Integrand size = 23, antiderivative size = 163 \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {2 b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {272, 45, 5346, 12, 587, 157, 95, 210} \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {2 b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}}+\frac {b c x \sqrt {c^2 x^2-1}}{3 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rule 12
Rule 45
Rule 95
Rule 157
Rule 210
Rule 272
Rule 587
Rule 5346
Rubi steps \begin{align*} \text {integral}& = \frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d-3 e x^2}{3 e^2 x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}} \\ & = \frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 d-3 e x^2}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2 \sqrt {c^2 x^2}} \\ & = \frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \text {Subst}\left (\int \frac {-2 d-3 e x}{x \sqrt {-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {d \left (c^2 d+e\right )}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = \frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2 \sqrt {c^2 x^2}} \\ & = \frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(2 b c x) \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^2 \sqrt {c^2 x^2}} \\ & = \frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \sec ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {2 b c x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )}{c^2 d+e}+a \left (2 d+3 e x^2\right )+\frac {b \sqrt {1+\frac {d}{e x^2}} \left (d+e x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )}{c x}+b \left (2 d+3 e x^2\right ) \sec ^{-1}(c x)}{3 e^2 \left (d+e x^2\right )^{3/2}} \]
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\[\int \frac {x^{3} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (137) = 274\).
Time = 0.39 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.07 \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\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} \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 \, \sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {-d} + 8 \, d^{2}}{x^{4}}\right ) + 2 \, {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\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 {\sqrt {c^{2} x^{2} - 1} {\left ({\left (c^{2} d - e\right )} x^{2} - 2 \, d\right )} \sqrt {e x^{2} + d} \sqrt {d}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) + {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (b d e^{2} x^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
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