Optimal. Leaf size=121 \[ \frac {3 (3-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}-\frac {3 (3-2 a) \text {ArcSin}(a+b x)}{2 b^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 121, 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 = {6298, 79, 52,
55, 633, 222} \begin {gather*} -\frac {3 (3-2 a) \text {ArcSin}(a+b x)}{2 b^2}+\frac {(1-a) (a+b x+1)^{5/2}}{b^2 \sqrt {-a-b x+1}}+\frac {(3-2 a) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{2 b^2}+\frac {3 (3-2 a) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 79
Rule 222
Rule 633
Rule 6298
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 {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}-\frac {(3 (3-2 a)) \int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx}{2 b}\\ &=\frac {3 (3-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}-\frac {(3 (3-2 a)) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b}\\ &=\frac {3 (3-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}-\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 {3 (3-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}+\frac {(3 (3-2 a)) \text {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 {3 (3-2 a) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^2}+\frac {(3-2 a) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{2 b^2}+\frac {(1-a) (1+a+b x)^{5/2}}{b^2 \sqrt {1-a-b x}}-\frac {3 (3-2 a) \sin ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 142, normalized size = 1.17 \begin {gather*} \frac {\frac {\sqrt {b} \sqrt {1+a+b x} \left (14-15 a+a^2-5 b x-b^2 x^2\right )}{\sqrt {1-a-b x}}+12 a \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {b}}\right )+18 \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {1-a-b x}}{\sqrt {2} \sqrt {-b}}\right )}{2 b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1117\) vs.
\(2(103)=206\).
time = 0.07, size = 1118, normalized size = 9.24
method | result | size |
risch | \(\frac {\left (-b x +a -6\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b \sqrt {b^{2}}}-\frac {9 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b \sqrt {b^{2}}}+\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}\, a}{b^{3} \left (x -\frac {1}{b}+\frac {a}{b}\right )}-\frac {4 \sqrt {-\left (x +\frac {-1+a}{b}\right )^{2} b^{2}-2 b \left (x +\frac {-1+a}{b}\right )}}{b^{3} \left (x -\frac {1}{b}+\frac {a}{b}\right )}\) | \(247\) |
default | \(b^{3} \left (-\frac {x^{3}}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {5 a \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2}}\right )}{2 b}+\frac {3 \left (-a^{2}+1\right ) \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{2 b^{2}}\right )+3 \left (1+a \right ) b^{2} \left (-\frac {x^{2}}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 a \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )}{b}+\frac {2 \left (-a^{2}+1\right ) \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2}}\right )+3 \left (1+a \right )^{2} b \left (\frac {x}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b}-\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 )}{b^{2} \sqrt {b^{2}}}\right )+\left (1+a \right )^{3} \left (\frac {1}{b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {2 a \left (-2 b^{2} x -2 b a \right )}{b \left (-4 b^{2} \left (-a^{2}+1\right )-4 b^{2} a^{2}\right ) \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )\) | \(1118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1137 vs.
\(2 (102) = 204\).
time = 0.48, size = 1137, normalized size = 9.40 \begin {gather*} \frac {15 \, a^{4} b x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {31 \, {\left (a^{2} - 1\right )} a^{2} b x}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {b x^{3}}{2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {5 \, {\left (a^{2} - 1\right )} a^{3}}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {6 \, {\left (a^{2} b + 2 \, a b + b\right )} a^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {18 \, {\left (a b^{2} + b^{2}\right )} a^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} + \frac {3 \, {\left (a^{2} - 1\right )}^{2} b x}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} a b x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {5 \, a x^{2}}{2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {3 \, {\left (a^{2} - 1\right )}^{2} a}{2 \, {\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {{\left (a^{3} + 3 \, a^{2} + 3 \, a + 1\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} - \frac {3 \, {\left (a^{2} b + 2 \, a b + b\right )} {\left (a^{2} - 1\right )} x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}} + \frac {15 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} a x}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} + \frac {15 \, 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 {3 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} + \frac {3 \, {\left (a^{2} b + 2 \, a b + b\right )} {\left (a^{2} - 1\right )} a}{{\left (a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b} - \frac {3 \, {\left (a b^{2} + b^{2}\right )} x^{2}}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} - \frac {3 \, {\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 {5 \, {\left (a^{2} - 1\right )} a}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} - \frac {9 \, {\left (a b^{2} + b^{2}\right )} a \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{4}} + \frac {3 \, {\left (a^{2} b + 2 \, a b + b\right )} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b^{3}} + \frac {a^{3} + 3 \, a^{2} + 3 \, a + 1}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{2}} - \frac {6 \, {\left (a b^{2} + b^{2}\right )} {\left (a^{2} - 1\right )}}{\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 131, normalized size = 1.08 \begin {gather*} -\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 )}} \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 \left (a + b x + 1\right )^{3}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 127, normalized size = 1.05 \begin {gather*} \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}\left (b\right )}{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 |}} \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 = -0.01 \begin {gather*} \int \frac {x\,{\left (a+b\,x+1\right )}^3}{{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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