Integrand size = 20, antiderivative size = 295 \[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (1+n)} \]
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Time = 0.17 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {102, 152, 72, 71} \[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{n+1} (c+d x)^{1-n} \left (a^2 d^2 \left (n^2-5 n+6\right )+2 a b c d \left (3-n^2\right )-2 b d x (a d (3-n)+b c (n+3))+b^2 c^2 \left (n^2+5 n+6\right )\right )}{24 b^3 d^3}-\frac {(a+b x)^{n+1} (c+d x)^{-n} \left (a^3 d^3 \left (-n^3+6 n^2-11 n+6\right )+3 a^2 b c d^2 \left (n^3-2 n^2-n+2\right )+3 a b^2 c^2 d \left (-n^3-2 n^2+n+2\right )+b^3 c^3 \left (n^3+6 n^2+11 n+6\right )\right ) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (n+1)}+\frac {x^2 (a+b x)^{n+1} (c+d x)^{1-n}}{4 b d} \]
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Rule 71
Rule 72
Rule 102
Rule 152
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {\int x (a+b x)^n (c+d x)^{-n} (-2 a c+(-a d (3-n)-b c (3+n)) x) \, dx}{4 b d} \\ & = \frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) \int (a+b x)^n (c+d x)^{-n} \, dx}{24 b^3 d^3} \\ & = \frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n\right ) \int (a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-n} \, dx}{24 b^3 d^3} \\ & = \frac {x^2 (a+b x)^{1+n} (c+d x)^{1-n}}{4 b d}+\frac {(a+b x)^{1+n} (c+d x)^{1-n} \left (2 a b c d \left (3-n^2\right )+a^2 d^2 \left (6-5 n+n^2\right )+b^2 c^2 \left (6+5 n+n^2\right )-2 b d (a d (3-n)+b c (3+n)) x\right )}{24 b^3 d^3}-\frac {\left (3 a b^2 c^2 d \left (2+n-2 n^2-n^3\right )+a^3 d^3 \left (6-11 n+6 n^2-n^3\right )+3 a^2 b c d^2 \left (2-n-2 n^2+n^3\right )+b^3 c^3 \left (6+11 n+6 n^2+n^3\right )\right ) (a+b x)^{1+n} (c+d x)^{-n} \left (\frac {b (c+d x)}{b c-a d}\right )^n \, _2F_1\left (n,1+n;2+n;-\frac {d (a+b x)}{b c-a d}\right )}{24 b^4 d^3 (1+n)} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.89 \[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\frac {(a+b x)^{1+n} (c+d x)^{-n} \left (b^3 d^2 (1+n) x^2 (c+d x)-(b c-a d)^2 (-a d (-3+n)+b c (3+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (-2+n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )+2 b c (b c-a d) (-a d (-2+n)+b c (3+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (-1+n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )-b^2 c^2 (-a d (-1+n)+b c (3+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{4 b^4 d^3 (1+n)} \]
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\[\int x^{3} \left (b x +a \right )^{n} \left (d x +c \right )^{-n}d x\]
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\[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}} \,d x } \]
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Exception generated. \[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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\[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}} \,d x } \]
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\[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x^{3}}{{\left (d x + c\right )}^{n}} \,d x } \]
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Timed out. \[ \int x^3 (a+b x)^n (c+d x)^{-n} \, dx=\int \frac {x^3\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n} \,d x \]
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