Integrand size = 23, antiderivative size = 279 \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}} \]
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Time = 0.31 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {272, 45, 5096, 12, 542, 537, 223, 212, 385, 209} \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \arctan \left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}}-\frac {b x \left (13 c^2 d-30 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e}+\frac {b x \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e} \]
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Rule 12
Rule 45
Rule 209
Rule 212
Rule 223
Rule 272
Rule 385
Rule 537
Rule 542
Rule 5096
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-\frac {(b c) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{1+c^2 x^2} \, dx}{35 e^2} \\ & = -\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-\frac {b \int \frac {\left (d+e x^2\right )^{3/2} \left (-d \left (12 c^2 d+5 e\right )+\left (13 c^2 d-30 e\right ) e x^2\right )}{1+c^2 x^2} \, dx}{210 c e^2} \\ & = -\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-\frac {b \int \frac {\sqrt {d+e x^2} \left (-3 d \left (16 c^4 d^2+11 c^2 d e-10 e^2\right )-3 e \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x^2\right )}{1+c^2 x^2} \, dx}{840 c^3 e^2} \\ & = \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}-\frac {b \int \frac {-3 d \left (32 c^6 d^3+19 c^4 d^2 e-74 c^2 d e^2+40 e^3\right )-3 e \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) x^2}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{1680 c^5 e^2} \\ & = \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{560 c^7 e} \\ & = \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}+\frac {\left (b \left (c^2 d-e\right )^3 \left (2 c^2 d+5 e\right )\right ) \text {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {\left (b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{560 c^7 e} \\ & = \frac {b \left (3 c^4 d^2+54 c^2 d e-40 e^2\right ) x \sqrt {d+e x^2}}{560 c^5 e}-\frac {b \left (13 c^2 d-30 e\right ) x \left (d+e x^2\right )^{3/2}}{840 c^3 e}-\frac {b x \left (d+e x^2\right )^{5/2}}{42 c e}-\frac {d \left (d+e x^2\right )^{5/2} (a+b \arctan (c x))}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} (a+b \arctan (c x))}{7 e^2}+\frac {b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{35 c^7 e^2}+\frac {b \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{560 c^7 e^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.50 \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=-\frac {c^2 \sqrt {d+e x^2} \left (48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \left (120 e^2-6 c^2 e \left (37 d+10 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )\right )+48 b c^7 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^{5/2} \arctan (c x)+24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (-\frac {140 i c^8 e^2 \left (c d-i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (i+c x)}\right )-24 i b \left (c^2 d-e\right )^{5/2} \left (2 c^2 d+5 e\right ) \log \left (\frac {140 i c^8 e^2 \left (c d+i e x+\sqrt {c^2 d-e} \sqrt {d+e x^2}\right )}{b \left (c^2 d-e\right )^{7/2} \left (2 c^2 d+5 e\right ) (-i+c x)}\right )-3 b \sqrt {e} \left (35 c^6 d^3+70 c^4 d^2 e-168 c^2 d e^2+80 e^3\right ) \log \left (e x+\sqrt {e} \sqrt {d+e x^2}\right )}{1680 c^7 e^2} \]
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\[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )d x\]
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Time = 8.76 (sec) , antiderivative size = 1566, normalized size of antiderivative = 5.61 \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\text {Too large to display} \]
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\[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \]
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Exception generated. \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int { {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arctan \left (c x\right ) + a\right )} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (d+e x^2\right )^{3/2} (a+b \arctan (c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2} \,d x \]
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