Optimal. Leaf size=172 \[ \frac {208 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 \sqrt {1-a x}}+\frac {26 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{35 x^2 \sqrt {1-a x}}-\frac {104 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)} \]
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Rubi [A]
time = 0.19, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6269, 6263,
893, 862, 47, 37} \begin {gather*} \frac {208 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}}{7 x^3 (1-a x)}-\frac {104 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{105 x \sqrt {1-a x}}+\frac {26 a \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{35 x^2 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 862
Rule 893
Rule 6263
Rule 6269
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 {(1-a x)^{3/2}}{x^{9/2} \sqrt {1-a^2 x^2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)}-\frac {\left (13 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1-a x}}{x^{7/2} \sqrt {1-a^2 x^2}} \, dx}{7 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)}-\frac {\left (13 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{7/2} \sqrt {1+a x}} \, dx}{7 \sqrt {1-a x}}\\ &=\frac {26 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{35 x^2 \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)}+\frac {\left (52 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{5/2} \sqrt {1+a x}} \, dx}{35 \sqrt {1-a x}}\\ &=\frac {26 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{35 x^2 \sqrt {1-a x}}-\frac {104 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)}-\frac {\left (104 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+a x}} \, dx}{105 \sqrt {1-a x}}\\ &=\frac {208 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 \sqrt {1-a x}}+\frac {26 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{35 x^2 \sqrt {1-a x}}-\frac {104 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{105 x \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{7 x^3 (1-a x)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 63, normalized size = 0.37 \begin {gather*} \frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x} \left (-15+39 a x-52 a^2 x^2+104 a^3 x^3\right )}{105 x^3 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 62, normalized size = 0.36
method | result | size |
gosper | \(-\frac {2 \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{105 x^{3} \left (a x -1\right )}\) | \(62\) |
default | \(-\frac {2 \left (104 a^{3} x^{3}-52 a^{2} x^{2}+39 a x -15\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{105 x^{3} \left (a x -1\right )}\) | \(62\) |
risch | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (104 a^{4} x^{4}+52 a^{3} x^{3}-13 a^{2} x^{2}+24 a x -15\right )}{105 \sqrt {-a^{2} x^{2}+1}\, x^{3} \sqrt {-a c x \left (a x +1\right )}}\) | \(98\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 66, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\left (104 \, a^{3} x^{3} - 52 \, a^{2} x^{2} + 39 \, 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 )}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 86, normalized size = 0.50 \begin {gather*} -\frac {152\,\sqrt {1-a^2\,x^2}\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3\,\left (a\,x-1\right )}-\frac {26\,\sqrt {1-a^2\,x^2}\,\left (8\,a^2\,x^2+4\,a\,x+7\right )\,\sqrt {\frac {c\,\left (a\,x-1\right )}{a\,x}}}{105\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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