Integrand size = 23, antiderivative size = 278 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {b \left (9 c^2 d-e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{5/2}}+\frac {8 b d^{3/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^3} \]
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Time = 0.76 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {272, 45, 6436, 12, 1629, 163, 65, 223, 209, 95, 213} \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (9 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{5/2}}+\frac {8 b d^{3/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^3}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2} \]
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Rule 12
Rule 45
Rule 65
Rule 95
Rule 163
Rule 209
Rule 213
Rule 223
Rule 272
Rule 1629
Rule 6436
Rubi steps \begin{align*} \text {integral}& = -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx \\ & = -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 e^3} \\ & = -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {-8 d^2-4 d e x+e^2 x^2}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e^3} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {8 c^2 d^2 e+\frac {1}{2} \left (9 c^2 d-e\right ) e^2 x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c^2 e^4} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {\left (4 b d^2 \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 )}{3 e^3}-\frac {\left (b \left (9 c^2 d-e\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 )}{12 c^2 e^2} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}-\frac {\left (8 b d^2 \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 )}{3 e^3}+\frac {\left (b \left (9 c^2 d-e\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 )}{6 c^4 e^2} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {8 b d^{3/2} \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 )}{3 e^3}+\frac {\left (b \left (9 c^2 d-e\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 )}{6 c^4 e^2} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{e^3 \sqrt {d+e x^2}}-\frac {2 d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^3}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^3}+\frac {b \left (9 c^2 d-e\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 )}{6 c^3 e^{5/2}}+\frac {8 b d^{3/2} \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 )}{3 e^3} \\ \end{align*}
Time = 23.34 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.57 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right )-2 a c^2 \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c^2 \left (8 d^2+4 d e x^2-e^2 x^4\right ) \text {sech}^{-1}(c x)}{6 c^2 e^3 \sqrt {d+e x^2}}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (-9 \left (-c^2\right )^{3/2} d \sqrt {-c^2 d-e} \sqrt {e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+\sqrt {-c^2} \sqrt {-c^2 d-e} e^{3/2} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \arcsin \left (\frac {\sqrt {-c^2} \sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {-c^2 d-e}}\right )+16 c^5 d^{3/2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{6 c^5 e^3 (-1+c x) \sqrt {d+e x^2}} \]
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\[\int \frac {x^{5} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (184) = 368\).
Time = 0.44 (sec) , antiderivative size = 1771, normalized size of antiderivative = 6.37 \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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