Optimal. Leaf size=195 \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}-\frac{b x \left (3 c^2 d+e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{c^2 x^2}} \]
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Rubi [A] time = 0.190292, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5236, 446, 102, 157, 63, 217, 206, 93, 204} \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{3 e \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}-\frac{b x \left (3 c^2 d+e\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5236
Rule 446
Rule 102
Rule 157
Rule 63
Rule 217
Rule 206
Rule 93
Rule 204
Rubi steps
\begin{align*} \int x \sqrt{d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2}}{x \sqrt{-1+c^2 x^2}} \, dx}{3 e \sqrt{c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x \sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{6 e \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{(b x) \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 e \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (b c d^2 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 \sqrt{c^2 x^2}}-\frac{\left (b \left (3 c^2 d+e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{12 c \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}-\frac{\left (b c d^2 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 \sqrt{c^2 x^2}}-\frac{\left (b \left (3 c^2 d+e\right ) 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^3 \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e \sqrt{c^2 x^2}}-\frac{\left (b \left (3 c^2 d+e\right ) 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^3 \sqrt{c^2 x^2}}\\ &=-\frac{b x \sqrt{-1+c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \sec ^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1+c^2 x^2}}\right )}{3 e \sqrt{c^2 x^2}}-\frac{b \left (3 c^2 d+e\right ) x \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.456565, size = 266, normalized size = 1.36 \[ \frac{\sqrt{d+e x^2} \left (2 a c \left (d+e x^2\right )-b e x \sqrt{1-\frac{1}{c^2 x^2}}+2 b c \sec ^{-1}(c x) \left (d+e x^2\right )\right )}{6 c e}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left (2 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right )+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left (3 c^2 d+e\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right )\right )}{6 c^4 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.386, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arcsec} \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(-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 = 6.69076, size = 2425, normalized size = 12.44 \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(-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 \sqrt{e x^{2} + d}{\left (b \operatorname{arcsec}\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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