Optimal. Leaf size=357 \[ \frac {c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {7}{24} c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a^3 c}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{72 a^3}+\frac {c \sqrt {a^2 c x^2+c}}{16 a^3} \]
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Rubi [A] time = 0.78, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4950, 4946, 4952, 261, 4890, 4886, 266, 43} \[ -\frac {i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{30 a^3 c}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{72 a^3}+\frac {c \sqrt {a^2 c x^2+c}}{16 a^3}+\frac {1}{6} a^2 c x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {7}{24} c x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{16 a^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 4886
Rule 4890
Rule 4946
Rule 4950
Rule 4952
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx &=c \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\left (a^2 c\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{4} c^2 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (a c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{6} \left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (a^3 c^2\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{8} c^2 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {c^2 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {1}{24} \left (a c^2\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{12} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{8 a^3}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}+\frac {c^2 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{16 a}-\frac {1}{48} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{8} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {1}{12} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c \sqrt {c+a^2 c x^2}}{48 a^3}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{36 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3 c}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {1}{48} \left (a c^2\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c \sqrt {c+a^2 c x^2}}{16 a^3}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{72 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3 c}+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 6.34, size = 576, normalized size = 1.61 \[ \frac {c \sqrt {a^2 c x^2+c} \left (\frac {3}{4} \left (a^2 x^2+1\right )^{5/2}+\frac {55}{8} \left (a^2 x^2+1\right )^3 \cos \left (3 \tan ^{-1}(a x)\right )-\frac {45}{8} \left (a^2 x^2+1\right )^3 \cos \left (5 \tan ^{-1}(a x)\right )+\frac {15}{16} \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \left (\frac {156 a x}{\sqrt {a^2 x^2+1}}+30 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-30 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-94 \sin \left (3 \tan ^{-1}(a x)\right )+6 \sin \left (5 \tan ^{-1}(a x)\right )+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )+45 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+18 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \cos \left (6 \tan ^{-1}(a x)\right )\right )-\frac {15}{2} \left (a^2 x^2+1\right )^2 \left (-\frac {2}{\sqrt {a^2 x^2+1}}+3 \tan ^{-1}(a x) \left (-\frac {14 a x}{\sqrt {a^2 x^2+1}}+3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2 \sin \left (3 \tan ^{-1}(a x)\right )+4 \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (2 \tan ^{-1}(a x)\right )+\left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right ) \cos \left (4 \tan ^{-1}(a x)\right )\right )-6 \cos \left (3 \tan ^{-1}(a x)\right )\right )-90 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+90 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )\right )}{1440 a^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.32, size = 221, normalized size = 0.62 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 \arctan \left (a x \right ) x^{5} a^{5}-24 a^{4} x^{4}+210 \arctan \left (a x \right ) x^{3} a^{3}-38 a^{2} x^{2}+45 \arctan \left (a x \right ) x a +31\right )}{720 a^{3}}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{16 a^{3} \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )\,{d x} \]
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 \[ \int x^2\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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 \[ \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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