Optimal. Leaf size=319 \[ \frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}-\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]
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Rubi [A] time = 0.279362, 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, Rules used = {180, 43, 68} \[ \frac{\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}-\frac{a^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac{e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac{(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac{c^4 (e+f x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac{e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac{2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac{(e+f x)^{n+3}}{b d f^3 (n+3)} \]
Antiderivative was successfully verified.
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Rule 180
Rule 43
Rule 68
Rubi steps
\begin{align*} \int \frac{x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx &=\int \left (\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^n}{b^3 d^3}-\frac{(b c+a d) x (e+f x)^n}{b^2 d^2}+\frac{x^2 (e+f x)^n}{b d}+\frac{a^4 (e+f x)^n}{b^3 (b c-a d) (a+b x)}+\frac{c^4 (e+f x)^n}{d^3 (-b c+a d) (c+d x)}\right ) \, dx\\ &=\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}+\frac{\int x^2 (e+f x)^n \, dx}{b d}+\frac{a^4 \int \frac{(e+f x)^n}{a+b x} \, dx}{b^3 (b c-a d)}-\frac{c^4 \int \frac{(e+f x)^n}{c+d x} \, dx}{d^3 (b c-a d)}-\frac{(b c+a d) \int x (e+f x)^n \, dx}{b^2 d^2}\\ &=\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac{a^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac{c^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)}+\frac{\int \left (\frac{e^2 (e+f x)^n}{f^2}-\frac{2 e (e+f x)^{1+n}}{f^2}+\frac{(e+f x)^{2+n}}{f^2}\right ) \, dx}{b d}-\frac{(b c+a d) \int \left (-\frac{e (e+f x)^n}{f}+\frac{(e+f x)^{1+n}}{f}\right ) \, dx}{b^2 d^2}\\ &=\frac{e^2 (e+f x)^{1+n}}{b d f^3 (1+n)}+\frac{(b c+a d) e (e+f x)^{1+n}}{b^2 d^2 f^2 (1+n)}+\frac{\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac{2 e (e+f x)^{2+n}}{b d f^3 (2+n)}-\frac{(b c+a d) (e+f x)^{2+n}}{b^2 d^2 f^2 (2+n)}+\frac{(e+f x)^{3+n}}{b d f^3 (3+n)}-\frac{a^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac{c^4 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac{d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)}\\ \end{align*}
Mathematica [A] time = 1.32229, size = 285, normalized size = 0.89 \[ \frac{(e+f x)^{n+1} \left (\frac{b^3 c^4 f^3 \left (n^2+5 n+6\right ) \, _2F_1\left (1,n+1;n+2;\frac{d (e+f x)}{d e-c f}\right )-(b c-a d) (c f-d e) \left (a^2 d^2 f^2 \left (n^2+5 n+6\right )+a b d f (n+3) (c f (n+2)+d (e-f (n+1) x))+b^2 \left (c^2 f^2 \left (n^2+5 n+6\right )+c d f (n+3) (e-f (n+1) x)+d^2 \left (2 e^2-2 e f (n+1) x+f^2 \left (n^2+3 n+2\right ) x^2\right )\right )\right )}{f^3 (n+2) (n+3) (a d-b c) (c f-d e)}-\frac{a^4 d^3 \, _2F_1\left (1,n+1;n+2;\frac{b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f)}\right )}{b^3 d^3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{n}{x}^{4}}{ \left ( bx+a \right ) \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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