Optimal. Leaf size=359 \[ \frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}-\frac {231 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}+\frac {231 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt {2} a}-\frac {231 \log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{1024 \sqrt {2} a} \]
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Rubi [A]
time = 0.23, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6275, 52, 65,
246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {231 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{512 \sqrt {2} a}-\frac {231 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{512 \sqrt {2} a}-\frac {(a x+1)^{11/4} (1-a x)^{13/4}}{6 a}-\frac {11 (a x+1)^{7/4} (1-a x)^{13/4}}{60 a}-\frac {77 (a x+1)^{3/4} (1-a x)^{13/4}}{480 a}+\frac {77 (a x+1)^{3/4} (1-a x)^{9/4}}{960 a}+\frac {231 (a x+1)^{3/4} (1-a x)^{5/4}}{1280 a}+\frac {231 (a x+1)^{3/4} \sqrt [4]{1-a x}}{512 a}+\frac {231 \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 )}{1024 \sqrt {2} a}-\frac {231 \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 )}{1024 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 6275
Rubi steps
\begin {align*} \int e^{\frac {1}{2} \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx &=\int (1-a x)^{9/4} (1+a x)^{11/4} \, dx\\ &=-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {11}{12} \int (1-a x)^{9/4} (1+a x)^{7/4} \, dx\\ &=-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {77}{120} \int (1-a x)^{9/4} (1+a x)^{3/4} \, dx\\ &=-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {77}{320} \int \frac {(1-a x)^{9/4}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231}{640} \int \frac {(1-a x)^{5/4}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231}{512} \int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231 \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx}{1024}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac {231 \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{256 a}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac {231 \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{256 a}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac {231 \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 )}{512 a}-\frac {231 \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 )}{512 a}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}-\frac {231 \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 )}{1024 a}-\frac {231 \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 )}{1024 a}+\frac {231 \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 )}{1024 \sqrt {2} a}+\frac {231 \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 )}{1024 \sqrt {2} a}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231 \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 )}{1024 \sqrt {2} a}-\frac {231 \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 )}{1024 \sqrt {2} a}-\frac {231 \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 )}{512 \sqrt {2} a}+\frac {231 \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 )}{512 \sqrt {2} a}\\ &=\frac {231 \sqrt [4]{1-a x} (1+a x)^{3/4}}{512 a}+\frac {231 (1-a x)^{5/4} (1+a x)^{3/4}}{1280 a}+\frac {77 (1-a x)^{9/4} (1+a x)^{3/4}}{960 a}-\frac {77 (1-a x)^{13/4} (1+a x)^{3/4}}{480 a}-\frac {11 (1-a x)^{13/4} (1+a x)^{7/4}}{60 a}-\frac {(1-a x)^{13/4} (1+a x)^{11/4}}{6 a}+\frac {231 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}-\frac {231 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{512 \sqrt {2} a}+\frac {231 \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 )}{1024 \sqrt {2} a}-\frac {231 \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 )}{1024 \sqrt {2} a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 42, normalized size = 0.12 \begin {gather*} -\frac {16\ 2^{3/4} (1-a x)^{13/4} \, _2F_1\left (-\frac {11}{4},\frac {13}{4};\frac {17}{4};\frac {1}{2} (1-a x)\right )}{13 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}\, \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}\, dx\]
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. 557 vs.
\(2 (274) = 548\).
time = 0.35, size = 557, normalized size = 1.55 \begin {gather*} -\frac {13860 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) + 13860 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) + 3465 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 3465 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, {\left (1280 \, a^{5} x^{5} + 128 \, a^{4} x^{4} - 4144 \, a^{3} x^{3} - 520 \, a^{2} x^{2} + 5174 \, a x + 1547\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{30720 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (1-a^2\,x^2\right )}^{5/2}\,\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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