Optimal. Leaf size=156 \[ -\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1} \left (18 a^2-2 (6 a+1) b x+10 a+7\right )}{24 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{8 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sin ^{-1}(a+b x)}{8 b^4}-\frac {x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b^2} \]
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Rubi [A] time = 0.17, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 100, 147, 50, 53, 619, 216} \[ -\frac {(-a-b x+1)^{3/2} \sqrt {a+b x+1} \left (18 a^2-2 (6 a+1) b x+10 a+7\right )}{24 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{8 b^4}-\frac {\left (8 a^3+12 a^2+12 a+3\right ) \sin ^{-1}(a+b x)}{8 b^4}-\frac {x^2 (-a-b x+1)^{3/2} \sqrt {a+b x+1}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 100
Rule 147
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 \sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx\\ &=-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {\int \frac {x \sqrt {1-a-b x} \left (-2 \left (1-a^2\right )+(1+6 a) b x\right )}{\sqrt {1+a+b x}} \, dx}{4 b^2}\\ &=-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {\sqrt {1-a-b x}}{\sqrt {1+a+b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{8 b^3}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}+\frac {\left (3+12 a+12 a^2+8 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{16 b^5}\\ &=-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 (1-a-b x)^{3/2} \sqrt {1+a+b x}}{4 b^2}-\frac {(1-a-b x)^{3/2} \sqrt {1+a+b x} \left (7+10 a+18 a^2-2 (1+6 a) b x\right )}{24 b^4}-\frac {\left (3+12 a+12 a^2+8 a^3\right ) \sin ^{-1}(a+b x)}{8 b^4}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 160, normalized size = 1.03 \[ \frac {\frac {6 \left (8 a^3+12 a^2+12 a+3\right ) \sqrt {b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {-b}}+\frac {\sqrt {a+b x+1} \left (6 a^4+38 a^3+5 a^2 (6 b x-1)+a \left (-18 b^2 x^2+50 b x-23\right )-6 b^4 x^4+14 b^3 x^3-17 b^2 x^2+25 b x-16\right )}{\sqrt {-a-b x+1}}}{24 b^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.68, size = 143, normalized size = 0.92 \[ \frac {3 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (6 \, b^{3} x^{3} - 2 \, {\left (3 \, a + 4\right )} b^{2} x^{2} - 6 \, a^{3} + {\left (6 \, a^{2} + 20 \, a + 9\right )} b x - 44 \, a^{2} - 39 \, a - 16\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{24 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 148, normalized size = 0.95 \[ \frac {1}{24} \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{b} - \frac {3 \, a b^{5} + 4 \, b^{5}}{b^{7}}\right )} + \frac {6 \, a^{2} b^{4} + 20 \, a b^{4} + 9 \, b^{4}}{b^{7}}\right )} x - \frac {6 \, a^{3} b^{3} + 44 \, a^{2} b^{3} + 39 \, a b^{3} + 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{8 \, b^{3} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 809, normalized size = 5.19 \[ \frac {3 a^{2} x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{3}}+\frac {3 a^{2} \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^{3} \sqrt {b^{2}}}+\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 )}{2 b^{3} \sqrt {b^{2}}}-\frac {\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 ) a^{3}}{b^{3} \sqrt {b^{2}}}-\frac {3 \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 ) a^{2}}{b^{3} \sqrt {b^{2}}}-\frac {3 \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 ) a}{b^{3} \sqrt {b^{2}}}+\frac {3 a x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{3}}-\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}}{b^{4}}+\frac {\left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{3 b^{4}}+\frac {3 a^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{4}}-\frac {\sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a^{3}}{b^{4}}-\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^{4}}-\frac {3 \sqrt {-\left (x +\frac {1+a}{b}\right )^{2} b^{2}+2 b \left (x +\frac {1+a}{b}\right )}\, a}{b^{4}}-\frac {\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^{3} \sqrt {b^{2}}}+\frac {5 a \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{8 b^{4}}-\frac {x \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{4 b^{3}}+\frac {3 a \left (-b^{2} x^{2}-2 a b x -a^{2}+1\right )^{\frac {3}{2}}}{4 b^{4}}+\frac {3 a^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{4}}+\frac {5 x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{8 b^{3}}+\frac {5 \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 )}{8 b^{3} \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 338, normalized size = 2.17 \[ \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} x}{2 \, b^{3}} - \frac {a^{3} \arcsin \left (b x + a\right )}{b^{4}} + \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{2 \, b^{4}} - \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x}{2 \, b^{3}} - \frac {3 \, a^{2} \arcsin \left (b x + a\right )}{2 \, b^{4}} + \frac {3 \, {\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}} a}{4 \, b^{4}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x}{8 \, b^{3}} - \frac {3 \, a \arcsin \left (b x + a\right )}{2 \, b^{4}} + \frac {{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{4}} - \frac {19 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{8 \, b^{4}} - \frac {3 \, \arcsin \left (b x + a\right )}{8 \, b^{4}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4}} \]
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 \[ \int \frac {x^3\,\sqrt {1-{\left (a+b\,x\right )}^2}}{a+b\,x+1} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}{a + b x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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