Optimal. Leaf size=119 \[ \frac {(a+1) (-a-b x+1)^{5/2}}{b^2 \sqrt {a+b x+1}}+\frac {(2 a+3) \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}+\frac {3 (2 a+3) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^2}+\frac {3 (2 a+3) \sin ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.09, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 78, 50, 53, 619, 216} \[ \frac {(a+1) (-a-b x+1)^{5/2}}{b^2 \sqrt {a+b x+1}}+\frac {(2 a+3) \sqrt {a+b x+1} (-a-b x+1)^{3/2}}{2 b^2}+\frac {3 (2 a+3) \sqrt {a+b x+1} \sqrt {-a-b x+1}}{2 b^2}+\frac {3 (2 a+3) \sin ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 78
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a+b x)} x \, dx &=\int \frac {x (1-a-b x)^{3/2}}{(1+a+b x)^{3/2}} \, dx\\ &=\frac {(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {(3+2 a) \int \frac {(1-a-b x)^{3/2}}{\sqrt {1+a+b x}} \, dx}{b}\\ &=\frac {(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {(3+2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}+\frac {(3 (3+2 a)) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{2 b}\\ &=\frac {(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {3 (3+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3+2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}+\frac {(3 (3+2 a)) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b}\\ &=\frac {(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {3 (3+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3+2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}+\frac {(3 (3+2 a)) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b}\\ &=\frac {(1+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {3 (3+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3+2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}-\frac {(3 (3+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+a) (1-a-b x)^{5/2}}{b^2 \sqrt {1+a+b x}}+\frac {3 (3+2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3+2 a) (1-a-b x)^{3/2} \sqrt {1+a+b x}}{2 b^2}+\frac {3 (3+2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 157, normalized size = 1.32 \[ \frac {6 (2 a+3) \sqrt {b} \sqrt {-a^2-2 a b x-b^2 x^2+1} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {-b}}\right )+\sqrt {-b} \left (-a^3-a^2 (b x+14)+a \left (b^2 x^2-20 b x+1\right )+b^3 x^3-6 b^2 x^2-9 b x+14\right )}{2 (-b)^{5/2} \sqrt {-((a+b x-1) (a+b x+1))}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.26, size = 130, normalized size = 1.09 \[ -\frac {3 \, {\left ({\left (2 \, a + 3\right )} b x + 2 \, a^{2} + 5 \, a + 3\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (b^{2} x^{2} - a^{2} - 5 \, b x - 15 \, a - 14\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, {\left (b^{3} x + {\left (a + 1\right )} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 127, normalized size = 1.07 \[ -\frac {1}{2} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (\frac {x}{b} - \frac {a b^{2} + 6 \, b^{2}}{b^{4}}\right )} - \frac {3 \, {\left (2 \, a + 3\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b {\left | b \right |}} - \frac {8 \, {\left (a + 1\right )}}{b {\left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} + 1\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 543, normalized size = 4.56 \[ \frac {\left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{3}}+\frac {3 \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{2}}+\frac {3 \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{b^{2}}+\frac {9 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, x}{2 b}+\frac {9 a \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{2 b^{2}}+\frac {9 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{2 b \sqrt {b^{2}}}+\frac {a \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{5} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{3}}+\frac {2 a \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {5}{2}}}{b^{4} \left (x +\frac {1}{b}+\frac {a}{b}\right )^{2}}+\frac {2 a \left (-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )\right )^{\frac {3}{2}}}{b^{2}}+\frac {3 a \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, x}{b}+\frac {3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{2}}{b^{2}}+\frac {3 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {1+a}{b}-\frac {1}{b}\right )}{\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}\right )}{b \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 299, normalized size = 2.51 \[ -\frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} + 2 \, b^{3} x + 2 \, a b^{2} + b^{2}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2} + 2 \, b^{3} x + 2 \, a b^{2} + b^{2}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (b^{3} x + a b^{2} + b^{2}\right )}} + \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3} x + a b^{2} + b^{2}} + \frac {3 \, a \arcsin \left (b x + a\right )}{b^{2}} + \frac {6 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3} x + a b^{2} + b^{2}} + \frac {9 \, \arcsin \left (b x + a\right )}{2 \, b^{2}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{2 \, b^{2}} \]
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 \[ \int \frac {x\,{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}}{{\left (a+b\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {x \left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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