Optimal. Leaf size=151 \[ -3 a^2 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {1}{2} a^2 c^2 (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {a c (a x+12) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2} \]
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Rubi [A] time = 0.33, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {6151, 1807, 813, 815, 844, 217, 203, 266, 63, 208} \[ -\frac {1}{2} a^2 c^2 (6 a x+1) \sqrt {c-a^2 c x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {a c (a x+12) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 813
Rule 815
Rule 844
Rule 1807
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx &=c \int \frac {(1+a x)^2 \left (c-a^2 c x^2\right )^{3/2}}{x^3} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac {1}{2} \int \frac {\left (-4 a c+a^2 c x\right ) \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx\\ &=-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}+\frac {1}{4} \int \frac {\left (-2 a^2 c^2-24 a^3 c^2 x\right ) \sqrt {c-a^2 c x^2}}{x} \, dx\\ &=-\frac {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac {\int \frac {4 a^4 c^4+24 a^5 c^4 x}{x \sqrt {c-a^2 c x^2}} \, dx}{8 a^2 c}\\ &=-\frac {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac {1}{2} \left (a^2 c^3\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx-\left (3 a^3 c^3\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-\frac {1}{4} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )-\left (3 a^3 c^3\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 {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} c^2 \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 {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-3 a^2 c^{5/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} a^2 c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 151, normalized size = 1.00 \[ \frac {1}{2} a^2 c^{5/2} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+3 a^2 c^{5/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )-\frac {1}{2} a^2 c^{5/2} \log (x)-\frac {c^2 \left (2 a^4 x^4+6 a^3 x^3-2 a^2 x^2+12 a x+3\right ) \sqrt {c-a^2 c x^2}}{6 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.68, size = 329, normalized size = 2.18 \[ \left [\frac {36 \, a^{2} c^{\frac {5}{2}} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 3 \, a^{2} c^{\frac {5}{2}} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{12 \, x^{2}}, \frac {3 \, a^{2} \sqrt {-c} c^{2} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 9 \, a^{2} \sqrt {-c} c^{2} x^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - {\left (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 302, normalized size = 2.00 \[ -\frac {a^{2} c^{3} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {3 \, a^{3} \sqrt {-c} c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} c^{2} - {\left (a^{4} c^{2} x + 3 \, a^{3} c^{2}\right )} x\right )} - \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{2} c^{3} {\left | a \right |} - 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{2} c^{4} {\left | a \right |} + 4 \, a^{3} \sqrt {-c} c^{4}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 399, normalized size = 2.64 \[ -\frac {a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{10}-\frac {a^{2} c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{6}+\frac {a^{2} c^{\frac {5}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2}-\frac {a^{2} \sqrt {-a^{2} c \,x^{2}+c}\, c^{2}}{2}-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{c x}-2 a^{3} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}-\frac {5 a^{3} c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2}-\frac {15 a^{3} c^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{4}-\frac {15 a^{3} c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{4 \sqrt {a^{2} c}}-\frac {2 a^{2} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{5}+\frac {a^{3} c \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{2}+\frac {3 a^{3} c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x}{4}+\frac {3 a^{3} c^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{4 \sqrt {a^{2} c}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}} \]
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 \[ -\int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{3}}\,{d x} \]
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 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{x^3\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 118.63, size = 401, normalized size = 2.66 \[ - a^{4} c^{2} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {\sqrt {c} x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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