Integrand size = 29, antiderivative size = 167 \[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=-\frac {d x^{1+m} \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {1}{2},1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}}+\frac {c x^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)} \]
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Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2187, 371, 525, 524} \[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\frac {c x^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac {d x^{m+1} \sqrt {\frac {b x^n}{a}+1} \operatorname {AppellF1}\left (\frac {m+1}{n},\frac {1}{2},1,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt {a+b x^n}} \]
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Rule 371
Rule 524
Rule 525
Rule 2187
Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {x^m}{a^2 c^2-a d^2+a b c^2 x^n} \, dx-(a d) \int \frac {x^m}{\sqrt {a+b x^n} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx \\ & = \frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)}-\frac {\left (a d \sqrt {1+\frac {b x^n}{a}}\right ) \int \frac {x^m}{\sqrt {1+\frac {b x^n}{a}} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx}{\sqrt {a+b x^n}} \\ & = -\frac {d x^{1+m} \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1+m}{n};\frac {1}{2},1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}}+\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.93 \[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\frac {x^{1+m} \left (-d \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {1}{2},1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )+c \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b c^2 x^n}{a c^2-d^2}\right )\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}} \]
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\[\int \frac {x^{m}}{a c +b c \,x^{n}+d \sqrt {a +b \,x^{n}}}d x\]
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\[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\int { \frac {x^{m}}{b c x^{n} + a c + \sqrt {b x^{n} + a} d} \,d x } \]
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\[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\int \frac {x^{m}}{a c + b c x^{n} + d \sqrt {a + b x^{n}}}\, dx \]
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\[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\int { \frac {x^{m}}{b c x^{n} + a c + \sqrt {b x^{n} + a} d} \,d x } \]
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\[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\int { \frac {x^{m}}{b c x^{n} + a c + \sqrt {b x^{n} + a} d} \,d x } \]
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Timed out. \[ \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx=\int \frac {x^m}{a\,c+d\,\sqrt {a+b\,x^n}+b\,c\,x^n} \,d x \]
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