Optimal. Leaf size=257 \[ -\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {1+a x}}-\frac {1312 a^4 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{45 \sqrt {1-a x}}-\frac {164 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{15 x^2 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{45 x \sqrt {1-a x}} \]
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Rubi [A]
time = 0.18, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6269, 6264, 91,
79, 47, 37} \begin {gather*} -\frac {1312 a^4 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 x \sqrt {1-a x}}-\frac {164 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{15 x^2 \sqrt {1-a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {a x+1}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {a x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 91
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}}}{x^5} \, 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^{11/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^2}{x^{11/2} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-14 a+\frac {9 a^2 x}{2}}{x^{9/2} (1+a x)^{3/2}} \, dx}{9 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (41 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{7/2} (1+a x)^{3/2}} \, dx}{9 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (82 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{7/2} \sqrt {1+a x}} \, dx}{3 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {1+a x}}-\frac {164 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{15 x^2 \sqrt {1-a x}}-\frac {\left (328 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{5/2} \sqrt {1+a x}} \, dx}{15 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {1+a x}}-\frac {164 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{15 x^2 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{45 x \sqrt {1-a x}}+\frac {\left (656 a^4 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+a x}} \, dx}{45 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {1+a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {1+a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {1+a x}}-\frac {1312 a^4 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{45 \sqrt {1-a x}}-\frac {164 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{15 x^2 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{45 x \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 74, normalized size = 0.29 \begin {gather*} -\frac {2 \sqrt {c-\frac {c}{a x}} \left (5-20 a x+41 a^2 x^2-82 a^3 x^3+328 a^4 x^4+656 a^5 x^5\right )}{45 x^4 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 85, normalized size = 0.33
method | result | size |
gosper | \(-\frac {2 \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{45 \left (a x +1\right )^{2} x^{4} \left (a x -1\right )^{2}}\) | \(85\) |
default | \(\frac {2 \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (656 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {-a^{2} x^{2}+1}}{45 x^{4} \left (a x -1\right ) \left (a x +1\right )}\) | \(85\) |
risch | \(-\frac {2 \left (476 a^{5} x^{5}+328 a^{4} x^{4}-82 a^{3} x^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{45 x^{4} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}-\frac {8 a^{5} x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}\) | \(175\) |
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 = 0.36, size = 84, normalized size = 0.33 \begin {gather*} \frac {2 \, {\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{45 \, {\left (a^{2} x^{6} - x^{4}\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 )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 139, normalized size = 0.54 \begin {gather*} \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {2\,\sqrt {1-a^2\,x^2}}{9\,a^2}+\frac {82\,x^2\,\sqrt {1-a^2\,x^2}}{45}-\frac {8\,x\,\sqrt {1-a^2\,x^2}}{9\,a}-\frac {164\,a\,x^3\,\sqrt {1-a^2\,x^2}}{45}+\frac {656\,a^2\,x^4\,\sqrt {1-a^2\,x^2}}{45}+\frac {1312\,a^3\,x^5\,\sqrt {1-a^2\,x^2}}{45}\right )}{x^6-\frac {x^4}{a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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