Optimal. Leaf size=309 \[ \frac{92 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{472 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1576 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{38 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.324192, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6176, 6181, 98, 152, 12, 93, 206} \[ \frac{92 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{472 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{1576 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{38 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6176
Rule 6181
Rule 98
Rule 152
Rule 12
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt{c-a c x} \, dx &=\frac{\sqrt{c-a c x} \int e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}} x^{7/2} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^{11/2} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{19}{2 a}-\frac{17 x}{2 a^2}}{x^{9/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{9 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{69}{2 a^2}+\frac{57 x}{2 a^3}}{x^{7/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{63 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{177}{2 a^3}-\frac{69 x}{a^4}}{x^{5/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{315 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (16 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\frac{591}{4 a^4}+\frac{177 x}{2 a^5}}{x^{3/2} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{945 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{\left (32 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int -\frac{945}{8 a^5 \sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{945 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^5 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{a^5 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{1576 \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{315 a^4 \sqrt{1-\frac{1}{a x}}}+\frac{472 \sqrt{1+\frac{1}{a x}} x \sqrt{c-a c x}}{315 a^3 \sqrt{1-\frac{1}{a x}}}+\frac{92 \sqrt{1+\frac{1}{a x}} x^2 \sqrt{c-a c x}}{105 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{38 \sqrt{1+\frac{1}{a x}} x^3 \sqrt{c-a c x}}{63 a \sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{1+\frac{1}{a x}} x^4 \sqrt{c-a c x}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{a^{9/2} \sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.108288, size = 130, normalized size = 0.42 \[ \frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} \left (35 a^4 x^4+95 a^3 x^3+138 a^2 x^2+236 a x+788\right )-630 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{315 a^{9/2} \sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.177, size = 161, normalized size = 0.5 \begin{align*} -{\frac{2\,ax-2}{ \left ( 315\,ax+315 \right ){a}^{4}}\sqrt{-c \left ( ax-1 \right ) } \left ( -35\,{x}^{4}{a}^{4}\sqrt{-c \left ( ax+1 \right ) }-95\,{x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }-138\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+630\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -236\,xa\sqrt{-c \left ( ax+1 \right ) }-788\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \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 c x + c} x^{3}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69984, size = 749, normalized size = 2.42 \begin{align*} \left [\frac{2 \,{\left (315 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{315 \,{\left (a^{5} x - a^{4}\right )}}, -\frac{2 \,{\left (630 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{315 \,{\left (a^{5} x - a^{4}\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 [C] time = 1.28071, size = 244, normalized size = 0.79 \begin{align*} -\frac{1260 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 2584 \, \sqrt{2} \sqrt{-c}}{315 \, a^{4} \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (630 \, \sqrt{2} c^{\frac{9}{2}} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right ) - 35 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} + 45 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c - 63 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{2} + 105 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{3} - 630 \, \sqrt{-a c x - c} c^{4}\right )}}{315 \, a^{4} c^{4} \mathrm{sgn}\left (-a c x - c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]