Optimal. Leaf size=72 \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.028386, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {345, 195, 217, 206} \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 345
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^m \sqrt{a+b x^{2+2 m}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac{x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{2 (1+m)}+\frac{a \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^{1+m}\right )}{2 (1+m)}\\ &=\frac{x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{2 (1+m)}+\frac{a \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{1+m}}{\sqrt{a+b x^{2+2 m}}}\right )}{2 (1+m)}\\ &=\frac{x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{2 (1+m)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{1+m}}{\sqrt{a+b x^{2 (1+m)}}}\right )}{2 \sqrt{b} (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0293429, size = 85, normalized size = 1.18 \[ \frac{x^{m+1} \sqrt{a+b x^{2 m+2}} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{2 m+2};\frac{m+1}{2 m+2}+1;-\frac{b x^{2 m+2}}{a}\right )}{(m+1) \sqrt{\frac{b x^{2 m+2}}{a}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{a+b{x}^{2+2\,m}}\, 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 \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{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 [C] time = 17.4511, size = 121, normalized size = 1.68 \begin{align*} \frac{\sqrt{\pi } a x x^{m}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{2} \\ \frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )} \end{matrix}\middle |{\frac{b x^{2} x^{2 m} e^{i \pi }}{a}} \right )}}{2 a^{\frac{m}{2 \left (m + 1\right )}} a^{\frac{1}{2 \left (m + 1\right )}} m \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right ) + 2 a^{\frac{m}{2 \left (m + 1\right )}} a^{\frac{1}{2 \left (m + 1\right )}} \Gamma \left (\frac{m}{2 \left (m + 1\right )} + 1 + \frac{1}{2 \left (m + 1\right )}\right )} \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 \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]