Optimal. Leaf size=123 \[ \frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}+\frac {8 x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x} \sqrt {a x+1}}-\frac {7 \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {1-a x}} \]
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Rubi [A] time = 0.16, antiderivative size = 123, 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 = {6134, 6129, 89, 80, 54, 215} \[ \frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x}}+\frac {8 x \sqrt {c-\frac {c}{a x}}}{\sqrt {1-a x} \sqrt {a x+1}}-\frac {7 \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 80
Rule 89
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {1-a x}}{\sqrt {x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^2}{\sqrt {x} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-a x} \sqrt {1+a x}}-\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\frac {3 a^2}{2}-\frac {a^3 x}{2}}{\sqrt {x} \sqrt {1+a x}} \, dx}{a^2 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {\left (7 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {\left (7 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {1-a x}}\\ &=\frac {8 \sqrt {c-\frac {c}{a x}} x}{\sqrt {1-a x} \sqrt {1+a x}}+\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{\sqrt {1-a x}}-\frac {7 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a} \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.65 \[ \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\sqrt {a} \sqrt {x} (a x+9)-7 \sqrt {a x+1} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{\sqrt {a} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 282, normalized size = 2.29 \[ \left [\frac {7 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} x^{2} - a\right )}}, \frac {7 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} + 9 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} x^{2} - a\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 140, normalized size = 1.14 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+7 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x a +18 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}+7 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \sqrt {a}\, \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-\frac {c}{a\,x}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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