Optimal. Leaf size=122 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{3/2}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}-\frac {3 \sqrt {1-a^2 x^2}}{4 a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6127, 663, 661, 208} \[ \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{3/2}}-\frac {3 \sqrt {1-a^2 x^2}}{4 a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 663
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{9/2}} \, dx\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}-\frac {1}{4} (3 c) \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{5/2}} \, dx\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{4 a (c-a c x)^{3/2}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}+\frac {3 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx}{8 c}\\ &=-\frac {3 \sqrt {1-a^2 x^2}}{4 a (c-a c x)^{3/2}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}-\frac {1}{4} (3 a) \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}}{4 a (c-a c x)^{3/2}}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{2 a (c-a c x)^{7/2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 91, normalized size = 0.75 \[ \frac {\sqrt {c-a c x} \left (10 a^2 x^2+8 a x+3 (a x-1)^2 \sqrt {2 a x+2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )-2\right )}{8 a c^2 (a x-1)^2 \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 312, normalized size = 2.56 \[ \left [\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (5 \, a x - 1\right )}}{16 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}, \frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (5 \, a x - 1\right )}}{8 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 73, normalized size = 0.60 \[ -\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {2 \, {\left (5 \, {\left (a c x + c\right )}^{\frac {3}{2}} - 6 \, \sqrt {a c x + c} c\right )}}{{\left (a c x - c\right )}^{2}}}{8 \, a {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 158, normalized size = 1.30 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x^{2} a^{2} c -6 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c +10 x a \sqrt {c \left (a x +1\right )}\, \sqrt {c}+3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -2 \sqrt {c \left (a x +1\right )}\, \sqrt {c}\right )}{8 c^{\frac {5}{2}} \left (a x -1\right )^{3} \sqrt {c \left (a x +1\right )}\, a} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (-a c x + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}\,{\left (c-a\,c\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a x + 1\right )^{3}}{\left (- c \left (a x - 1\right )\right )^{\frac {3}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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