Optimal. Leaf size=329 \[ -\frac {149}{120} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {29}{120} a c^2 x \sqrt {a^2 c x^2+c}+c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {1}{20} a c x \left (a^2 c x^2+c\right )^{3/2}+\frac {1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]
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Rubi [A] time = 0.55, antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4950, 4946, 4958, 4954, 217, 206, 4930, 195} \[ \frac {i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {29}{120} a c^2 x \sqrt {a^2 c x^2+c}+c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {149}{120} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )-\frac {2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {1}{20} a c x \left (a^2 c x^2+c\right )^{3/2}+\frac {1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4930
Rule 4946
Rule 4950
Rule 4954
Rule 4958
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx\\ &=\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac {1}{5} (a c) \int \left (c+a^2 c x^2\right )^{3/2} \, dx+c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac {1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac {1}{20} \left (3 a c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx-\frac {1}{3} \left (a c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx+c^3 \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {29}{120} a c^2 x \sqrt {c+a^2 c x^2}-\frac {1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac {1}{40} \left (3 a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\left (a 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 {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {29}{120} a c^2 x \sqrt {c+a^2 c x^2}-\frac {1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {1}{40} \left (3 a 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}{6} \left (a 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 {29}{120} a c^2 x \sqrt {c+a^2 c x^2}-\frac {1}{20} a c x \left (c+a^2 c x^2\right )^{3/2}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149}{120} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 268, normalized size = 0.81 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (-35 a x \sqrt {a^2 x^2+1}+88 a^2 x^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+184 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+24 a^4 x^4 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)-6 a^3 x^3 \sqrt {a^2 x^2+1}+120 i \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )-120 i \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )-29 \sinh ^{-1}(a x)+120 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-120 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+120 \log \left (\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-120 \log \left (\sin \left (\frac {1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )\right )}{120 \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 198, normalized size = 0.60 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 \arctan \left (a x \right ) x^{4} a^{4}-6 a^{3} x^{3}+88 \arctan \left (a x \right ) x^{2} a^{2}-35 a x +184 \arctan \left (a x \right )\right )}{120}+\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (149 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+60 i \dilog \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-60 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+60 i \dilog \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 \sqrt {a^{2} x^{2}+1}} \]
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 \[ \frac {2}{3} \, {\left (a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c} \arctan \left (a x\right ) - \frac {1}{3} \, {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} {\left (a c^{2} x \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right ) + 2 \, c^{2} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, -a^{2} x^{2} + 3\right )\right )\right )} \sqrt {c} - \frac {1}{120} \, {\left ({\left (a {\left (\frac {3 \, {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )}}{a^{4}}\right )} - 8 \, {\left (\frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \arctan \left (a x\right )\right )} a^{4} c^{2} - 20 \, c^{2} \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) + 2, a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) - 20 \, c^{2} \arctan \left ({\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right ) - 2, -a x + {\left (a^{4} x^{4} + 10 \, a^{2} x^{2} + 9\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (4 \, a x, a^{2} x^{2} - 3\right )\right )\right ) - 120 \, c^{2} \int \frac {\sqrt {a^{2} x^{2} + 1} \arctan \left (a x\right )}{x}\,{d x}\right )} \sqrt {c} \]
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 \[ \int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \]
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 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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