Optimal. Leaf size=166 \[ -\frac {316 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x}}+\frac {158 a^2 \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {a x+1}}+\frac {32 a \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.23, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6134, 6129, 89, 78, 45, 37} \[ -\frac {316 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x}}+\frac {158 a^2 \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {a x+1}}+\frac {32 a \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 89
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^3} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{7/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^2}{x^{7/2} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-8 a+\frac {5 a^2 x}{2}}{x^{5/2} (1+a x)^{3/2}} \, dx}{5 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {32 a \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (79 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} (1+a x)^{3/2}} \, dx}{15 \sqrt {1-a x}}\\ &=\frac {158 a^2 \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {32 a \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (158 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+a x}} \, dx}{15 \sqrt {1-a x}}\\ &=\frac {158 a^2 \sqrt {c-\frac {c}{a x}}}{15 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {32 a \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x} \sqrt {1+a x}}-\frac {316 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{15 \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 0.35 \[ -\frac {2 \left (158 a^3 x^3+79 a^2 x^2-16 a x+3\right ) \sqrt {c-\frac {c}{a x}}}{15 x^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 68, normalized size = 0.41 \[ \frac {2 \, {\left (158 \, a^{3} x^{3} + 79 \, a^{2} x^{2} - 16 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{15 \, {\left (a^{2} x^{4} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 69, normalized size = 0.42 \[ -\frac {2 \left (158 x^{3} a^{3}+79 a^{2} x^{2}-16 a x +3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{15 \left (a x +1\right )^{2} x^{2} \left (a x -1\right )^{2}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 99, normalized size = 0.60 \[ \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {2\,\sqrt {1-a^2\,x^2}}{5\,a^2}+\frac {158\,x^2\,\sqrt {1-a^2\,x^2}}{15}-\frac {32\,x\,\sqrt {1-a^2\,x^2}}{15\,a}+\frac {316\,a\,x^3\,\sqrt {1-a^2\,x^2}}{15}\right )}{x^4-\frac {x^2}{a^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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