Optimal. Leaf size=110 \[ \frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3} \]
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Rubi [A] time = 0.23, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {\sqrt {c-a c x}}{4 x^4} \]
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^5} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx\\ &=-\int \frac {(1+a x) \sqrt {c-a c x}}{x^5 (1-a x)} \, dx\\ &=-\left (c \int \frac {1+a x}{x^5 \sqrt {c-a c x}} \, dx\right )\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {1}{8} (15 a c) \int \frac {1}{x^4 \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {1}{16} \left (25 a^2 c\right ) \int \frac {1}{x^3 \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {1}{64} \left (75 a^3 c\right ) \int \frac {1}{x^2 \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{128} \left (75 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{64} \left (75 a^3\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}}{4 x^4}+\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {75}{64} a^4 \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 = 71, normalized size = 0.65 \[ \frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+\frac {\left (75 a^3 x^3+50 a^2 x^2+40 a x+16\right ) \sqrt {c-a c x}}{64 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 149, normalized size = 1.35 \[ \left [\frac {75 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{128 \, x^{4}}, -\frac {75 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{64 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 131, normalized size = 1.19 \[ -\frac {\frac {75 \, a^{5} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {75 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{5} c + 275 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{5} c^{2} - 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{5} c^{3} + 181 \, \sqrt {-a c x + c} a^{5} c^{4}}{a^{4} c^{4} x^{4}}}{64 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 93, normalized size = 0.85 \[ 2 c^{4} a^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{x^{4} a^{4} c^{4}}+\frac {75 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 163, normalized size = 1.48 \[ -\frac {1}{128} \, a^{4} c^{4} {\left (\frac {2 \, {\left (75 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 275 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 181 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {75 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 91, normalized size = 0.83 \[ \frac {181\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {365\,{\left (c-a\,c\,x\right )}^{3/2}}{64\,c\,x^4}+\frac {275\,{\left (c-a\,c\,x\right )}^{5/2}}{64\,c^2\,x^4}-\frac {75\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,75{}\mathrm {i}}{64} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 28.19, size = 639, normalized size = 5.81 \[ \frac {558 a^{4} c^{8} \sqrt {- a c x + c}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac {1022 a^{4} c^{7} \left (- a c x + c\right )^{\frac {3}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac {770 a^{4} c^{6} \left (- a c x + c\right )^{\frac {5}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac {66 a^{4} c^{6} \sqrt {- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac {210 a^{4} c^{5} \left (- a c x + c\right )^{\frac {7}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac {80 a^{4} c^{5} \left (- a c x + c\right )^{\frac {3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac {35 a^{4} c^{5} \sqrt {\frac {1}{c^{9}}} \log {\left (- c^{5} \sqrt {\frac {1}{c^{9}}} + \sqrt {- a c x + c} \right )}}{128} + \frac {35 a^{4} c^{5} \sqrt {\frac {1}{c^{9}}} \log {\left (c^{5} \sqrt {\frac {1}{c^{9}}} + \sqrt {- a c x + c} \right )}}{128} - \frac {30 a^{4} c^{4} \left (- a c x + c\right )^{\frac {5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} - \frac {5 a^{4} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (- c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} + \frac {5 a^{4} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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