Optimal. Leaf size=84 \[ -\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}-\frac {(1-2 a) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^2}+\frac {(1-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.06, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6163, 80, 50, 53, 619, 216} \[ -\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}-\frac {(1-2 a) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^2}+\frac {(1-2 a) \sin ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a+b x)} x \, dx &=\int \frac {x \sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx\\ &=-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-2 a) \int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx}{2 b}\\ &=-\frac {(1-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-2 a) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b}\\ &=-\frac {(1-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-2 a) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b}\\ &=-\frac {(1-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}-\frac {(1-2 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^3}\\ &=-\frac {(1-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 130, normalized size = 1.55 \[ \frac {\sqrt {b} \sqrt {-a^2-2 a b x-b^2 x^2+1} (a-b x-2)+2 \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )+4 a \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {-b}}\right )}{2 b^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 92, normalized size = 1.10 \[ \frac {{\left (2 \, a - 1\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x - a + 2\right )}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 59, normalized size = 0.70 \[ -\frac {1}{2} \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (\frac {x}{b} - \frac {a b - 2 \, b}{b^{3}}\right )} + \frac {{\left (2 \, a - 1\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 178, normalized size = 2.12 \[ -\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b}+\frac {a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}+\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b \sqrt {b^{2}}}-\frac {\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{b^{2}}-\frac {a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 209, normalized size = 2.49 \[ \frac {{\left (a + 1\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{2}} - \frac {3 \, a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{2 \, b} + \frac {{\left (a^{2} - 1\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{2}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )}}{b^{2}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{2 \, b^{2}} \]
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 \[ \int \frac {x\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \]
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 \[ \int \frac {x \left (a + b x + 1\right )}{\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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