Optimal. Leaf size=84 \[ \frac {4 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x}} \]
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Rubi [A] time = 0.22, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6134, 6128, 848, 45, 37} \[ \frac {4 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{15 x \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{5 x^2 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 848
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{\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^{\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 {\sqrt {1-a^2 x^2}}{x^{7/2} \sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{7/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{5 x^2 \sqrt {1-a x}}-\frac {\left (2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{5/2}} \, dx}{5 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{5 x^2 \sqrt {1-a x}}+\frac {4 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{15 x \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.56 \[ \frac {2 (a x+1)^{3/2} (2 a x-3) \sqrt {c-\frac {c}{a x}}}{15 x^2 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 58, normalized size = 0.69 \[ -\frac {2 \, {\left (2 \, a^{2} x^{2} - a x - 3\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{15 \, {\left (a x^{3} - x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 0.55 \[ \frac {2 \left (a x +1\right )^{2} \left (2 a x -3\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{15 x^{2} \sqrt {-a^{2} x^{2}+1}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 52, normalized size = 0.62 \[ -\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (-\frac {4\,a^3\,x^3}{15}-\frac {2\,a^2\,x^2}{15}+\frac {8\,a\,x}{15}+\frac {2}{5}\right )}{x^2\,\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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