Integrand size = 24, antiderivative size = 163 \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {(d x)^{1+m} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {3}{2},\frac {3}{2},\frac {1+m+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a d (1+m) \sqrt {a+b x^n+c x^{2 n}}} \]
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Time = 0.10 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1399, 524} \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {(d x)^{m+1} \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {m+1}{n},\frac {3}{2},\frac {3}{2},\frac {m+n+1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a d (m+1) \sqrt {a+b x^n+c x^{2 n}}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {(d x)^m}{\left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{3/2}} \, dx}{a \sqrt {a+b x^n+c x^{2 n}}} \\ & = \frac {(d x)^{1+m} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1+m}{n};\frac {3}{2},\frac {3}{2};\frac {1+m+n}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a d (1+m) \sqrt {a+b x^n+c x^{2 n}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(428\) vs. \(2(163)=326\).
Time = 1.45 (sec) , antiderivative size = 428, normalized size of antiderivative = 2.63 \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {x (d x)^m \left (\left (-4 a c (1+m-n)+b^2 (2+2 m-n)\right ) (1+m+n) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1+m}{n},\frac {1}{2},\frac {1}{2},\frac {1+m+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-2 (1+m) \left ((1+m+n) \left (b^2-2 a c+b c x^n\right )-b c (1+m) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1+m+n}{n},\frac {1}{2},\frac {1}{2},\frac {1+m+2 n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{a \left (-b^2+4 a c\right ) (1+m) n (1+m+n) \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int \frac {\left (d x \right )^{m}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \]
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