Optimal. Leaf size=137 \[ \frac {c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}+\frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(35 a x+27) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2} \]
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Rubi [A] time = 0.22, 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 = {6151, 1809, 780, 195, 217, 203} \[ \frac {c^2 x \sqrt {c-a^2 c x^2}}{8 a}+\frac {c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a^2}+\frac {c x \left (c-a^2 c x^2\right )^{3/2}}{12 a}-\frac {1}{7} x^2 \left (c-a^2 c x^2\right )^{5/2}-\frac {(35 a x+27) \left (c-a^2 c x^2\right )^{5/2}}{105 a^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 780
Rule 1809
Rule 6151
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 \operatorname {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 = 0.15, size = 115, normalized size = 0.84 \[ -\frac {c^2 \left (105 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\left (120 a^6 x^6+280 a^5 x^5-24 a^4 x^4-490 a^3 x^3-312 a^2 x^2+105 a x+216\right ) \sqrt {c-a^2 c x^2}\right )}{840 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 263, normalized size = 1.92 \[ \left [\frac {105 \, \sqrt {-c} c^{2} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, a^{2}}, -\frac {105 \, c^{\frac {5}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (120 \, a^{6} c^{2} x^{6} + 280 \, a^{5} c^{2} x^{5} - 24 \, a^{4} c^{2} x^{4} - 490 \, a^{3} c^{2} x^{3} - 312 \, a^{2} c^{2} x^{2} + 105 \, a c^{2} x + 216 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 131, normalized size = 0.96 \[ \frac {1}{840} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (156 \, c^{2} + {\left (245 \, a c^{2} + 4 \, {\left (3 \, a^{2} c^{2} - 5 \, {\left (3 \, a^{4} c^{2} x + 7 \, a^{3} c^{2}\right )} x\right )} x\right )} x\right )} x - \frac {105 \, c^{2}}{a}\right )} x - \frac {216 \, c^{2}}{a^{2}}\right )} - \frac {c^{3} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, a \sqrt {-c} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 284, normalized size = 2.07 \[ \frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{7 a^{2} c}-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 a}-\frac {5 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{12 a}-\frac {5 c^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{8 a}-\frac {5 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{8 a \sqrt {a^{2} c}}-\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5 a^{2}}+\frac {c \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{2 a}+\frac {3 c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{4 a}+\frac {3 c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{4 a \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 193, normalized size = 1.41 \[ -\frac {1}{840} \, {\left (\frac {280 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x}{a^{2}} - \frac {70 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c x}{a^{2}} - \frac {630 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2} x}{a^{2}} + \frac {525 \, \sqrt {-a^{2} c x^{2} + c} c^{2} x}{a^{2}} + \frac {525 \, c^{\frac {5}{2}} \arcsin \left (a x\right )}{a^{3}} + \frac {336 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}{a^{3}} - \frac {120 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{a^{3} c} + \frac {1260 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{2}}{a^{3}} - \frac {630 \, c^{4} \arcsin \left (a x - 2\right )}{a^{6} \left (-\frac {c}{a^{2}}\right )^{\frac {3}{2}}}\right )} a \]
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 \[ \int -\frac {x\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.64, size = 586, normalized size = 4.28 \[ - a^{4} c^{2} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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