Optimal. Leaf size=148 \[ \frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6167, 6130, 21, 98, 151, 156, 63, 208, 206} \[ \frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {\sqrt {c-a c x}}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 63
Rule 98
Rule 151
Rule 156
Rule 206
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\\ &=-\frac {\int \frac {(c-a c x)^{3/2}}{x^5 (1+a x)} \, dx}{c}\\ &=\frac {\sqrt {c-a c x}}{4 x^4}+\frac {\int \frac {\frac {17 a c^2}{2}-\frac {15}{2} a^2 c^2 x}{x^4 (1+a x) \sqrt {c-a c x}} \, dx}{4 c}\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {\int \frac {\frac {107 a^2 c^3}{4}-\frac {85}{4} a^3 c^3 x}{x^3 (1+a x) \sqrt {c-a c x}} \, dx}{12 c^2}\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {\int \frac {\frac {447 a^3 c^4}{8}-\frac {321}{8} a^4 c^4 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{24 c^3}\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {\int \frac {\frac {1089 a^4 c^5}{16}-\frac {447}{16} a^5 c^5 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{24 c^4}\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{128} \left (363 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx+\left (4 a^5 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{64} \left (363 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 )-\left (8 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 109, normalized size = 0.74 \[ \frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {\left (-447 a^3 x^3+214 a^2 x^2-136 a x+48\right ) \sqrt {c-a c x}}{192 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 236, normalized size = 1.59 \[ \left [\frac {768 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 1089 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{384 \, x^{4}}, \frac {768 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 1089 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{192 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 160, normalized size = 1.08 \[ \frac {4 \, \sqrt {2} a^{4} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {363 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c}} - \frac {447 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{4} c + 1127 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{2} - 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{3} + 321 \, \sqrt {-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 123, normalized size = 0.83 \[ 2 c^{4} a^{4} \left (-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}-\frac {\frac {-\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}+\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}-\frac {1049 \left (-a c x +c \right )^{\frac {3}{2}} c^{2}}{384}+\frac {107 \sqrt {-a c x +c}\, c^{3}}{128}}{x^{4} a^{4} c^{4}}-\frac {363 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 212, normalized size = 1.43 \[ \frac {1}{384} \, a^{4} c^{4} {\left (\frac {2 \, {\left (447 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 1127 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 321 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {768 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} - \frac {1089 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.26, size = 122, normalized size = 0.82 \[ \frac {1049\,{\left (c-a\,c\,x\right )}^{3/2}}{192\,c\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,363{}\mathrm {i}}{64}-\frac {107\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {1127\,{\left (c-a\,c\,x\right )}^{5/2}}{192\,c^2\,x^4}+\frac {149\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}+\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 32.76, size = 991, normalized size = 6.70 \[ \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 {198 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 {240 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 {90 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 {40 a^{4} 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 {15 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 {15 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 {24 a^{4} 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^{4} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{2} + \frac {3 a^{4} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{2} + 2 a^{4} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )} - 2 a^{4} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )} - \frac {8 a^{4} c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 \sqrt {2} a^{4} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 a^{3} \sqrt {- a c x + c}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________