Optimal. Leaf size=137 \[ \frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c^{5/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2} \]
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Rubi [A]
time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6286, 1823,
794, 201, 223, 209} \begin {gather*} \frac {c^{5/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}+\frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(35 a x+27) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 794
Rule 1823
Rule 6286
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^{5/2} \, dx &=c \int x (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {\int x \left (-9 a^2 c-14 a^3 c x\right ) \left (c-a^2 c x^2\right )^{3/2} \, dx}{7 a^2}\\ &=-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c \int \left (c-a^2 c x^2\right )^{3/2} \, dx}{3 a}\\ &=\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c^2 \int \sqrt {c-a^2 c x^2} \, dx}{4 a}\\ &=\frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c^3 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{8 a}\\ &=\frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c^3 \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}\\ &=\frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(27+35 a x) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2}+\frac {c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 115, normalized size = 0.84 \begin {gather*} -\frac {c^2 \left (\sqrt {c-a^2 c x^2} \left (216+105 a x-312 a^2 x^2-490 a^3 x^3-24 a^4 x^4+280 a^5 x^5+120 a^6 x^6\right )+105 \sqrt {c} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )\right )}{840 a^2} \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. \(322\) vs.
\(2(113)=226\).
time = 0.06, size = 323, normalized size = 2.36
method | result | size |
risch | \(\frac {\left (120 x^{6} a^{6}+280 x^{5} a^{5}-24 a^{4} x^{4}-490 a^{3} x^{3}-312 a^{2} x^{2}+105 a x +216\right ) \left (a^{2} x^{2}-1\right ) c^{3}}{840 a^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {\arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{3}}{8 a \sqrt {c \,a^{2}}}\) | \(117\) |
default | \(\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{7 c \,a^{2}}-\frac {2 \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6}\right )}{a}-\frac {2 \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )\right )}{a^{2}}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 193, normalized size = 1.41 \begin {gather*} -\frac {1}{840} \, {\left (\frac {280 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{2}} - \frac {70 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x}{a^{2}} - \frac {630 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x}{a^{2}} + \frac {525 \, \sqrt {-a^{2} c x^{2} + c} c^{2} x}{a^{2}} + \frac {525 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{a^{3}} + \frac {336 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{3}} - \frac {120 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{3} c} + \frac {1260 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{a^{3}} - \frac {630 \, c^{4} \arcsin \left (a x - 2\right )}{a^{6} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 263, normalized size = 1.92 \begin {gather*} \left [\frac {105 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, a^{2}}, -\frac {105 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, a^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.38, size = 586, normalized size = 4.28 \begin {gather*} - a^{4} c^{2} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \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^{2} \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 ) + c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 131, normalized size = 0.96 \begin {gather*} \frac {1}{840} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (156 \, c^{2} + {\left (245 \, a c^{2} + 4 \, {\left (3 \, a^{2} c^{2} - 5 \, {\left (3 \, a^{4} c^{2} x + 7 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {105 \, c^{2}}{a}\right )} x - \frac {216 \, c^{2}}{a^{2}}\right )} - \frac {c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a \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\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\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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