Optimal. Leaf size=372 \[ \frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (1-a x)^2}+\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (1-a x)}-\frac {13 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{40 (1-a x)^3}-\frac {57 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac {25 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}}+\frac {41 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{24 (1-a x)^3 (a x+1)^2}+\frac {57 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{80 (1-a x)^3 (a x+1)}+\frac {11 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{30 (1-a x)^3} \]
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Rubi [A] time = 0.54, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ -\frac {57 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac {41 a^5 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{24 (1-a x)^3 (a x+1)^2}+\frac {57 a^4 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{80 (1-a x)^3 (a x+1)}+\frac {11 a^3 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{30 (1-a x)^3}-\frac {13 a^2 x^3 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{40 (1-a x)^3}+\frac {a x^2 (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{15 (1-a x)^2}+\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}{6 (1-a x)}+\frac {2 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac {25 a^6 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 149
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{7/2} \, dx\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{7/2} (1+a x)^{7/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{5/2} (1+a x)^{9/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1-a x)^{3/2} (1+a x)^{7/2} \left (2 a-7 a^2 x\right )}{x^6} \, dx}{6 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1-a x} (1+a x)^{7/2} \left (-39 a^2+33 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{7/2} \left (132 a^3-93 a^4 x\right )}{x^4 \sqrt {1-a x}} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{5/2} \left (513 a^4-411 a^5 x\right )}{x^3 \sqrt {1-a x}} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {(1+a x)^{3/2} \left (1230 a^5-1335 a^6 x\right )}{x^2 \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}+\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {\sqrt {1+a x} \left (1125 a^6-2565 a^7 x\right )}{x \sqrt {1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {-1125 a^7+1440 a^8 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{720 a (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}-\frac {\left (25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{16 (1-a x)^{7/2} (1+a x)^{7/2}}+\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {\left (25 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}+\frac {\left (2 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac {11 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^4}{30 (1-a x)^3}-\frac {57 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac {41 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}{24 (1-a x)^3 (1+a x)^2}+\frac {57 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^5}{80 (1-a x)^3 (1+a x)}-\frac {13 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^3 (1+a x)}{40 (1-a x)^3}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^2 (1+a x)}{15 (1-a x)^2}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^{7/2} x (1+a x)}{6 (1-a x)}+\frac {2 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \sin ^{-1}(a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}+\frac {25 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 150, normalized size = 0.40 \[ \frac {c^3 \sqrt {c-\frac {c}{a^2 x^2}} \left (480 a^6 x^6 \log \left (\sqrt {a^2 x^2-1}+a x\right )+375 a^6 x^6 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (240 a^6 x^6-736 a^5 x^5+105 a^4 x^4+352 a^3 x^3+70 a^2 x^2-96 a x-40\right )\right )}{240 a^6 x^5 \sqrt {a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.76, size = 438, normalized size = 1.18 \[ \left [-\frac {960 \, a^{5} \sqrt {-c} c^{3} x^{5} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 375 \, a^{5} \sqrt {-c} c^{3} x^{5} \log \left (-\frac {a^{2} c x^{2} - 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, \frac {375 \, a^{5} c^{\frac {7}{2}} x^{5} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 240 \, a^{5} c^{\frac {7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (240 \, a^{6} c^{3} x^{6} - 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} + 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} - 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 113.35, size = 561, normalized size = 1.51 \[ -\frac {1}{120} \, {\left (\frac {375 \, c^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} + \frac {240 \, c^{\frac {7}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {120 \, \sqrt {a^{2} c x^{2} - c} c^{3} \mathrm {sgn}\relax (x)}{a^{2}} + \frac {105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\relax (x) + 1440 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{10} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} c^{5} {\left | a \right |} \mathrm {sgn}\relax (x) + 4320 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{8} a c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} c^{6} {\left | a \right |} \mathrm {sgn}\relax (x) + 7360 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a c^{\frac {13}{2}} \mathrm {sgn}\relax (x) + 150 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{5} c^{7} {\left | a \right |} \mathrm {sgn}\relax (x) + 6720 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a c^{\frac {15}{2}} \mathrm {sgn}\relax (x) - 595 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{8} {\left | a \right |} \mathrm {sgn}\relax (x) + 2976 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {17}{2}} \mathrm {sgn}\relax (x) - 105 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{9} {\left | a \right |} \mathrm {sgn}\relax (x) + 736 \, a c^{\frac {19}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 795, normalized size = 2.14 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {7}{2}} x \left (-2016 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{9} c +2016 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{5} a^{9}+375 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{8} c -480 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{6} a^{8} c +105 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{4} a^{8}+2352 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{7} c^{2}-560 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {5}{2}} x^{7} a^{7} c^{2}+224 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{3} a^{7}-525 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{6} c^{2}-2940 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{5} c^{3}+700 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{7} a^{5} c^{3}+630 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{2} a^{6}+875 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{4} c^{3}+672 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}\, x \,a^{5}+4410 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{7} a^{3} c^{4}-1050 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{7} a^{3} c^{4}+280 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {9}{2}} \sqrt {-\frac {c}{a^{2}}}-2625 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{2} c^{4}-4410 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{2}} x^{6} a +1050 \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {9}{2}} x^{6} a -2625 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{6} c^{5}\right )}{1680 a^{2} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \sqrt {-\frac {c}{a^{2}}}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}{a x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 33.43, size = 1059, normalized size = 2.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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