Optimal. Leaf size=168 \[ \frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{6 b^3}+\frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^3}-\frac {\left (6 a^2-18 a+11\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac {\sqrt {-a-b x+1} (a+b x+1)^{5/2}}{3 b^3}+\frac {(1-a)^2 (a+b x+1)^{5/2}}{b^3 \sqrt {-a-b x+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6163, 89, 80, 50, 53, 619, 216} \[ \frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} (a+b x+1)^{3/2}}{6 b^3}+\frac {\left (6 a^2-18 a+11\right ) \sqrt {-a-b x+1} \sqrt {a+b x+1}}{2 b^3}-\frac {\left (6 a^2-18 a+11\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac {\sqrt {-a-b x+1} (a+b x+1)^{5/2}}{3 b^3}+\frac {(1-a)^2 (a+b x+1)^{5/2}}{b^3 \sqrt {-a-b x+1}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 80
Rule 89
Rule 216
Rule 619
Rule 6163
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1+a+b x)^{3/2}}{(1-a-b x)^{3/2}} \, dx\\ &=\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}-\frac {\int \frac {(1+a+b x)^{3/2} \left ((3-2 a) (1-a) b+b^2 x\right )}{\sqrt {1-a-b x}} \, dx}{b^3}\\ &=\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {(1+a+b x)^{3/2}}{\sqrt {1-a-b x}} \, dx}{3 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {\sqrt {1+a+b x}}{\sqrt {1-a-b x}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {1}{\sqrt {1-a-b x} \sqrt {1+a+b x}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \int \frac {1}{\sqrt {(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}+\frac {\left (11-18 a+6 a^2\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 )}{4 b^4}\\ &=\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} \sqrt {1+a+b x}}{2 b^3}+\frac {\left (11-18 a+6 a^2\right ) \sqrt {1-a-b x} (1+a+b x)^{3/2}}{6 b^3}+\frac {(1-a)^2 (1+a+b x)^{5/2}}{b^3 \sqrt {1-a-b x}}+\frac {\sqrt {1-a-b x} (1+a+b x)^{5/2}}{3 b^3}-\frac {\left (11-18 a+6 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 170, normalized size = 1.01 \[ \frac {6 \left (6 a^2+11\right ) \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {-b}}\right )-\frac {\sqrt {b} \sqrt {a+b x+1} \left (2 a^3-53 a^2+a (103-16 b x)+2 b^3 x^3+7 b^2 x^2+19 b x-52\right )}{\sqrt {-a-b x+1}}+108 a \sqrt {-b} \sinh ^{-1}\left (\frac {\sqrt {-b} \sqrt {-a-b x+1}}{\sqrt {2} \sqrt {b}}\right )}{6 b^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.74, size = 159, normalized size = 0.95 \[ \frac {3 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} - 18 \, a + 11\right )} b x - 24 \, a^{2} + 29 \, a - 11\right )} \arctan \left (\frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} {\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (2 \, b^{3} x^{3} + 7 \, b^{2} x^{2} + 2 \, a^{3} - {\left (16 \, a - 19\right )} b x - 53 \, a^{2} + 103 \, a - 52\right )} \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \, {\left (b^{4} x + {\left (a - 1\right )} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 148, normalized size = 0.88 \[ \frac {1}{6} \, \sqrt {-{\left (b x + a\right )}^{2} + 1} {\left (x {\left (\frac {2 \, x}{b} - \frac {2 \, a b^{6} - 9 \, b^{6}}{b^{8}}\right )} + \frac {2 \, a^{2} b^{5} - 27 \, a b^{5} + 28 \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} - 18 \, a + 11\right )} \arcsin \left (-b x - a\right ) \mathrm {sgn}\relax (b)}{2 \, b^{2} {\left | b \right |}} + \frac {8 \, {\left (a^{2} - 2 \, a + 1\right )}}{b^{2} {\left (\frac {\sqrt {-{\left (b x + a\right )}^{2} + 1} {\left | b \right |} + b}{b^{2} x + a b} - 1\right )} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 552, normalized size = 3.29 \[ -\frac {3 a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}+\frac {23 a^{2} x}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {3 x^{3}}{2 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {3 a \,x^{2}}{2 b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{3} x}{3 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {53 a x}{3 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {25 a^{2}}{3 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {13 x^{2}}{3 b \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {9 a \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{b^{2} \sqrt {b^{2}}}-\frac {17 a}{2 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {17 a^{3}}{2 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {b \,x^{4}}{3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a \,x^{3}}{3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {a^{4}}{3 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}+\frac {11 x}{2 b^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}-\frac {11 \arctan \left (\frac {\sqrt {b^{2}}\, \left (x +\frac {a}{b}\right )}{\sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}}\right )}{2 b^{2} \sqrt {b^{2}}}+\frac {26}{3 b^{3} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 1645, normalized size = 9.79 \[ \text {result too large to display} \]
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 \[ \int \frac {x^2\,{\left (a+b\,x+1\right )}^3}{{\left (1-{\left (a+b\,x\right )}^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (a + b x + 1\right )^{3}}{\left (- \left (a + b x - 1\right ) \left (a + b x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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