Integrand size = 18, antiderivative size = 127 \[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{1+m}}{b^3 (1+m)}+\frac {d^2 (3 b c-a d) x^{2+m}}{b^2 (2+m)}+\frac {d^3 x^{3+m}}{b (3+m)}+\frac {(b c-a d)^3 x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1,1-m,\frac {a}{a+b x}\right )}{b^3 m (a+b x)} \]
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Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.35, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 66, 45} \[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\frac {x^{m+1} (b c-a d)^3 \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a b^3 (m+1)}+\frac {d x^{m+1} (b c-a d)^2}{b^3 (m+1)}+\frac {d^2 x^{m+2} (b c-a d)}{b^2 (m+2)}+\frac {c d x^{m+1} (b c-a d)}{b^2 (m+1)}+\frac {c^2 d x^{m+1}}{b (m+1)}+\frac {2 c d^2 x^{m+2}}{b (m+2)}+\frac {d^3 x^{m+3}}{b (m+3)} \]
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Rule 45
Rule 66
Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (b c-a d)^2 x^m}{b^3}+\frac {(b c-a d)^3 x^m}{b^3 (a+b x)}+\frac {d (b c-a d) x^m (c+d x)}{b^2}+\frac {d x^m (c+d x)^2}{b}\right ) \, dx \\ & = \frac {d (b c-a d)^2 x^{1+m}}{b^3 (1+m)}+\frac {d \int x^m (c+d x)^2 \, dx}{b}+\frac {(d (b c-a d)) \int x^m (c+d x) \, dx}{b^2}+\frac {(b c-a d)^3 \int \frac {x^m}{a+b x} \, dx}{b^3} \\ & = \frac {d (b c-a d)^2 x^{1+m}}{b^3 (1+m)}+\frac {(b c-a d)^3 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b^3 (1+m)}+\frac {d \int \left (c^2 x^m+2 c d x^{1+m}+d^2 x^{2+m}\right ) \, dx}{b}+\frac {(d (b c-a d)) \int \left (c x^m+d x^{1+m}\right ) \, dx}{b^2} \\ & = \frac {c^2 d x^{1+m}}{b (1+m)}+\frac {c d (b c-a d) x^{1+m}}{b^2 (1+m)}+\frac {d (b c-a d)^2 x^{1+m}}{b^3 (1+m)}+\frac {2 c d^2 x^{2+m}}{b (2+m)}+\frac {d^2 (b c-a d) x^{2+m}}{b^2 (2+m)}+\frac {d^3 x^{3+m}}{b (3+m)}+\frac {(b c-a d)^3 x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {b x}{a}\right )}{a b^3 (1+m)} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.93 \[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\frac {x^{1+m} \left (d \left (\frac {a^2 d^2}{1+m}+a b d \left (-\frac {3 c}{1+m}-\frac {d x}{2+m}\right )+b^2 \left (\frac {3 c^2}{1+m}+\frac {3 c d x}{2+m}+\frac {d^2 x^2}{3+m}\right )\right )+\frac {(b c-a d)^3 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a (1+m)}\right )}{b^3} \]
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\[\int \frac {x^{m} \left (d x +c \right )^{3}}{b x +a}d x\]
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\[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )}^{3} x^{m}}{b x + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.30 \[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\frac {c^{3} m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {c^{3} x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a \Gamma \left (m + 2\right )} + \frac {3 c^{2} d m x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {6 c^{2} d x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a \Gamma \left (m + 3\right )} + \frac {3 c d^{2} m x^{m + 3} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac {9 c d^{2} x^{m + 3} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 3\right ) \Gamma \left (m + 3\right )}{a \Gamma \left (m + 4\right )} + \frac {d^{3} m x^{m + 4} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} + \frac {4 d^{3} x^{m + 4} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 4\right ) \Gamma \left (m + 4\right )}{a \Gamma \left (m + 5\right )} \]
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\[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )}^{3} x^{m}}{b x + a} \,d x } \]
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\[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\int { \frac {{\left (d x + c\right )}^{3} x^{m}}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {x^m (c+d x)^3}{a+b x} \, dx=\int \frac {x^m\,{\left (c+d\,x\right )}^3}{a+b\,x} \,d x \]
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