Integrand size = 22, antiderivative size = 76 \[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\frac {C (c x)^{1+m}}{b c (1+m)}+\frac {(A b-a C) (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a b c (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {470, 371} \[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\frac {(c x)^{m+1} (A b-a C) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a b c (m+1)}+\frac {C (c x)^{m+1}}{b c (m+1)} \]
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Rule 371
Rule 470
Rubi steps \begin{align*} \text {integral}& = \frac {C (c x)^{1+m}}{b c (1+m)}-\frac {(-A b (1+m)+a C (1+m)) \int \frac {(c x)^m}{a+b x^2} \, dx}{b (1+m)} \\ & = \frac {C (c x)^{1+m}}{b c (1+m)}+\frac {(A b-a C) (c x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a b c (1+m)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74 \[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\frac {x (c x)^m \left (a C+(A b-a C) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )\right )}{a b (1+m)} \]
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\[\int \frac {\left (c x \right )^{m} \left (C \,x^{2}+A \right )}{b \,x^{2}+a}d x\]
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\[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\int { \frac {{\left (C x^{2} + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.82 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.64 \[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\frac {A c^{m} m x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A c^{m} x^{m + 1} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {C c^{m} m x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {3 C c^{m} x^{m + 3} \Phi \left (\frac {b x^{2} e^{i \pi }}{a}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{4 a \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]
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\[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\int { \frac {{\left (C x^{2} + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \]
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\[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\int { \frac {{\left (C x^{2} + A\right )} \left (c x\right )^{m}}{b x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {(c x)^m \left (A+C x^2\right )}{a+b x^2} \, dx=\int \frac {\left (C\,x^2+A\right )\,{\left (c\,x\right )}^m}{b\,x^2+a} \,d x \]
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