Integrand size = 30, antiderivative size = 329 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {D (c+d x)^{1+n}}{b^3 d (1+n)}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac {\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}-\frac {\left (b^3 \left (2 c^2 C+2 B c d n-A d^2 (1-n) n\right )-a^3 d^2 D \left (6+5 n+n^2\right )+a^2 b d (2+n) (6 c D+C d (1+n))-a b^2 \left (6 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )\right ) (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{2 b^3 (b c-a d)^3 (1+n)} \]
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Time = 0.40 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1635, 963, 81, 70} \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=-\frac {(c+d x)^{n+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^3 (a+b x)^2 (b c-a d)}-\frac {(c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (c+d x)}{b c-a d}\right ) \left (a^3 \left (-d^2\right ) D \left (n^2+5 n+6\right )+a^2 b d (n+2) (6 c D+C d (n+1))-a b^2 \left (B d^2 n (n+1)+6 c^2 D+4 c C d (n+1)\right )+b^3 \left (-A d^2 (1-n) n+2 B c d n+2 c^2 C\right )\right )}{2 b^3 (n+1) (b c-a d)^3}-\frac {(c+d x)^{n+1} \left (a^3 (-d) D (n+5)+a^2 b (6 c D+C d (n+3))-a b^2 (B d (n+1)+4 c C)+b^3 (2 B c-A d (1-n))\right )}{2 b^3 (a+b x) (b c-a d)^2}+\frac {D (c+d x)^{n+1}}{b^3 d (n+1)} \]
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Rule 70
Rule 81
Rule 963
Rule 1635
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac {\int \frac {(c+d x)^n \left (-2 B c+A d (1-n)+\frac {a^3 d D (1+n)}{b^3}+\frac {a (2 c C+B d (1+n))}{b}-\frac {a^2 (2 c D+C d (1+n))}{b^2}-\frac {2 (b c-a d) (b C-a D) x}{b^2}-2 \left (c-\frac {a d}{b}\right ) D x^2\right )}{(a+b x)^2} \, dx}{2 (b c-a d)} \\ & = -\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac {\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}+\frac {\int \frac {(c+d x)^n \left (2 c^2 C+2 B c d n-A d^2 (1-n) n-\frac {a^3 d^2 D \left (4+5 n+n^2\right )}{b^3}-\frac {a \left (4 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )}{b}+\frac {a^2 d \left (2 c D (4+3 n)+C d \left (2+3 n+n^2\right )\right )}{b^2}+\frac {2 (b c-a d)^2 D x}{b^2}\right )}{a+b x} \, dx}{2 (b c-a d)^2} \\ & = \frac {D (c+d x)^{1+n}}{b^3 d (1+n)}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac {\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}+\frac {\left (b^3 \left (2 c^2 C+2 B c d n-A d^2 (1-n) n\right )-a^3 d^2 D \left (6+5 n+n^2\right )+a^2 b d (2+n) (6 c D+C d (1+n))-a b^2 \left (6 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )\right ) \int \frac {(c+d x)^n}{a+b x} \, dx}{2 b^3 (b c-a d)^2} \\ & = \frac {D (c+d x)^{1+n}}{b^3 d (1+n)}-\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d) (a+b x)^2}-\frac {\left (b^3 (2 B c-A d (1-n))-a^3 d D (5+n)-a b^2 (4 c C+B d (1+n))+a^2 b (6 c D+C d (3+n))\right ) (c+d x)^{1+n}}{2 b^3 (b c-a d)^2 (a+b x)}-\frac {\left (b^3 \left (2 c^2 C+2 B c d n-A d^2 (1-n) n\right )-a^3 d^2 D \left (6+5 n+n^2\right )+a^2 b d (2+n) (6 c D+C d (1+n))-a b^2 \left (6 c^2 D+4 c C d (1+n)+B d^2 n (1+n)\right )\right ) (c+d x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (c+d x)}{b c-a d}\right )}{2 b^3 (b c-a d)^3 (1+n)} \\ \end{align*}
Time = 1.31 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.57 \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {(c+d x)^{1+n} \left (\frac {D}{d}-\frac {(b C-3 a D) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{b c-a d}+\frac {d \left (b^2 B-2 a b C+3 a^2 D\right ) \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^2}-\frac {d^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,\frac {b (c+d x)}{b c-a d}\right )}{(b c-a d)^3}\right )}{b^3 (1+n)} \]
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\[\int \frac {\left (d x +c \right )^{n} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (b x +a \right )^{3}}d x\]
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\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int { \frac {{\left (\mathit {capitalD} x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int \frac {\left (c + d x\right )^{n} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (a + b x\right )^{3}}\, dx \]
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\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{n}}{{\left (b x + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^n \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int \frac {{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^3} \,d x \]
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