Optimal. Leaf size=59 \[ \frac {16 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{315 b^3}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2168, 2157, 30} \[ \frac {16 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{315 b^3}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2157
Rule 2168
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {4 \int x \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx}{5 b}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {8 \int \tanh ^{-1}(\tanh (a+b x))^{7/2} \, dx}{35 b^2}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {8 \operatorname {Subst}\left (\int x^{7/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{35 b^3}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^2}+\frac {16 \tanh ^{-1}(\tanh (a+b x))^{9/2}}{315 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 0.83 \[ \frac {2 \tanh ^{-1}(\tanh (a+b x))^{5/2} \left (-36 b x \tanh ^{-1}(\tanh (a+b x))+8 \tanh ^{-1}(\tanh (a+b x))^2+63 b^2 x^2\right )}{315 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 53, normalized size = 0.90 \[ \frac {2 \, {\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 168, normalized size = 2.85 \[ \frac {\sqrt {2} {\left (\frac {21 \, \sqrt {2} {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{2}}{b^{2}} + \frac {18 \, \sqrt {2} {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a}{b^{2}} + \frac {\sqrt {2} {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )}}{b^{2}}\right )}}{315 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 69, normalized size = 1.17 \[ \frac {\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {9}{2}}}{9}+\frac {2 \left (-2 \arctanh \left (\tanh \left (b x +a \right )\right )+2 b x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {5}{2}}}{5}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 42, normalized size = 0.71 \[ \frac {2 \, {\left (35 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} - 12 \, a^{2} b x + 8 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{315 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 1153, normalized size = 19.54 \[ \text {result too large to display} \]
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 \[ \int x^{2} \operatorname {atanh}^{\frac {3}{2}}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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