Optimal. Leaf size=163 \[ \frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {a x+1} (c-a c x)^{3/2}}-\frac {47 a^2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{4 (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {a x+1} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {a x+1} (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6130, 23, 89, 78, 51, 63, 208} \[ \frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {a x+1} (c-a c x)^{3/2}}-\frac {47 a^2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{4 (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {a x+1} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {a x+1} (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 51
Rule 63
Rule 78
Rule 89
Rule 208
Rule 6130
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^3} \, dx &=\int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x^3 (1+a x)^{3/2}} \, dx\\ &=\frac {(1-a x)^{3/2} \int \frac {(c-a c x)^2}{x^3 (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {(1-a x)^{3/2} \int \frac {-\frac {13 a c^2}{2}+2 a^2 c^2 x}{x^2 (1+a x)^{3/2}} \, dx}{2 (c-a c x)^{3/2}}\\ &=-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {\left (47 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac {1}{x (1+a x)^{3/2}} \, dx}{8 (c-a c x)^{3/2}}\\ &=\frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {\left (47 a^2 c^2 (1-a x)^{3/2}\right ) \int \frac {1}{x \sqrt {1+a x}} \, dx}{8 (c-a c x)^{3/2}}\\ &=\frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {\left (47 a c^2 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {1+a x}\right )}{4 (c-a c x)^{3/2}}\\ &=\frac {47 a^2 c^2 (1-a x)^{3/2}}{4 \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{2 x^2 \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {13 a c^2 (1-a x)^{3/2}}{4 x \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {47 a^2 c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{4 (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 65, normalized size = 0.40 \[ \frac {c \sqrt {1-a x} \left (47 a^2 x^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};a x+1\right )+13 a x-2\right )}{4 x^2 \sqrt {a x+1} \sqrt {c-a c x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 252, normalized size = 1.55 \[ \left [\frac {47 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) - 2 \, {\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{8 \, {\left (a^{2} x^{4} - x^{2}\right )}}, -\frac {47 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (47 \, a^{2} x^{2} + 13 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{4 \, {\left (a^{2} x^{4} - x^{2}\right )}}\right ] \]
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 = 0.06, size = 100, normalized size = 0.61 \[ \frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (47 \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) x^{2} a^{2} \sqrt {c \left (a x +1\right )}-47 x^{2} a^{2} \sqrt {c}-13 x a \sqrt {c}+2 \sqrt {c}\right )}{4 \sqrt {c}\, \left (a x -1\right ) \left (a x +1\right ) x^{2}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x^3\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{3} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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