Optimal. Leaf size=257 \[ -\frac {1312 a^4 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 x \sqrt {1-a x}}-\frac {164 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{15 x^2 \sqrt {1-a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {a x+1}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.25, 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 = {6134, 6129, 89, 78, 45, 37} \[ -\frac {164 a^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{15 x^2 \sqrt {1-a x}}+\frac {82 a^2 \sqrt {c-\frac {c}{a x}}}{9 x^2 \sqrt {1-a x} \sqrt {a x+1}}-\frac {1312 a^4 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 \sqrt {1-a x}}+\frac {656 a^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{45 x \sqrt {1-a x}}+\frac {8 a \sqrt {c-\frac {c}{a x}}}{9 x^3 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{9 x^4 \sqrt {1-a x} \sqrt {a x+1}} \]
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^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 = 0.04, size = 74, normalized size = 0.29 \[ -\frac {2 \left (656 a^5 x^5+328 a^4 x^4-82 a^3 x^3+41 a^2 x^2-20 a x+5\right ) \sqrt {c-\frac {c}{a x}}}{45 x^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 84, normalized size = 0.33 \[ \frac {2 \, {\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{45 \, {\left (a^{2} x^{6} - x^{4}\right )}} \]
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 = 85, normalized size = 0.33 \[ -\frac {2 \left (656 x^{5} a^{5}+328 x^{4} a^{4}-82 x^{3} a^{3}+41 a^{2} x^{2}-20 a x +5\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{45 \left (a x +1\right )^{2} x^{4} \left (a x -1\right )^{2}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x^{5}}\,{d x} \]
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 \[ \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {2\,\sqrt {1-a^2\,x^2}}{9\,a^2}+\frac {82\,x^2\,\sqrt {1-a^2\,x^2}}{45}-\frac {8\,x\,\sqrt {1-a^2\,x^2}}{9\,a}-\frac {164\,a\,x^3\,\sqrt {1-a^2\,x^2}}{45}+\frac {656\,a^2\,x^4\,\sqrt {1-a^2\,x^2}}{45}+\frac {1312\,a^3\,x^5\,\sqrt {1-a^2\,x^2}}{45}\right )}{x^6-\frac {x^4}{a^2}} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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