Integrand size = 22, antiderivative size = 148 \[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {x^4 \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {4}{n},-\frac {1}{2},-\frac {1}{2},\frac {4+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.12 (sec) , antiderivative size = 148, 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} \[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {x^4 \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {4}{n},-\frac {1}{2},-\frac {1}{2},\frac {n+4}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+b x^n+c x^{2 n}} \int x^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}}} \, 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 {x^4 \sqrt {a+b x^n+c x^{2 n}} F_1\left (\frac {4}{n};-\frac {1}{2},-\frac {1}{2};\frac {4+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 )}{4 \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. \(365\) vs. \(2(148)=296\).
Time = 0.61 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.47 \[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\frac {x^4 \left (4 (4+n) \left (a+x^n \left (b+c x^n\right )\right )+a n (4+n) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{n},\frac {1}{2},\frac {1}{2},\frac {4+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+2 b n x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4+n}{n},\frac {1}{2},\frac {1}{2},2+\frac {4}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )}{4 (4+n)^2 \sqrt {a+x^n \left (b+c x^n\right )}} \]
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\[\int x^{3} \sqrt {a +b \,x^{n}+c \,x^{2 n}}d x\]
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Exception generated. \[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\int x^{3} \sqrt {a + b x^{n} + c x^{2 n}}\, dx \]
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\[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\int { \sqrt {c x^{2 \, n} + b x^{n} + a} x^{3} \,d x } \]
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\[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\int { \sqrt {c x^{2 \, n} + b x^{n} + a} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \sqrt {a+b x^n+c x^{2 n}} \, dx=\int x^3\,\sqrt {a+b\,x^n+c\,x^{2\,n}} \,d x \]
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