Optimal. Leaf size=132 \[ \frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{e \sqrt{c^2 x^2}}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.145577, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {5236, 446, 105, 63, 217, 206, 93, 204} \[ \frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{e \sqrt{c^2 x^2}}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5236
Rule 446
Rule 105
Rule 63
Rule 217
Rule 206
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{x \left (a+b \sec ^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{(b c x) \int \frac{\sqrt{d+e x^2}}{x \sqrt{-1+c^2 x^2}} \, dx}{e \sqrt{c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{2 e \sqrt{c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 \sqrt{c^2 x^2}}-\frac{(b c d x) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e \sqrt{c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{(b x) \operatorname{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 )}{c \sqrt{c^2 x^2}}-\frac{(b c d 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 )}{e \sqrt{c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{e \sqrt{c^2 x^2}}-\frac{(b x) \operatorname{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 )}{c \sqrt{c^2 x^2}}\\ &=\frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}+\frac{b c \sqrt{d} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{e \sqrt{c^2 x^2}}-\frac{b x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{\sqrt{e} \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.219815, size = 211, normalized size = 1.6 \[ \frac{\sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )\right )}{c^2 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.591, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\rm arcsec} \left (cx\right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, 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 [A] time = 2.59961, size = 1925, normalized size = 14.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \operatorname{asec}{\left (c x \right )}\right )}{\sqrt{d + e x^{2}}}\, dx \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}{\sqrt{e x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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