Optimal. Leaf size=160 \[ \frac{x^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^2}+\frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^3}+\frac{7 x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{24 a^3}+\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^3}-\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt{1-a x} \sqrt{a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.557536, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {6167, 6159, 6129, 90, 80, 50, 41, 216} \[ \frac{x^2 (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^2}+\frac{x (a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{6 a^3}+\frac{7 x (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{24 a^3}+\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}}}{8 a^3}-\frac{7 x \sqrt{c-\frac{c}{a^2 x^2}} \sin ^{-1}(a x)}{8 a^3 \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6159
Rule 6129
Rule 90
Rule 80
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int e^{2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^3 \, dx\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{2 \tanh ^{-1}(a x)} x^2 \sqrt{1-a x} \sqrt{1+a x} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=-\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{x^2 (1+a x)^{3/2}}{\sqrt{1-a x}} \, dx}{\sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}+\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(-1-2 a x) (1+a x)^{3/2}}{\sqrt{1-a x}} \, dx}{4 a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{6 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}-\frac{\left (7 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{(1+a x)^{3/2}}{\sqrt{1-a x}} \, dx}{12 a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{24 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{6 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}-\frac{\left (7 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{\sqrt{1+a x}}{\sqrt{1-a x}} \, dx}{8 a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x}{8 a^3}+\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{24 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{6 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}-\frac{\left (7 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{8 a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x}{8 a^3}+\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{24 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{6 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}-\frac{\left (7 \sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{8 a^2 \sqrt{1-a x} \sqrt{1+a x}}\\ &=\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x}{8 a^3}+\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)}{24 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x (1+a x)^2}{6 a^3}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2 (1+a x)^2}{4 a^2}-\frac{7 \sqrt{c-\frac{c}{a^2 x^2}} x \sin ^{-1}(a x)}{8 a^3 \sqrt{1-a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [A] time = 0.0911736, size = 93, normalized size = 0.58 \[ \frac{x \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (6 a^3 x^3+16 a^2 x^2+21 a x+32\right )+21 \log \left (\sqrt{a^2 x^2-1}+a x\right )\right )}{24 a^3 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.191, size = 196, normalized size = 1.2 \begin{align*} -{\frac{x}{24\,c{a}^{4}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ( -6\,x \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{a}^{4}-16\, \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}} \right ) ^{3/2}{a}^{3}-27\,\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}x{a}^{2}c+27\,{c}^{3/2}\ln \left ( x\sqrt{c}+\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}} \right ) -48\,{c}^{3/2}\ln \left ({\frac{1}{\sqrt{c}} \left ( \sqrt{c}\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}+cx \right ) } \right ) -48\,\sqrt{{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) c}{{a}^{2}}}}ac \right ){\frac{1}{\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}}}}}}} \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{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}} x^{3}}{a x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6971, size = 487, normalized size = 3.04 \begin{align*} \left [\frac{2 \,{\left (6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} + 21 \, \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{48 \, a^{4}}, \frac{{\left (6 \, a^{4} x^{4} + 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 32 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{24 \, a^{4}}\right ] \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^{3} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{a x - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15859, size = 173, normalized size = 1.08 \begin{align*} \frac{1}{48} \,{\left (2 \, \sqrt{a^{2} c x^{2} - c}{\left ({\left (2 \, x{\left (\frac{3 \, x \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{8 \, \mathrm{sgn}\left (x\right )}{a^{3}}\right )} + \frac{21 \, \mathrm{sgn}\left (x\right )}{a^{4}}\right )} x + \frac{32 \, \mathrm{sgn}\left (x\right )}{a^{5}}\right )} - \frac{42 \, \sqrt{c} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a^{4}{\left | a \right |}} + \frac{{\left (21 \, a \sqrt{c} \log \left ({\left | c \right |}\right ) - 64 \, \sqrt{-c}{\left | a \right |}\right )} \mathrm{sgn}\left (x\right )}{a^{5}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]