Optimal. Leaf size=126 \[ -\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {5 \sqrt {a} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6269, 6264, 91,
81, 56, 221} \begin {gather*} -\frac {5 \sqrt {a} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}+\frac {a x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2}}-\frac {2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 81
Rule 91
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{-\tanh ^{-1}(a x)} (1-a x)^{3/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {(1-a x)^2}{x^{3/2} \sqrt {1+a x}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {\left (2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {-a+\frac {a^2 x}{2}}{\sqrt {x} \sqrt {1+a x}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {\left (5 a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {\left (5 a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1+a x}}{(1-a x)^{3/2}}+\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 \sqrt {1+a x}}{(1-a x)^{3/2}}-\frac {5 \sqrt {a} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 71, normalized size = 0.56 \begin {gather*} -\frac {c \sqrt {c-\frac {c}{a x}} \left ((-2+a x) \sqrt {1+a x}-5 \sqrt {a} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.95, size = 109, normalized size = 0.87
method | result | size |
default | \(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \sqrt {-a^{2} x^{2}+1}\, \left (2 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a x -4 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{2 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}}\) | \(109\) |
risch | \(-\frac {\left (a^{2} x^{2}-a x -2\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}\, a}+\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right ) c \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{2 \sqrt {a^{2} c}\, \sqrt {-a^{2} x^{2}+1}}\) | \(176\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 268, normalized size = 2.13 \begin {gather*} \left [\frac {5 \, {\left (a c x - c\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} x - a\right )}}, -\frac {5 \, {\left (a c x - c\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (a c x - 2 \, c\right )} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} x - a\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\sqrt {1-a^2\,x^2}}{a\,x+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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