Optimal. Leaf size=218 \[ \frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {119 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}} \]
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Rubi [A]
time = 0.18, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6269, 6264, 91,
81, 52, 56, 221} \begin {gather*} -\frac {119 \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}+\frac {119 x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{8 a^2 \sqrt {1-a x}}+\frac {x^3 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x}}+\frac {8 x^3 \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x} \sqrt {a x+1}}-\frac {119 x^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{12 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 91
Rule 221
Rule 6264
Rule 6269
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{-3 \tanh ^{-1}(a x)} x^{3/2} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2} (1-a x)^2}{(1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2} \left (\frac {19 a^2}{2}-\frac {a^3 x}{2}\right )}{\sqrt {1+a x}} \, dx}{a^2 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {\left (119 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2}}{\sqrt {1+a x}} \, dx}{6 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}+\frac {\left (119 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x}}{\sqrt {1+a x}} \, dx}{8 a \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {119 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {\left (119 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{16 a^2 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {119 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {\left (119 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{8 a^2 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x^3}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {119 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{8 a^2 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{12 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^3 \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {119 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{8 a^{5/2} \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 100, normalized size = 0.46 \begin {gather*} \frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \left (\sqrt {a} \sqrt {x} \left (357+119 a x-38 a^2 x^2+8 a^3 x^3\right )-357 \sqrt {1+a x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{24 a^{5/2} \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.90, size = 176, normalized size = 0.81
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}-76 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+238 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+357 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a x +714 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}+357 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{48 a^{\frac {5}{2}} \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )}\) | \(176\) |
risch | \(\frac {\left (8 a^{2} x^{2}-46 a x +165\right ) x \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{24 a^{2} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (-\frac {119 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{16 a^{2} \sqrt {a^{2} c}}-\frac {8 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+\left (x +\frac {1}{a}\right ) a c}}{a^{4} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\) | \(225\) |
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 = 0.40, size = 320, normalized size = 1.47 \begin {gather*} \left [\frac {357 \, {\left (a^{2} x^{2} - 1\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 \, {\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{5} x^{2} - a^{3}\right )}}, \frac {357 \, {\left (a^{2} x^{2} - 1\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 \, {\left (8 \, a^{4} x^{4} - 38 \, a^{3} x^{3} + 119 \, a^{2} x^{2} + 357 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{5} x^{2} - a^{3}\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 {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, 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 = -0.00 \begin {gather*} \int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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