Optimal. Leaf size=305 \[ -\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {55 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}-\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}+\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3} \]
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Rubi [A]
time = 0.16, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6261, 91, 81,
52, 65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {55 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{8 \sqrt {2} a^3}+\frac {55 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{8 \sqrt {2} a^3}-\frac {(a x+1)^{3/4} (1-a x)^{9/4}}{3 a^3}-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{a x+1}}-\frac {11 (a x+1)^{3/4} (1-a x)^{5/4}}{4 a^3}-\frac {55 (a x+1)^{3/4} \sqrt [4]{1-a x}}{8 a^3}-\frac {55 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt {2} a^3}+\frac {55 \log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{16 \sqrt {2} a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6261
Rubi steps
\begin {align*} \int e^{-\frac {5}{2} \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-a x)^{5/4}}{(1+a x)^{5/4}} \, dx\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}+\frac {2 \int \frac {(1-a x)^{5/4} \left (-\frac {5 a}{2}+\frac {a^2 x}{2}\right )}{\sqrt [4]{1+a x}} \, dx}{a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {11 \int \frac {(1-a x)^{5/4}}{\sqrt [4]{1+a x}} \, dx}{2 a^2}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {55 \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx}{8 a^2}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {55 \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{16 a^2}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}+\frac {55 \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{4 a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}+\frac {55 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{4 a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}+\frac {55 \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^3}+\frac {55 \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}+\frac {55 \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 a^3}+\frac {55 \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 a^3}-\frac {55 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}-\frac {55 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}+\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}+\frac {55 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}-\frac {55 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}\\ &=-\frac {2 (1-a x)^{9/4}}{a^3 \sqrt [4]{1+a x}}-\frac {55 \sqrt [4]{1-a x} (1+a x)^{3/4}}{8 a^3}-\frac {11 (1-a x)^{5/4} (1+a x)^{3/4}}{4 a^3}-\frac {(1-a x)^{9/4} (1+a x)^{3/4}}{3 a^3}-\frac {55 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}+\frac {55 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{8 \sqrt {2} a^3}-\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}+\frac {55 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{16 \sqrt {2} a^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 70, normalized size = 0.23 \begin {gather*} \frac {(1-a x)^{9/4} \left (-3 (7+a x)+11\ 2^{3/4} \sqrt [4]{1+a x} \, _2F_1\left (\frac {1}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-a x)\right )\right )}{9 a^3 \sqrt [4]{1+a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 589 vs.
\(2 (234) = 468\).
time = 0.36, size = 589, normalized size = 1.93 \begin {gather*} \frac {660 \, \sqrt {2} {\left (a^{4} x + a^{3}\right )} \frac {1}{a^{12}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {\frac {\sqrt {2} {\left (a^{10} x - a^{9}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {3}{4}} + {\left (a^{7} x - a^{6}\right )} \sqrt {\frac {1}{a^{12}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {1}{4}} - 1\right ) + 660 \, \sqrt {2} {\left (a^{4} x + a^{3}\right )} \frac {1}{a^{12}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a^{3} \sqrt {-\frac {\sqrt {2} {\left (a^{10} x - a^{9}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {3}{4}} - {\left (a^{7} x - a^{6}\right )} \sqrt {\frac {1}{a^{12}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {1}{4}} - \sqrt {2} a^{3} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {1}{4}} + 1\right ) + 165 \, \sqrt {2} {\left (a^{4} x + a^{3}\right )} \frac {1}{a^{12}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{10} x - a^{9}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {3}{4}} + {\left (a^{7} x - a^{6}\right )} \sqrt {\frac {1}{a^{12}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 165 \, \sqrt {2} {\left (a^{4} x + a^{3}\right )} \frac {1}{a^{12}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{10} x - a^{9}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{12}}^{\frac {3}{4}} - {\left (a^{7} x - a^{6}\right )} \sqrt {\frac {1}{a^{12}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 61 \, a x + 287\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{96 \, {\left (a^{4} x + a^{3}\right )}} \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^{2}}{\left (\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x^2}{{\left (\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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