Integrand size = 26, antiderivative size = 350 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {45 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{144 a^4 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.46 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5060, 5050, 5018, 5025, 5024, 3385, 3433, 5091, 5090, 3393, 3377} \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {45 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{16 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{144 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {5 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {a^2 c x^2+c}} \]
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Rule 3377
Rule 3385
Rule 3393
Rule 3433
Rule 5018
Rule 5024
Rule 5025
Rule 5050
Rule 5060
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = \frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5}{12} \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {2 \int \frac {x \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c} \\ & = \frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^3 c}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \int \frac {x^3 \sqrt {\arctan (a x)}}{\left (1+a^2 x^2\right )^{5/2}} \, dx}{12 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {5 \sqrt {\arctan (a x)}}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{4 a^3 c}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sqrt {x} \sin ^3(x) \, dx,x,\arctan (a x)\right )}{12 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {5 \sqrt {\arctan (a x)}}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{4} \sqrt {x} \sin (x)-\frac {1}{4} \sqrt {x} \sin (3 x)\right ) \, dx,x,\arctan (a x)\right )}{12 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \sqrt {\arctan (a x)}} \, dx}{4 a^3 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {5 \sqrt {\arctan (a x)}}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sqrt {x} \sin (3 x) \, dx,x,\arctan (a x)\right )}{48 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \sqrt {x} \sin (x) \, dx,x,\arctan (a x)\right )}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{4 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{288 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{32 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{16 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {45 \sqrt {\arctan (a x)}}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 x^3 \arctan (a x)^{3/2}}{18 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {5 x \arctan (a x)^{3/2}}{3 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^{5/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 \arctan (a x)^{5/2}}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \sqrt {\arctan (a x)} \cos (3 \arctan (a x))}{144 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {45 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{16 a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{144 a^4 c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {4800 \arctan (a x)+5040 a^2 x^2 \arctan (a x)+2880 a x \arctan (a x)^2+3360 a^3 x^3 \arctan (a x)^2-1152 \arctan (a x)^3-1728 a^2 x^2 \arctan (a x)^3+1215 i \left (1+a^2 x^2\right )^{3/2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-1215 i \left (1+a^2 x^2\right )^{3/2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )-5 i \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )-5 i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+5 i \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )+5 i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )}{1728 a^4 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \]
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\[\int \frac {x^{3} \arctan \left (a x \right )^{\frac {5}{2}}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^{5/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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