Optimal. Leaf size=130 \[ -\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7}{8} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x} \]
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Rubi [A] time = 0.31, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1807, 835, 807, 266, 63, 208} \[ -\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {7}{8} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 1807
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx &=c \int \frac {(1+a x)^2}{x^5 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {1}{4} \int \frac {-8 a c-7 a^2 c x}{x^4 \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}+\frac {\int \frac {21 a^2 c^2+16 a^3 c^2 x}{x^3 \sqrt {c-a^2 c x^2}} \, dx}{12 c}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\int \frac {-32 a^3 c^3-21 a^4 c^3 x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{24 c^2}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{8} \left (7 a^4 c\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}+\frac {1}{16} \left (7 a^4 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {1}{8} \left (7 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )\\ &=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {7}{8} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 95, normalized size = 0.73 \[ \frac {7}{8} a^4 \sqrt {c} \log (x)-\frac {7}{8} a^4 \sqrt {c} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )-\frac {\left (32 a^3 x^3+21 a^2 x^2+16 a x+6\right ) \sqrt {c-a^2 c x^2}}{24 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.68, size = 180, normalized size = 1.38 \[ \left [\frac {21 \, a^{4} \sqrt {c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac {21 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 324, normalized size = 2.49 \[ \frac {7 \, a^{4} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} + 96 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} - 128 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} {\left | a \right |} + 21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{4} + 32 \, a^{3} \sqrt {-c} c^{4} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 287, normalized size = 2.21 \[ -\frac {7 \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right ) a^{4}}{8}+\frac {7 \sqrt {-a^{2} c \,x^{2}+c}\, a^{4}}{8}-\frac {2 a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{5} x \sqrt {-a^{2} c \,x^{2}+c}-\frac {2 a^{5} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}-2 a^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}+\frac {2 a^{5} c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}}-\frac {9 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{8 c \,x^{2}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}} \]
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 \[ -\int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{x^5\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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