Optimal. Leaf size=176 \[ -\frac {77 c^{9/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a}-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a} \]
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Rubi [A] time = 0.17, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {77 c^{9/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{9/2}}{10 a}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a} \]
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 )^{9/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx\\ &=-\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{7/2} \, dx\right )\\ &=\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{10} (11 c) \int (1+a x) \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{10} (11 c) \int \left (c-a^2 c x^2\right )^{7/2} \, dx\\ &=-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{80} \left (77 c^2\right ) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{96} \left (77 c^3\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{128} \left (77 c^4\right ) \int \sqrt {c-a^2 c x^2} \, dx\\ &=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{256} \left (77 c^5\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {1}{256} \left (77 c^5\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 {77}{256} c^4 x \sqrt {c-a^2 c x^2}-\frac {77}{384} c^3 x \left (c-a^2 c x^2\right )^{3/2}-\frac {77}{480} c^2 x \left (c-a^2 c x^2\right )^{5/2}-\frac {11}{80} c x \left (c-a^2 c x^2\right )^{7/2}+\frac {11 \left (c-a^2 c x^2\right )^{9/2}}{90 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{9/2}}{10 a}-\frac {77 c^{9/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{256 a}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 167, normalized size = 0.95 \[ \frac {c^4 \sqrt {c-a^2 c x^2} \left (\sqrt {a x+1} \left (-1152 a^{10} x^{10}-1408 a^9 x^9+5584 a^8 x^8+7216 a^7 x^7-10552 a^6 x^6-15048 a^5 x^5+9210 a^4 x^4+16390 a^3 x^3-2185 a^2 x^2-10615 a x+2560\right )+6930 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{11520 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 329, normalized size = 1.87 \[ \left [\frac {3465 \, \sqrt {-c} c^{4} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{23040 \, a}, \frac {3465 \, c^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + {\left (1152 \, a^{9} c^{4} x^{9} + 2560 \, a^{8} c^{4} x^{8} - 3024 \, a^{7} c^{4} x^{7} - 10240 \, a^{6} c^{4} x^{6} + 312 \, a^{5} c^{4} x^{5} + 15360 \, a^{4} c^{4} x^{4} + 6150 \, a^{3} c^{4} x^{3} - 10240 \, a^{2} c^{4} x^{2} - 8055 \, a c^{4} x + 2560 \, c^{4}\right )} \sqrt {-a^{2} c x^{2} + c}}{11520 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 164, normalized size = 0.93 \[ \frac {77 \, c^{5} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{256 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{11520} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {2560 \, c^{4}}{a} - {\left (8055 \, c^{4} + 2 \, {\left (5120 \, a c^{4} - {\left (3075 \, a^{2} c^{4} + 4 \, {\left (1920 \, a^{3} c^{4} + {\left (39 \, a^{4} c^{4} - 2 \, {\left (640 \, a^{5} c^{4} + {\left (189 \, a^{6} c^{4} - 8 \, {\left (9 \, a^{8} c^{4} x + 20 \, a^{7} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 350, normalized size = 1.99 \[ \frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}{10}+\frac {9 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{80}+\frac {21 c^{2} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{160}+\frac {21 c^{3} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{128}+\frac {63 c^{4} x \sqrt {-a^{2} c \,x^{2}+c}}{256}+\frac {63 c^{5} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{256 \sqrt {a^{2} c}}+\frac {2 \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {9}{2}}}{9 a}-\frac {c \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {7}{2}} x}{4}-\frac {7 c^{2} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}} x}{24}-\frac {35 c^{3} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{96}-\frac {35 c^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{64}-\frac {35 c^{5} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{64 \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 192, normalized size = 1.09 \[ \frac {1}{10} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} x - \frac {11}{80} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c x - \frac {77}{480} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2} x - \frac {77}{384} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3} x - \frac {35}{64} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4} x + \frac {63}{256} \, \sqrt {-a^{2} c x^{2} + c} c^{4} x - \frac {35 \, c^{6} \arcsin \left (a x - 2\right )}{64 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {63 \, c^{\frac {9}{2}} \arcsin \left (a x\right )}{256 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}{9 \, a} + \frac {35 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{4}}{32 \, a} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^{9/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 36.35, size = 1341, normalized size = 7.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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