Optimal. Leaf size=136 \[ \frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {3 c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6286, 1823,
847, 794, 201, 223, 209} \begin {gather*} -\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}+\frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {3 c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a^3}-\frac {(45 a x+32) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 847
Rule 1823
Rule 6286
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^{3/2} \, dx &=c \int x^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {\int x^2 \left (-9 a^2 c-12 a^3 c x\right ) \sqrt {c-a^2 c x^2} \, dx}{6 a^2}\\ &=-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {\int x \left (24 a^3 c^2+45 a^4 c^2 x\right ) \sqrt {c-a^2 c x^2} \, dx}{30 a^4 c}\\ &=-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {(3 c) \int \sqrt {c-a^2 c x^2} \, dx}{8 a^2}\\ &=\frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{16 a^2}\\ &=\frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{16 a^2}\\ &=\frac {3 c x \sqrt {c-a^2 c x^2}}{16 a^2}-\frac {2 x^2 \left (c-a^2 c x^2\right )^{3/2}}{5 a}-\frac {1}{6} x^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {(32+45 a x) \left (c-a^2 c x^2\right )^{3/2}}{120 a^3}+\frac {3 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 105, normalized size = 0.77 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-64-45 a x-32 a^2 x^2+50 a^3 x^3+96 a^4 x^4+40 a^5 x^5\right )-45 c^{3/2} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )}{240 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs.
\(2(112)=224\).
time = 0.06, size = 266, normalized size = 1.96
method | result | size |
risch | \(-\frac {\left (40 x^{5} a^{5}+96 a^{4} x^{4}+50 a^{3} x^{3}-32 a^{2} x^{2}-45 a x -64\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{240 a^{3} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {3 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{16 a^{2} \sqrt {c \,a^{2}}}\) | \(109\) |
default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,a^{2}}-\frac {13 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6 a^{2}}+\frac {2 \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5 a^{3} c}-\frac {2 \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )}{a^{3}}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 189, normalized size = 1.39 \begin {gather*} -\frac {1}{240} \, a {\left (\frac {130 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x}{a^{3}} - \frac {40 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{3} c} - \frac {240 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c x}{a^{3}} + \frac {195 \, \sqrt {-a^{2} c x^{2} + c} c x}{a^{3}} + \frac {195 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{a^{4}} + \frac {160 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{a^{4}} - \frac {96 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{4} c} + \frac {480 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c}{a^{4}} - \frac {240 \, c^{3} \arcsin \left (a x - 2\right )}{a^{7} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 216, normalized size = 1.59 \begin {gather*} \left [\frac {45 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, a^{3}}, -\frac {45 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (40 \, a^{5} c x^{5} + 96 \, a^{4} c x^{4} + 50 \, a^{3} c x^{3} - 32 \, a^{2} c x^{2} - 45 \, a c x - 64 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, a^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 11.74, size = 515, normalized size = 3.79 \begin {gather*} a^{2} c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 107, normalized size = 0.79 \begin {gather*} \frac {1}{240} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, a^{2} c x + 12 \, a c\right )} x + 25 \, c\right )} x - \frac {16 \, c}{a}\right )} x - \frac {45 \, c}{a^{2}}\right )} x - \frac {64 \, c}{a^{3}}\right )} - \frac {3 \, c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{16 \, a^{2} \sqrt {-c} {\left | a \right |}} \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 = -0.01 \begin {gather*} \int -\frac {x^2\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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