Optimal. Leaf size=130 \[ -\frac {7 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a}-\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a} \]
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Rubi [A] time = 0.14, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6167, 6141, 671, 641, 195, 217, 203} \[ -\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{5/2}}{6 a}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rule 671
Rule 6141
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2} \, dx\right )\\ &=\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {1}{6} (7 c) \int (1+a x) \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {1}{6} (7 c) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {1}{8} \left (7 c^2\right ) \int \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {1}{16} \left (7 c^3\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {1}{16} \left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=-\frac {7}{16} c^2 x \sqrt {c-a^2 c x^2}-\frac {7}{24} c x \left (c-a^2 c x^2\right )^{3/2}+\frac {7 \left (c-a^2 c x^2\right )^{5/2}}{30 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{5/2}}{6 a}-\frac {7 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 135, normalized size = 1.04 \[ \frac {c^2 \sqrt {c-a^2 c x^2} \left (\sqrt {a x+1} \left (-40 a^6 x^6-56 a^5 x^5+106 a^4 x^4+182 a^3 x^3-57 a^2 x^2-231 a x+96\right )+210 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.64, size = 241, normalized size = 1.85 \[ \left [\frac {105 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{480 \, a}, \frac {105 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (40 \, a^{5} c^{2} x^{5} + 96 \, a^{4} c^{2} x^{4} - 10 \, a^{3} c^{2} x^{3} - 192 \, a^{2} c^{2} x^{2} - 135 \, a c^{2} x + 96 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{240 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 116, normalized size = 0.89 \[ \frac {7 \, c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{16 \, \sqrt {-c} {\left | a \right |}} - \frac {1}{240} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (135 \, c^{2} + 2 \, {\left (96 \, a c^{2} + {\left (5 \, a^{2} c^{2} - 4 \, {\left (5 \, a^{4} c^{2} x + 12 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {96 \, c^{2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 242, normalized size = 1.86 \[ \frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{24}+\frac {5 c^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{16}+\frac {5 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{16 \sqrt {a^{2} c}}+\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5 a}-\frac {c \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{2}-\frac {3 c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{4}-\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{4 \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 154, normalized size = 1.18 \[ \frac {1}{6} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x - \frac {7}{24} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x - \frac {3}{4} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x + \frac {5}{16} \, \sqrt {-a^{2} c x^{2} + c} c^{2} x - \frac {3 \, c^{4} \arcsin \left (a x - 2\right )}{4 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {5 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{16 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{5 \, a} + \frac {3 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{2 \, a} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.02, size = 478, normalized size = 3.68 \[ a^{4} c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 2 a^{3} c^{2} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 a c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) - c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {\sqrt {c} x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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