Integrand size = 21, antiderivative size = 154 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 d^{3/2} e} \]
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Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6434, 531, 457, 98, 95, 213} \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 d^{3/2} e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rule 95
Rule 98
Rule 213
Rule 457
Rule 531
Rule 6434
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \left (d+e x^2\right )^{3/2}} \, dx}{3 e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e} \\ & = -\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {\left (b \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 )}{6 d e} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}-\frac {\left (b \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 d e} \\ & = -\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{3 e \left (d+e x^2\right )^{3/2}}+\frac {b \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 d^{3/2} e} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.32 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {-a d \left (c^2 d+e\right )-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right )-b d \left (c^2 d+e\right ) \text {sech}^{-1}(c x)}{3 d e \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \sqrt {-d-e x^2} \arctan \left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )}{3 d^{3/2} e (-1+c x) \sqrt {d+e x^2}} \]
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\[\int \frac {x \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (94) = 188\).
Time = 0.34 (sec) , antiderivative size = 692, normalized size of antiderivative = 4.49 \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {4 \, {\left (b c^{2} d^{3} + b d^{2} e\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} - 4 \, {\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 4 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{12 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c^{2} d^{3} + {\left (b c^{2} d e^{2} + b e^{3}\right )} x^{4} + b d^{2} e + 2 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (-\frac {{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - 2 \, {\left (b c^{2} d^{3} + b d^{2} e\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - 2 \, {\left (a c^{2} d^{3} + a d^{2} e + {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{5} e + d^{4} e^{2} + {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{4} + 2 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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