Integrand size = 22, antiderivative size = 149 \[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\frac {a x^3 \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {3}{n},-\frac {3}{2},-\frac {3}{2},\frac {3+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.09 (sec) , antiderivative size = 149, 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 = {1399, 524} \[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\frac {a x^3 \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {3}{n},-\frac {3}{2},-\frac {3}{2},\frac {n+3}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{3 \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a+b x^n+c x^{2 n}}\right ) \int x^2 \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}{\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}}}} \\ & = \frac {a x^3 \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {3}{n};-\frac {3}{2},-\frac {3}{2};\frac {3+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 )}{3 \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}}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(475\) vs. \(2(149)=298\).
Time = 1.05 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.19 \[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\frac {x^3 \left (2 (3+n) \left (3 b^2 n^2+4 a c \left (9+18 n+8 n^2\right )+2 b c \left (18+27 n+7 n^2\right ) x^n+4 c^2 \left (9+9 n+2 n^2\right ) x^{2 n}\right ) \left (a+x^n \left (b+c x^n\right )\right )+2 a n^2 (3+n) \left (-3 b^2+4 a c (3+2 n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3}{n},\frac {1}{2},\frac {1}{2},\frac {3+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-3 b n^2 \left (-12 a c (2+n)+b^2 (6+n)\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3+n}{n},\frac {1}{2},\frac {1}{2},2+\frac {3}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{24 c (1+n) (3+n)^2 (3+2 n) \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int x^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\int x^{2} \left (a + b x^{n} + c x^{2 n}\right )^{\frac {3}{2}}\, dx \]
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\[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\int { {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
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\[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\int { {\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b x^n+c x^{2 n}\right )^{3/2} \, dx=\int x^2\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2} \,d x \]
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