Optimal. Leaf size=135 \[ -\frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {1-a x}}+\frac {x^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{2 \sqrt {1-a x}}+\frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{4 a \sqrt {1-a x}} \]
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Rubi [A] time = 0.19, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6134, 6128, 848, 50, 54, 215} \[ -\frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {1-a x}}+\frac {x^2 \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{2 \sqrt {1-a x}}+\frac {x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{4 a \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 215
Rule 848
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{\tanh ^{-1}(a x)} \sqrt {x} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x} \sqrt {1-a^2 x^2}}{\sqrt {1-a x}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \sqrt {x} \sqrt {1+a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x}}{\sqrt {1+a x}} \, dx}{4 \sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{4 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{8 a \sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{4 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}-\frac {\left (\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 )}{4 a \sqrt {1-a x}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{4 a \sqrt {1-a x}}+\frac {\sqrt {c-\frac {c}{a x}} x^2 \sqrt {1+a x}}{2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {1-a x}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.59 \[ \frac {\sqrt {x} \sqrt {c-\frac {c}{a x}} \left (\sqrt {a} \sqrt {x} \sqrt {a x+1} (2 a x+1)-\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{4 a^{3/2} \sqrt {1-a x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 272, normalized size = 2.01 \[ \left [\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{3} x - a^{2}\right )}}, \frac {{\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{8 \, {\left (a^{3} x - a^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 105, normalized size = 0.78 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (4 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}+\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right )}{8 a^{\frac {3}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (a x + 1\right )} \sqrt {c - \frac {c}{a x}} x}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x\,\sqrt {c-\frac {c}{a\,x}}\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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