Optimal. Leaf size=130 \[ \frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.308009, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6152, 1807, 835, 807, 266, 63, 208} \[ \frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6152
Rule 1807
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x^5} \, dx &=c \int \frac{(1-a x)^2}{x^5 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}-\frac{1}{4} \int \frac{8 a c-7 a^2 c x}{x^4 \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}+\frac{\int \frac{21 a^2 c^2-16 a^3 c^2 x}{x^3 \sqrt{c-a^2 c x^2}} \, dx}{12 c}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}-\frac{\int \frac{32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt{c-a^2 c x^2}} \, dx}{24 c^2}\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{1}{8} \left (7 a^4 c\right ) \int \frac{1}{x \sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}+\frac{1}{16} \left (7 a^4 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{1}{8} \left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c-a^2 c x^2}\right )\\ &=-\frac{\sqrt{c-a^2 c x^2}}{4 x^4}+\frac{2 a \sqrt{c-a^2 c x^2}}{3 x^3}-\frac{7 a^2 \sqrt{c-a^2 c x^2}}{8 x^2}+\frac{4 a^3 \sqrt{c-a^2 c x^2}}{3 x}-\frac{7}{8} a^4 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a^2 c x^2}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.134225, size = 95, normalized size = 0.73 \[ \frac{\left (32 a^3 x^3-21 a^2 x^2+16 a x-6\right ) \sqrt{c-a^2 c x^2}}{24 x^4}-\frac{7}{8} a^4 \sqrt{c} \log \left (\sqrt{c} \sqrt{c-a^2 c x^2}+c\right )+\frac{7}{8} a^4 \sqrt{c} \log (x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.052, size = 279, normalized size = 2.2 \begin{align*} -{\frac{1}{4\,c{x}^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{a}^{2}}{8\,c{x}^{2}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{a}^{4}}{8}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{-{a}^{2}c{x}^{2}+c} \right ) } \right ) }+{\frac{7\,{a}^{4}}{8}\sqrt{-{a}^{2}c{x}^{2}+c}}+2\,{\frac{{a}^{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{3/2}}{cx}}+2\,{a}^{5}x\sqrt{-{a}^{2}c{x}^{2}+c}+2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-{a}^{2}c{x}^{2}+c}}} \right ) }-2\,{a}^{4}\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }-2\,{\frac{{a}^{5}c}{\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) }+{\frac{2\,a}{3\,c{x}^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{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{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 1\right )}}{{\left (a x + 1\right )}^{2} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.4791, size = 417, normalized size = 3.21 \begin{align*} \left [\frac{21 \, a^{4} \sqrt{c} x^{4} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{c} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac{21 \, a^{4} \sqrt{-c} x^{4} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) -{\left (32 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 16 \, a x - 6\right )} \sqrt{-a^{2} c x^{2} + c}}{24 \, x^{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{\sqrt{- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\, dx - \int \frac{a x \sqrt{- a^{2} c x^{2} + c}}{a x^{6} + x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17048, size = 348, normalized size = 2.68 \begin{align*} \frac{1}{192} \,{\left (\frac{336 \, a^{3} c \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{-c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} - \frac{4 \,{\left (21 \, \pi a^{3} c - 64 \, a^{3} c\right )} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{\sqrt{-c}} + \frac{75 \, a^{3}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{3} c \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 83 \, a^{3}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{2} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + 21 \, a^{3} c^{4} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 77 \, a^{3} c^{3}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )}{{\left (c - \frac{c}{a x + 1}\right )}^{4}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]