Optimal. Leaf size=179 \[ -\frac{a^2 x^3 \sqrt{a x+1} (7 a x+66) \left (c-\frac{c}{a x}\right )^{7/2}}{5 (1-a x)^{7/2}}-\frac{9 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{2 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{(1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.166558, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6134, 6129, 98, 150, 143, 54, 215} \[ -\frac{a^2 x^3 \sqrt{a x+1} (7 a x+66) \left (c-\frac{c}{a x}\right )^{7/2}}{5 (1-a x)^{7/2}}-\frac{9 a^{5/2} x^{7/2} \left (c-\frac{c}{a x}\right )^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}+\frac{2 a x^2 \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{(1-a x)^{3/2}}-\frac{2 x \sqrt{a x+1} \left (c-\frac{c}{a x}\right )^{7/2}}{5 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6134
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^{7/2} \, dx &=\frac{\left (\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{e^{-\tanh ^{-1}(a x)} (1-a x)^{7/2}}{x^{7/2}} \, dx}{(1-a x)^{7/2}}\\ &=\frac{\left (\left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x)^4}{x^{7/2} \sqrt{1+a x}} \, dx}{(1-a x)^{7/2}}\\ &=-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{\left (2 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x)^2 \left (\frac{15 a}{2}-\frac{3 a^2 x}{2}\right )}{x^{5/2} \sqrt{1+a x}} \, dx}{5 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{(1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{\left (4 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{(1-a x) \left (-\frac{99 a^2}{4}-\frac{21 a^3 x}{4}\right )}{x^{3/2} \sqrt{1+a x}} \, dx}{15 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{(1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (66+7 a x)}{5 (1-a x)^{7/2}}-\frac{\left (9 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{2 (1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{(1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (66+7 a x)}{5 (1-a x)^{7/2}}-\frac{\left (9 a^3 \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{(1-a x)^{7/2}}\\ &=\frac{2 a \left (c-\frac{c}{a x}\right )^{7/2} x^2 \sqrt{1+a x}}{(1-a x)^{3/2}}-\frac{2 \left (c-\frac{c}{a x}\right )^{7/2} x \sqrt{1+a x}}{5 \sqrt{1-a x}}-\frac{a^2 \left (c-\frac{c}{a x}\right )^{7/2} x^3 \sqrt{1+a x} (66+7 a x)}{5 (1-a x)^{7/2}}-\frac{9 a^{5/2} \left (c-\frac{c}{a x}\right )^{7/2} x^{7/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{(1-a x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0891843, size = 95, normalized size = 0.53 \[ -\frac{c^3 \sqrt{c-\frac{c}{a x}} \left (\sqrt{a x+1} \left (5 a^3 x^3-92 a^2 x^2+16 a x-2\right )-45 a^{5/2} x^{5/2} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{5 a^3 x^2 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.164, size = 154, normalized size = 0.9 \begin{align*}{\frac{{c}^{3}}{10\,{x}^{2} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 10\,{a}^{7/2}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}+45\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ){x}^{3}{a}^{3}-184\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+32\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}-4\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x} \right ){a}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.52005, size = 744, normalized size = 4.16 \begin{align*} \left [\frac{45 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (5 \, a^{3} c^{3} x^{3} - 92 \, a^{2} c^{3} x^{2} + 16 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{20 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, -\frac{45 \,{\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \,{\left (5 \, a^{3} c^{3} x^{3} - 92 \, a^{2} c^{3} x^{2} + 16 \, a c^{3} x - 2 \, c^{3}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{10 \,{\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]