Optimal. Leaf size=156 \[ -\frac {\left (3-12 a+12 a^2-8 a^3\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{8 b^4}-\frac {x^2 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \left (7-10 a+18 a^2+2 (1-6 a) b x\right )}{24 b^4}+\frac {\left (3-12 a+12 a^2-8 a^3\right ) \text {ArcSin}(a+b x)}{8 b^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6298, 102, 152,
52, 55, 633, 222} \begin {gather*} -\frac {\sqrt {-a-b x+1} (a+b x+1)^{3/2} \left (18 a^2+2 (1-6 a) b x-10 a+7\right )}{24 b^4}+\frac {\left (-8 a^3+12 a^2-12 a+3\right ) \text {ArcSin}(a+b x)}{8 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 {x^2 \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 55
Rule 102
Rule 152
Rule 222
Rule 633
Rule 6298
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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \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 ) \text {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 \sqrt {1-a-b x} (1+a+b x)^{3/2}}{4 b^2}-\frac {\sqrt {1-a-b x} (1+a+b x)^{3/2} \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 = 0.17, size = 106, normalized size = 0.68 \begin {gather*} \frac {-\sqrt {1-a^2-2 a b x-b^2 x^2} \left (16-39 a+44 a^2-6 a^3+\left (9-20 a+6 a^2\right ) b x+2 (4-3 a) b^2 x^2+6 b^3 x^3\right )+3 \left (3-12 a+12 a^2-8 a^3\right ) \text {ArcSin}(a+b x)}{24 b^4} \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. \(793\) vs.
\(2(136)=272\).
time = 0.08, size = 794, normalized size = 5.09
method | result | size |
risch | \(-\frac {\left (-6 b^{3} x^{3}+6 a \,b^{2} x^{2}-6 a^{2} b x -8 b^{2} x^{2}+6 a^{3}+20 a b x -44 a^{2}-9 b x +39 a -16\right ) \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{24 b^{4} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{3}}{b^{3} \sqrt {b^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a^{2}}{2 b^{3} \sqrt {b^{2}}}-\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right ) a}{2 b^{3} \sqrt {b^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{8 b^{3} \sqrt {b^{2}}}\) | \(293\) |
default | \(b \left (-\frac {x^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{4 b^{2}}-\frac {7 a \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\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}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\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}}}\right )}{3 b^{2}}\right )}{4 b}+\frac {3 \left (-a^{2}+1\right ) \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\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}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )+\left (1+a \right ) \left (-\frac {x^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{3 b^{2}}-\frac {5 a \left (-\frac {x \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}{2 b^{2}}-\frac {3 a \left (-\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}}}\right )}{2 b}+\frac {\left (-a^{2}+1\right ) \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^{2} \sqrt {b^{2}}}\right )}{3 b}+\frac {2 \left (-a^{2}+1\right ) \left (-\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}}}\right )}{3 b^{2}}\right )\) | \(794\) |
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. 540 vs.
\(2 (136) = 272\).
time = 0.47, size = 540, normalized size = 3.46 \begin {gather*} -\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} x^{3}}{4 \, b} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} x^{2}}{3 \, b^{2}} + \frac {7 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a x^{2}}{12 \, b^{2}} + \frac {5 \, {\left (a + 1\right )} a^{3} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{2 \, b^{4}} - \frac {35 \, a^{4} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{8 \, b^{4}} + \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} a x}{6 \, b^{3}} - \frac {35 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2} x}{24 \, b^{3}} - \frac {3 \, {\left (a^{2} - 1\right )} {\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 )}{2 \, b^{4}} + \frac {15 \, {\left (a^{2} - 1\right )} a^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{4 \, b^{4}} - \frac {5 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a + 1\right )} a^{2}}{2 \, b^{4}} + \frac {35 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{3}}{8 \, b^{4}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} x}{8 \, b^{3}} - \frac {3 \, {\left (a^{2} - 1\right )}^{2} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{8 \, b^{4}} + \frac {2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} {\left (a + 1\right )}}{3 \, b^{4}} - \frac {55 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (a^{2} - 1\right )} a}{24 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 144, normalized size = 0.92 \begin {gather*} \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}} \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^{3} \left (a + b x + 1\right )}{\sqrt {- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 148, normalized size = 0.95 \begin {gather*} -\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}\left (b\right )}{8 \, b^{3} {\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^3\,\left (a+b\,x+1\right )}{\sqrt {1-{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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