Integrand size = 23, antiderivative size = 329 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac {2 b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^2} \]
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Time = 0.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 6436, 12, 587, 159, 163, 65, 223, 209, 95, 213} \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac {2 b d^{5/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt {d+e x^2}}{120 c^4 e} \]
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 159
Rule 163
Rule 209
Rule 213
Rule 223
Rule 272
Rule 587
Rule 6436
Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{x \sqrt {1-c^2 x^2}} \, dx}{15 e^2} \\ & = -\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{30 e^2} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (4 c^2 d^2-\frac {1}{2} e \left (c^2 d+9 e\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{60 c^2 e^2} \\ & = -\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {-4 c^4 d^3-\frac {1}{4} e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{60 c^4 e^2} \\ & = -\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{15 e^2}-\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{240 c^4 e} \\ & = -\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}-\frac {\left (2 b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{15 e^2}+\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d+\frac {e}{c^2}-\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{120 c^6 e} \\ & = -\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {2 b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^2}+\frac {\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1-c^2 x^2}}{\sqrt {d+e x^2}}\right )}{120 c^6 e} \\ & = -\frac {b \left (c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{120 c^4 e}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^2}+\frac {b \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac {2 b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{15 e^2} \\ \end{align*}
Time = 23.45 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.11 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=-\frac {\sqrt {d+e x^2} \left (8 a c^4 \left (2 d^2-d e x^2-3 e^2 x^4\right )+b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (9 e+c^2 \left (7 d+6 e x^2\right )\right )+8 b c^4 \left (2 d^2-d e x^2-3 e^2 x^4\right ) \text {sech}^{-1}(c x)\right )}{120 c^4 e^2}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+16 c^7 d^{5/2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{120 c^7 e^2 (-1+c x) \sqrt {d+e x^2}} \]
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\[\int x^{3} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
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Time = 0.74 (sec) , antiderivative size = 1669, normalized size of antiderivative = 5.07 \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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\[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
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Exception generated. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^3\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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