = 262, normalized size = 0.82 \[ \frac{6}{5} e x^5 (e+f x)^n \left (\frac{a b F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{b x}{a}\right )}{(a+b x) (b c-a d) \left (6 a e F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{b x}{a}\right )+a f n x F_1\left (6;1-n,1;7;-\frac{f x}{e},-\frac{b x}{a}\right )-b e x F_1\left (6;-n,2;7;-\frac{f x}{e},-\frac{b x}{a}\right )\right )}+\frac{c d F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{d x}{c}\right )}{(c+d x) (a d-b c) \left (6 c e F_1\left (5;-n,1;6;-\frac{f x}{e},-\frac{d x}{c}\right )+c f n x F_1\left (6;1-n,1;7;-\frac{f x}{e},-\frac{d x}{c}\right )-d e x F_1\left (6;-n,2;7;-\frac{f x}{e},-\frac{d x}{c}\right )\right )}\right ) \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x^4*(e + f*x)^n)/((a + b*x)*(c + d*x)),x]
[Out]
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Maple [F] time = 0.171, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4} \left ( fx+e \right ) ^{n}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(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. \[ \int \frac{{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^4/((b*x + a)*(d*x + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (f x + e\right )}^{n} x^{4}}{b d x^{2} + a c +{\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^4/((b*x + a)*(d*x + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (e + f x\right )^{n}}{\left (a + b x\right ) \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(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. \[ \int \frac{{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x + e)^n*x^4/((b*x + a)*(d*x + c)),x, algorithm="giac")
[Out]