Optimal. Leaf size=89 \[ -\frac{11 a^2 \sqrt{c-a c x}}{8 x}-\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-\frac{11 a \sqrt{c-a c x}}{12 x^2}-\frac{\sqrt{c-a c x}}{3 x^3} \]
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Rubi [A] time = 0.128482, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6130, 21, 78, 51, 63, 208} \[ -\frac{11 a^2 \sqrt{c-a c x}}{8 x}-\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )-\frac{11 a \sqrt{c-a c x}}{12 x^2}-\frac{\sqrt{c-a c x}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6130
Rule 21
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^4} \, dx &=\int \frac{(1+a x) \sqrt{c-a c x}}{x^4 (1-a x)} \, dx\\ &=c \int \frac{1+a x}{x^4 \sqrt{c-a c x}} \, dx\\ &=-\frac{\sqrt{c-a c x}}{3 x^3}+\frac{1}{6} (11 a c) \int \frac{1}{x^3 \sqrt{c-a c x}} \, dx\\ &=-\frac{\sqrt{c-a c x}}{3 x^3}-\frac{11 a \sqrt{c-a c x}}{12 x^2}+\frac{1}{8} \left (11 a^2 c\right ) \int \frac{1}{x^2 \sqrt{c-a c x}} \, dx\\ &=-\frac{\sqrt{c-a c x}}{3 x^3}-\frac{11 a \sqrt{c-a c x}}{12 x^2}-\frac{11 a^2 \sqrt{c-a c x}}{8 x}+\frac{1}{16} \left (11 a^3 c\right ) \int \frac{1}{x \sqrt{c-a c x}} \, dx\\ &=-\frac{\sqrt{c-a c x}}{3 x^3}-\frac{11 a \sqrt{c-a c x}}{12 x^2}-\frac{11 a^2 \sqrt{c-a c x}}{8 x}-\frac{1}{8} \left (11 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a c}} \, dx,x,\sqrt{c-a c x}\right )\\ &=-\frac{\sqrt{c-a c x}}{3 x^3}-\frac{11 a \sqrt{c-a c x}}{12 x^2}-\frac{11 a^2 \sqrt{c-a c x}}{8 x}-\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0563852, size = 63, normalized size = 0.71 \[ -\frac{\left (33 a^2 x^2+22 a x+8\right ) \sqrt{c-a c x}}{24 x^3}-\frac{11}{8} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{c}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 80, normalized size = 0.9 \begin{align*} 2\,{a}^{3}{c}^{3} \left ( -{\frac{1}{{a}^{3}{c}^{3}{x}^{3}} \left ({\frac{11\, \left ( -acx+c \right ) ^{5/2}}{16\,{c}^{2}}}-{\frac{11\, \left ( -acx+c \right ) ^{3/2}}{6\,c}}+{\frac{21\,\sqrt{-acx+c}}{16}} \right ) }-{\frac{11}{16\,{c}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{-acx+c}}{\sqrt{c}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84949, size = 331, normalized size = 3.72 \begin{align*} \left [\frac{33 \, a^{3} \sqrt{c} x^{3} \log \left (\frac{a c x + 2 \, \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{x}\right ) - 2 \,{\left (33 \, a^{2} x^{2} + 22 \, a x + 8\right )} \sqrt{-a c x + c}}{48 \, x^{3}}, \frac{33 \, a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{-c}}{c}\right ) -{\left (33 \, a^{2} x^{2} + 22 \, a x + 8\right )} \sqrt{-a c x + c}}{24 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 42.3672, size = 439, normalized size = 4.93 \begin{align*} \frac{66 a^{3} c^{6} \sqrt{- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{80 a^{3} c^{5} \left (- a c x + c\right )^{\frac{3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac{30 a^{3} c^{4} \left (- a c x + c\right )^{\frac{5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac{10 a^{3} c^{4} \sqrt{- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (- c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} - \frac{5 a^{3} c^{4} \sqrt{\frac{1}{c^{7}}} \log{\left (c^{4} \sqrt{\frac{1}{c^{7}}} + \sqrt{- a c x + c} \right )}}{16} + \frac{6 a^{3} c^{3} \left (- a c x + c\right )^{\frac{3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac{3 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (- c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} - \frac{3 a^{3} c^{3} \sqrt{\frac{1}{c^{5}}} \log{\left (c^{3} \sqrt{\frac{1}{c^{5}}} + \sqrt{- a c x + c} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21523, size = 134, normalized size = 1.51 \begin{align*} \frac{11 \, a^{3} c \arctan \left (\frac{\sqrt{-a c x + c}}{\sqrt{-c}}\right )}{8 \, \sqrt{-c}} - \frac{33 \,{\left (a c x - c\right )}^{2} \sqrt{-a c x + c} a^{3} c - 88 \,{\left (-a c x + c\right )}^{\frac{3}{2}} a^{3} c^{2} + 63 \, \sqrt{-a c x + c} a^{3} c^{3}}{24 \, a^{3} c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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