Optimal. Leaf size=148 \[ \frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{a+b x^n+c x^{2 n}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.110621, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1385, 510} \[ \frac{x^2 \sqrt{\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1} F_1\left (\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 \sqrt{a+b x^n+c x^{2 n}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1385
Rule 510
Rubi steps
\begin{align*} \int \frac{x}{\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{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}}}} \, dx}{\sqrt{a+b x^n+c x^{2 n}}}\\ &=\frac{x^2 \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{2}{n};\frac{1}{2},\frac{1}{2};\frac{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 )}{2 \sqrt{a+b x^n+c x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.140502, size = 175, normalized size = 1.18 \[ \frac{x^2 \sqrt{\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}} \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}} F_1\left (\frac{2}{n};\frac{1}{2},\frac{1}{2};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{2 \sqrt{a+x^n \left (b+c x^n\right )}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{x{\frac{1}{\sqrt{a+b{x}^{n}+c{x}^{2\,n}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a + b x^{n} + c x^{2 n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{c x^{2 \, n} + b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]