Optimal. Leaf size=106 \[ -\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.26, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1807, 807, 266, 47, 63, 208} \[ \frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^5} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {1}{4} \int \frac {\left (-8 a c-5 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^4} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{4} \left (5 a^2 c\right ) \int \frac {\sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^2 c\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c-a^2 c x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac {1}{16} \left (5 a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {1}{8} \left (5 a^2 c\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 {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 96, normalized size = 0.91 \[ -\frac {5}{8} a^4 c^{3/2} \log (x)+\frac {5}{8} a^4 c^{3/2} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+\frac {c \left (16 a^3 x^3-9 a^2 x^2-16 a x-6\right ) \sqrt {c-a^2 c x^2}}{24 x^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 191, normalized size = 1.80 \[ \left [\frac {15 \, a^{4} c^{\frac {3}{2}} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac {15 \, a^{4} \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 371, normalized size = 3.50 \[ -\frac {5 \, a^{4} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c^{2} + 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{3} - 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{3} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{4} + 16 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{4} {\left | a \right |} + 9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{5} - 16 \, a^{3} \sqrt {-c} c^{5} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 364, normalized size = 3.43 \[ -\frac {5 a^{4} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{24}+\frac {5 a^{4} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}-\frac {5 a^{4} \sqrt {-a^{2} c \,x^{2}+c}\, c}{8}-\frac {2 a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c x}-\frac {2 a^{5} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}-a^{5} c x \sqrt {-a^{2} c \,x^{2}+c}-\frac {a^{5} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{4} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a^{5} c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x +\frac {a^{5} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}}-\frac {7 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{8 c \,x^{2}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}} \]
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 \[ -\int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{5}}\,{d x} \]
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 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^5\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 14.70, size = 447, normalized size = 4.22 \[ a^{2} c \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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