Optimal. Leaf size=115 \[ \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}} \]
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Rubi [A] time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6127, 663, 665, 661, 208} \[ \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 663
Rule 665
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\sqrt {c-a c x}} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{7/2}} \, dx\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac {1}{2} (3 c) \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-3 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}+(6 a c) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=\frac {3 \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{a (c-a c x)^{5/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.50 \[ \frac {(a x+1)^{5/2} (c-a c x)^{3/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {1}{2} (a x+1)\right )}{10 a c^2 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 259, normalized size = 2.25 \[ \left [-\frac {4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x - 2\right )} - \frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \log \left (-\frac {a^{2} x^{2} + 2 \, a x + \frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a^{2} x^{2} - 2 \, a x + 1}\right )}{\sqrt {c}}}{2 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{2} c x^{2} - 2 \, a c x + c\right )} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a^{2} x^{2} - 1}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x - 2\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 70, normalized size = 0.61 \[ \frac {\frac {3 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + 2 \, \sqrt {a c x + c} - \frac {2 \, \sqrt {a c x + c} c}{a c x - c}}{a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 127, normalized size = 1.10 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c -2 x a \sqrt {c \left (a x +1\right )}\, \sqrt {c}-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +4 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{c^{\frac {3}{2}} \left (a x -1\right )^{2} \sqrt {c \left (a x +1\right )}\, a} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}\,{d x} \]
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 \[ \int \frac {{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}} \,d x \]
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 \[ \int \frac {\left (a x + 1\right )^{3}}{\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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