Optimal. Leaf size=59 \[ \frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac {16 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{693 b^3} \]
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Rubi [A]
time = 0.04, 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 = {2199, 2188, 30}
\begin {gather*} \frac {16 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{693 b^3}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2188
Rule 2199
Rubi steps
\begin {align*} \int x^2 \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac {4 \int x \tanh ^{-1}(\tanh (a+b x))^{7/2} \, dx}{7 b}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac {8 \int \tanh ^{-1}(\tanh (a+b x))^{9/2} \, dx}{63 b^2}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac {8 \text {Subst}\left (\int x^{9/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{63 b^3}\\ &=\frac {2 x^2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 b}-\frac {8 x \tanh ^{-1}(\tanh (a+b x))^{9/2}}{63 b^2}+\frac {16 \tanh ^{-1}(\tanh (a+b x))^{11/2}}{693 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 49, normalized size = 0.83 \begin {gather*} \frac {2 \tanh ^{-1}(\tanh (a+b x))^{7/2} \left (99 b^2 x^2-44 b x \tanh ^{-1}(\tanh (a+b x))+8 \tanh ^{-1}(\tanh (a+b x))^2\right )}{693 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 69, normalized size = 1.17
method | result | size |
default | \(\frac {\frac {2 \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {11}{2}}}{11}+\frac {2 \left (-2 \arctanh \left (\tanh \left (b x +a \right )\right )+2 b x \right ) \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {9}{2}}}{9}+\frac {2 \left (b x -\arctanh \left (\tanh \left (b x +a \right )\right )\right )^{2} \arctanh \left (\tanh \left (b x +a \right )\right )^{\frac {7}{2}}}{7}}{b^{3}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 42, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (63 \, b^{3} x^{3} + 35 \, a b^{2} x^{2} - 20 \, a^{2} b x + 8 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {5}{2}}}{693 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 64, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (63 \, b^{5} x^{5} + 161 \, a b^{4} x^{4} + 113 \, a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} - 4 \, a^{4} b x + 8 \, a^{5}\right )} \sqrt {b x + a}}{693 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 62.07, size = 71, normalized size = 1.20 \begin {gather*} \begin {cases} \frac {2 x^{2} \operatorname {atanh}^{\frac {7}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{7 b} - \frac {8 x \operatorname {atanh}^{\frac {9}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{63 b^{2}} + \frac {16 \operatorname {atanh}^{\frac {11}{2}}{\left (\tanh {\left (a + b x \right )} \right )}}{693 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {atanh}^{\frac {5}{2}}{\left (\tanh {\left (a \right )} \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs.
\(2 (47) = 94\).
time = 0.39, size = 248, normalized size = 4.20 \begin {gather*} \frac {\sqrt {2} {\left (\frac {231 \, \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^{3}}{b^{2}} + \frac {297 \, \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^{2}}{b^{2}} + \frac {33 \, \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 )} a}{b^{2}} + \frac {5 \, \sqrt {2} {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )}}{b^{2}}\right )}}{3465 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 1789, normalized size = 30.32 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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