Optimal. Leaf size=221 \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 \sqrt{e}} \]
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Rubi [A] time = 0.357428, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {6299, 517, 446, 102, 157, 63, 217, 203, 93, 207} \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (3 c^2 d+e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 6299
Rule 517
Rule 446
Rule 102
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 207
Rubi steps
\begin{align*} \int x \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2}}{x \sqrt{1-c x} \sqrt{1+c x}} \, dx}{3 e}\\ &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2}}{x \sqrt{1-c^2 x^2}} \, dx}{3 e}\\ &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x \sqrt{1-c^2 x}} \, 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} \sqrt{d+e x^2}}{6 c^2}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{-c^2 d^2-\frac{1}{2} e \left (3 c^2 d+e\right ) x}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 c^2 e}\\ &=-\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}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 e}+\frac{\left (b \left (3 c^2 d+e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{12 c^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}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}+\frac{\left (b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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}-\frac{\left (b \left (3 c^2 d+e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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 )}{6 c^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}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{b d^{3/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e}-\frac{\left (b \left (3 c^2 d+e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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 )}{6 c^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}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e}-\frac{b \left (3 c^2 d+e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 \sqrt{e}}-\frac{b d^{3/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 1.28663, size = 307, normalized size = 1.39 \[ \frac{\sqrt{d+e x^2} \left (2 a c^2 \left (d+e x^2\right )+2 b c^2 \text{sech}^{-1}(c x) \left (d+e x^2\right )-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right )}{6 c^2 e}+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left (2 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )+\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left (3 c^2 d+e\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right )\right )}{6 c^5 e (c x-1) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.395, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )}{e} - 3 \, \int \frac{\sqrt{e x^{2} + d}{\left (6 \,{\left (c^{2} e x^{2} - e\right )} x \log \left (\sqrt{x}\right ) + 3 \,{\left (c^{2} e x^{2} \log \left (c\right ) - e \log \left (c\right )\right )} x +{\left (6 \,{\left (c^{2} e x^{2} - e\right )} x \log \left (\sqrt{x}\right ) +{\left ({\left (3 \, e \log \left (c\right ) + e\right )} c^{2} x^{2} + c^{2} d - 3 \, e \log \left (c\right )\right )} x\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}\right )}}{3 \,{\left (c^{2} e x^{2} +{\left (c^{2} e x^{2} - e\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} - e\right )}}\,{d x}\right )} b + \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} a}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.03671, size = 3011, normalized size = 13.62 \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 x \left (a + b \operatorname{asech}{\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 \sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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