Optimal. Leaf size=181 \[ \frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^5 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{4 \sqrt {1-a x} \sqrt {a x+1}}+\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x} \]
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Rubi [A] time = 0.58, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6167, 6159, 6129, 98, 151, 12, 92, 208} \[ \frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {3 a^5 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{4 \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 98
Rule 151
Rule 208
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^6} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1+a x)^{3/2}}{x^6 \sqrt {1-a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-10 a-9 a^2 x}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{5 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {36 a^2+30 a^3 x}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{20 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {-90 a^3-72 a^4 x}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{60 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {144 a^4+90 a^5 x}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int -\frac {90 a^5}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (3 a^6 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 102, normalized size = 0.56 \[ \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (24 a^4 x^4+15 a^3 x^3+12 a^2 x^2+10 a x+4\right )-15 a^5 x^5 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{20 x^4 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 233, normalized size = 1.29 \[ \left [\frac {15 \, a^{4} \sqrt {-c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (24 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, -\frac {15 \, a^{4} \sqrt {c} x^{4} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (24 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.61, size = 362, normalized size = 2.00 \[ \frac {1}{10} \, {\left (15 \, a^{3} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm {sgn}\relax (x) - 40 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 200 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm {sgn}\relax (x) - 120 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm {sgn}\relax (x) - 24 \, a^{2} c^{\frac {11}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 447, normalized size = 2.47 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-40 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{4} c +40 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{4} a^{4}+15 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{5} a^{3} c +40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{5} a^{2}-40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{5} a^{2}-40 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{5} a^{3} c +25 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{3}+15 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{5} a \,c^{2}+16 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{2}+10 a \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x +4 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{20 x^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x - 1\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a\,x+1\right )}{x^5\,\left (a\,x-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{5} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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