Optimal. Leaf size=139 \[ \frac{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{a+b x^n+c x^{2 n}}} \]
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Rubi [A] time = 0.083524, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1348, 429} \[ \frac{x \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1348
Rule 429
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x^n+c x^{2 n}}} \, dx &=\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{1}{\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{a+b x^n+c x^{2 n}}}\\ &=\frac{x \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}{n};\frac{1}{2},\frac{1}{2};1+\frac{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 )}{\sqrt{a+b x^n+c x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0866659, size = 166, normalized size = 1.19 \[ \frac{x \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{1}{n};\frac{1}{2},\frac{1}{2};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{a+x^n \left (b+c x^n\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a + b x^{n} + c x^{2 n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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