Integrand size = 23, antiderivative size = 447 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac {8 b d^{7/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3} \]
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Time = 0.97 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 6436, 12, 1629, 159, 163, 65, 223, 209, 95, 213} \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{1680 c^7 e^{5/2}}-\frac {8 b d^{7/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{105 e^3}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (29 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^4 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {d+e x^2}}{1680 c^6 e^2} \]
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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 1629
Rule 6436
Rubi steps \begin{align*} \text {integral}& = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{3/2} \left (8 d^2-12 d e x^2+15 e^2 x^4\right )}{105 e^3 x \sqrt {1-c^2 x^2}} \, dx \\ & = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\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 (8 d^2-12 d e x^2+15 e^2 x^4\right )}{x \sqrt {1-c^2 x^2}} \, dx}{105 e^3} \\ & = \frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{3/2} \left (8 d^2-12 d e x+15 e^2 x^2\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{210 e^3} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{3/2} \left (-24 c^2 d^2 e+\frac {3}{2} \left (29 c^2 d-25 e\right ) e^2 x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{630 c^2 e^4} \\ & = \frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (48 c^4 d^3 e-\frac {3}{4} e^2 \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{1260 c^4 e^4} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {-48 c^6 d^4 e-\frac {3}{8} e^2 \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{1260 c^6 e^4} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (4 b d^4 \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 )}{105 e^3}+\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\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 )}{3360 c^6 e^2} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}+\frac {\left (8 b d^4 \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 )}{105 e^3}-\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\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 )}{1680 c^8 e^2} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {8 b d^{7/2} \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 )}{105 e^3}-\frac {\left (b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\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 )}{1680 c^8 e^2} \\ & = \frac {b \left (23 c^4 d^2+12 c^2 d e-75 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{1680 c^6 e^2}+\frac {b \left (29 c^2 d-25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^4 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 )^{5/2}}{42 c^2 e^2}+\frac {d^2 \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {2 d \left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e^3}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {sech}^{-1}(c x)\right )}{7 e^3}-\frac {b \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\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 )}{1680 c^7 e^{5/2}}-\frac {8 b d^{7/2} \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 )}{105 e^3} \\ \end{align*}
Time = 37.75 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.76 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {\sqrt {d+e x^2} \left (16 a c^6 \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right )-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (75 e^2+2 c^2 e \left (19 d+25 e x^2\right )+c^4 \left (-41 d^2+22 d e x^2+40 e^2 x^4\right )\right )+16 b c^6 \left (8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right ) \text {sech}^{-1}(c x)\right )}{1680 c^6 e^3}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {-1+c^2 x^2} \left (128 c^7 d^{7/2} \arctan \left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{1680 c^7 e^3 (-1+c x)} \]
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\[\int x^{5} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]
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Time = 1.45 (sec) , antiderivative size = 1995, normalized size of antiderivative = 4.46 \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]
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\[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^{5} \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]
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Exception generated. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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\[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5} \,d x } \]
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Timed out. \[ \int x^5 \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int x^5\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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