Optimal. Leaf size=196 \[ -\frac {5 c^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}+\frac {49 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{15 a \sqrt {c-\frac {c}{a x}}}+\frac {31 c^3 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}}{15 a}+\frac {7 c^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{3/2}}{5 a}+c x \sqrt {1-\frac {1}{a^2 x^2}} \left (c-\frac {c}{a x}\right )^{5/2} \]
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Rubi [A] time = 0.15, antiderivative size = 221, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6182, 6179, 97, 153, 147, 63, 208} \[ \frac {x \left (a-\frac {1}{x}\right )^3 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (80 a-\frac {31}{x}\right ) \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (1-\frac {1}{a x}\right )^{7/2}} \]
Warning: Unable to verify antiderivative.
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Rule 63
Rule 97
Rule 147
Rule 153
Rule 208
Rule 6179
Rule 6182
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{7/2} \, dx &=\frac {\left (c-\frac {c}{a x}\right )^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{7/2} \, dx}{\left (1-\frac {1}{a x}\right )^{7/2}}\\ &=-\frac {\left (c-\frac {c}{a x}\right )^{7/2} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (c-\frac {c}{a x}\right )^{7/2} \operatorname {Subst}\left (\int \frac {\left (-\frac {5}{2 a}-\frac {7 x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^2}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (2 a \left (c-\frac {c}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {25}{4 a^2}-\frac {31 x}{4 a^3}\right ) \left (1-\frac {x}{a}\right )}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5 \left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (5 \left (c-\frac {c}{a x}\right )^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}}\\ &=\frac {\left (80 a-\frac {31}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{15 a^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {7 \left (a-\frac {1}{x}\right )^2 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{7/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {5 \left (c-\frac {c}{a x}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 101, normalized size = 0.52 \[ \frac {c^3 \sqrt {c-\frac {c}{a x}} \left (\sqrt {\frac {1}{a x}+1} \left (15 a^3 x^3+56 a^2 x^2-28 a x+6\right )-75 a^2 x^2 \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )\right )}{15 a^3 x^2 \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.71, size = 415, normalized size = 2.12 \[ \left [\frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{60 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}, \frac {75 \, {\left (a^{3} c^{3} x^{3} - a^{2} c^{3} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (15 \, a^{4} c^{3} x^{4} + 71 \, a^{3} c^{3} x^{3} + 28 \, a^{2} c^{3} x^{2} - 22 \, a c^{3} x + 6 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{30 \, {\left (a^{4} x^{3} - a^{3} x^{2}\right )}}\right ] \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 149, normalized size = 0.76 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{3} \left (30 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}+112 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-75 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x^{3} a^{3}-56 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+12 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{30 \sqrt {\frac {a x -1}{a x +1}}\, x^{2} a^{\frac {7}{2}} \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{7/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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