Integrand size = 25, antiderivative size = 319 \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {e^2 (e+f x)^{1+n}}{b d f^3 (1+n)}+\frac {(b c+a d) e (e+f x)^{1+n}}{b^2 d^2 f^2 (1+n)}+\frac {\left (b^2 c^2+a b c d+a^2 d^2\right ) (e+f x)^{1+n}}{b^3 d^3 f (1+n)}-\frac {2 e (e+f x)^{2+n}}{b d f^3 (2+n)}-\frac {(b c+a d) (e+f x)^{2+n}}{b^2 d^2 f^2 (2+n)}+\frac {(e+f x)^{3+n}}{b d f^3 (3+n)}-\frac {a^4 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{b^3 (b c-a d) (b e-a f) (1+n)}+\frac {c^4 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{d^3 (b c-a d) (d e-c f) (1+n)} \]
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Time = 0.20 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {186, 45, 70} \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=-\frac {a^4 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{b^3 (n+1) (b c-a d) (b e-a f)}+\frac {\left (a^2 d^2+a b c d+b^2 c^2\right ) (e+f x)^{n+1}}{b^3 d^3 f (n+1)}+\frac {e (a d+b c) (e+f x)^{n+1}}{b^2 d^2 f^2 (n+1)}-\frac {(a d+b c) (e+f x)^{n+2}}{b^2 d^2 f^2 (n+2)}+\frac {c^4 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{d^3 (n+1) (b c-a d) (d e-c f)}+\frac {e^2 (e+f x)^{n+1}}{b d f^3 (n+1)}-\frac {2 e (e+f x)^{n+2}}{b d f^3 (n+2)}+\frac {(e+f x)^{n+3}}{b d f^3 (n+3)} \]
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Rule 45
Rule 70
Rule 186
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.88 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.89 \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n} \left (-\frac {a^4 d^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{(b c-a d) (b e-a f)}+\frac {-\left ((b c-a d) (-d e+c f) \left (a^2 d^2 f^2 \left (6+5 n+n^2\right )+a b d f (3+n) (c f (2+n)+d (e-f (1+n) x))+b^2 \left (c^2 f^2 \left (6+5 n+n^2\right )+c d f (3+n) (e-f (1+n) x)+d^2 \left (2 e^2-2 e f (1+n) x+f^2 \left (2+3 n+n^2\right ) x^2\right )\right )\right )\right )+b^3 c^4 f^3 \left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{(-b c+a d) f^3 (-d e+c f) (2+n) (3+n)}\right )}{b^3 d^3 (1+n)} \]
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\[\int \frac {x^{4} \left (f x +e \right )^{n}}{\left (b x +a \right ) \left (d x +c \right )}d x\]
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\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{4}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^4 (e+f x)^n}{(a+b x) (c+d x)} \, dx=\int \frac {x^4\,{\left (e+f\,x\right )}^n}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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