Optimal. Leaf size=128 \[ -\frac {16 a^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{105 x \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x}}+\frac {8 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x}} \]
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Rubi [A] time = 0.23, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {16 a^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{105 x \sqrt {1-a x}}+\frac {8 a (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x}}-\frac {2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{7 x^3 \sqrt {1-a x}} \]
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^4} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{\tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{9/2}} \, 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^{9/2} \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^{9/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{7 x^3 \sqrt {1-a x}}-\frac {\left (4 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{7/2}} \, dx}{7 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{7 x^3 \sqrt {1-a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{35 x^2 \sqrt {1-a x}}+\frac {\left (8 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x}}{x^{5/2}} \, dx}{35 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{7 x^3 \sqrt {1-a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{35 x^2 \sqrt {1-a x}}-\frac {16 a^2 \sqrt {c-\frac {c}{a x}} (1+a x)^{3/2}}{105 x \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.43 \[ -\frac {2 (a x+1)^{3/2} \left (8 a^2 x^2-12 a x+15\right ) \sqrt {c-\frac {c}{a x}}}{105 x^3 \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 66, normalized size = 0.52 \[ \frac {2 \, {\left (8 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 3 \, a x + 15\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{105 \, {\left (a x^{4} - x^{3}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 54, normalized size = 0.42 \[ -\frac {2 \left (a x +1\right )^{2} \left (8 a^{2} x^{2}-12 a x +15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}}{105 x^{3} \sqrt {-a^{2} x^{2}+1}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}}}{\sqrt {-a^{2} x^{2} + 1} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 60, normalized size = 0.47 \[ -\frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {16\,a^4\,x^4}{105}+\frac {8\,a^3\,x^3}{105}-\frac {2\,a^2\,x^2}{105}+\frac {12\,a\,x}{35}+\frac {2}{7}\right )}{x^3\,\sqrt {1-a^2\,x^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{x^{4} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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