Optimal. Leaf size=82 \[ \frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^m \sqrt {c-a^2 c x^2}}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6192, 6193, 43} \[ \frac {x^m \sqrt {c-a^2 c x^2}}{a (m+1) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{m+1} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6192
Rule 6193
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{\coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^{1+m} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int x^m (1+a x) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (x^m+a x^{1+m}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {x^m \sqrt {c-a^2 c x^2}}{a (1+m) \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x^{1+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 56, normalized size = 0.68 \[ \frac {x^m \sqrt {c-a^2 c x^2} (a m x+a x+m+2)}{a (m+1) (m+2) \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 74, normalized size = 0.90 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left ({\left (a m + a\right )} x^{2} + {\left (m + 2\right )} x\right )} x^{m} \sqrt {\frac {a x - 1}{a x + 1}}}{m^{2} - {\left (a m^{2} + 3 \, a m + 2 \, a\right )} x + 3 \, m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 62, normalized size = 0.76 \[ \frac {x^{1+m} \left (a m x +a x +m +2\right ) \sqrt {-a^{2} c \,x^{2}+c}}{\left (2+m \right ) \left (1+m \right ) \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 54, normalized size = 0.66 \[ \frac {{\left (a \sqrt {-c} {\left (m + 1\right )} x^{2} + \sqrt {-c} {\left (m + 2\right )} x\right )} {\left (a x + 1\right )} x^{m}}{{\left (m^{2} + 3 \, m + 2\right )} a x + m^{2} + 3 \, m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.59, size = 93, normalized size = 1.13 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (\frac {x^m\,x^2\,\sqrt {c-a^2\,c\,x^2}\,\left (m+1\right )}{m^2+3\,m+2}+\frac {x\,x^m\,\sqrt {c-a^2\,c\,x^2}\,\left (m+2\right )}{a\,\left (m^2+3\,m+2\right )}\right )}{x-\frac {1}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________