Optimal. Leaf size=163 \[ -\frac{a+b \sec ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{2 b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 \sqrt{d} e^2 \sqrt{c^2 x^2}}+\frac{b c x \sqrt{c^2 x^2-1}}{3 e \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.242974, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {266, 43, 5238, 12, 573, 152, 93, 204} \[ -\frac{a+b \sec ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left (a+b \sec ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac{2 b c x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 \sqrt{d} e^2 \sqrt{c^2 x^2}}+\frac{b c x \sqrt{c^2 x^2-1}}{3 e \sqrt{c^2 x^2} \left (c^2 d+e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 5238
Rule 12
Rule 573
Rule 152
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\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) \operatorname{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) \operatorname{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) \operatorname{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) \operatorname{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 \tan ^{-1}\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*}
Mathematica [A] time = 0.256064, size = 172, normalized size = 1.06 \[ \frac{-a \left (c^2 d+e\right ) \left (2 d+3 e x^2\right )+b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left (d+e x^2\right )-b \left (c^2 d+e\right ) \sec ^{-1}(c x) \left (2 d+3 e x^2\right )}{3 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}+\frac{2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )}{3 \sqrt{d} e^2 \sqrt{c^2 x^2-1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.631, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{\rm arcsec} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.34775, size = 1380, normalized size = 8.47 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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