Optimal. Leaf size=89 \[ \frac {11}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {11 a^2 \sqrt {c-a c x}}{8 x}+\frac {\sqrt {c-a c x}}{3 x^3}+\frac {11 a \sqrt {c-a c x}}{12 x^2} \]
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Rubi [A] time = 0.22, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6167, 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 21
Rule 51
Rule 63
Rule 78
Rule 208
Rule 6130
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx &=-\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\\ &=-\left (c \int \frac {1+a x}{x^4 \sqrt {c-a c x}} \, dx\right )\\ &=\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*}
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Mathematica [A] time = 0.07, size = 63, normalized size = 0.71 \[ \frac {11}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {\left (33 a^2 x^2+22 a x+8\right ) \sqrt {c-a c x}}{24 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 133, normalized size = 1.49 \[ \left [\frac {33 \, a^{3} \sqrt {c} x^{3} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (33 \, a^{2} x^{2} + 22 \, a x + 8\right )} \sqrt {-a c x + c}}{48 \, x^{3}}, -\frac {33 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (33 \, a^{2} x^{2} + 22 \, a x + 8\right )} \sqrt {-a c x + c}}{24 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 104, normalized size = 1.17 \[ -\frac {\frac {33 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {33 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{2} + 63 \, \sqrt {-a c x + c} a^{4} c^{3}}{a^{3} c^{3} x^{3}}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.90 \[ -2 c^{3} a^{3} \left (-\frac {\frac {11 \left (-a c x +c \right )^{\frac {5}{2}}}{16 c^{2}}-\frac {11 \left (-a c x +c \right )^{\frac {3}{2}}}{6 c}+\frac {21 \sqrt {-a c x +c}}{16}}{x^{3} a^{3} c^{3}}-\frac {11 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{16 c^{\frac {5}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 134, normalized size = 1.51 \[ \frac {1}{48} \, a^{3} c^{3} {\left (\frac {2 \, {\left (33 \, {\left (-a c x + c\right )}^{\frac {5}{2}} - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 63 \, \sqrt {-a c x + c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c^{2} + 3 \, {\left (a c x - c\right )}^{2} c^{3} + 3 \, {\left (a c x - c\right )} c^{4} + c^{5}} - \frac {33 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 74, normalized size = 0.83 \[ \frac {21\,\sqrt {c-a\,c\,x}}{8\,x^3}-\frac {11\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,c\,x^3}+\frac {11\,{\left (c-a\,c\,x\right )}^{5/2}}{8\,c^2\,x^3}-\frac {a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,11{}\mathrm {i}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 17.81, size = 439, normalized size = 4.93 \[ - \frac {66 a^{3} c^{6} \sqrt {- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac {80 a^{3} c^{5} \left (- a c x + c\right )^{\frac {3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac {30 a^{3} c^{4} \left (- a c x + c\right )^{\frac {5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac {10 a^{3} c^{4} \sqrt {- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {5 a^{3} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (- c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} + \frac {5 a^{3} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} - \frac {6 a^{3} c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {3 a^{3} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + \frac {3 a^{3} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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