Optimal. Leaf size=329 \[ -\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac{2 b d^{5/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{15 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt{d+e x^2}}{120 c^4 e} \]
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Rubi [A] time = 0.427168, antiderivative size = 329, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 573, 154, 157, 63, 217, 203, 93, 207} \[ -\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{120 c^5 e^{3/2}}+\frac{2 b d^{5/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{15 e^2}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (c^2 d+9 e\right ) \sqrt{d+e x^2}}{120 c^4 e} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6301
Rule 12
Rule 573
Rule 154
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 207
Rubi steps
\begin{align*} \int x^3 \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 x \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{x \sqrt{1-c^2 x^2}} \, dx}{15 e^2}\\ &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{30 e^2}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (4 c^2 d^2-\frac{1}{2} e \left (c^2 d+9 e\right ) x\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{60 c^2 e^2}\\ &=-\frac{b \left (c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{120 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{-4 c^4 d^3-\frac{1}{4} e \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) x}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{60 c^4 e^2}\\ &=-\frac{b \left (c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{120 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}-\frac{\left (b d^3 \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 )}{15 e^2}-\frac{\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{240 c^4 e}\\ &=-\frac{b \left (c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{120 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}-\frac{\left (2 b d^3 \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 )}{15 e^2}+\frac{\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{120 c^6 e}\\ &=-\frac{b \left (c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{120 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{2 b d^{5/2} \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 )}{15 e^2}+\frac{\left (b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{120 c^6 e}\\ &=-\frac{b \left (c^2 d+9 e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{120 c^4 e}-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2 e}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{sech}^{-1}(c x)\right )}{5 e^2}+\frac{b \left (15 c^4 d^2-10 c^2 d e-9 e^2\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 )}{120 c^5 e^{3/2}}+\frac{2 b d^{5/2} \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 )}{15 e^2}\\ \end{align*}
Mathematica [A] time = 1.45193, size = 365, normalized size = 1.11 \[ -\frac{\sqrt{d+e x^2} \left (8 a c^4 \left (2 d^2-d e x^2-3 e^2 x^4\right )+8 b c^4 \text{sech}^{-1}(c x) \left (2 d^2-d e x^2-3 e^2 x^4\right )+b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^2 \left (7 d+6 e x^2\right )+9 e\right )\right )}{120 c^4 e^2}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left (\sqrt{-c^2} \sqrt{e} \sqrt{c^2 (-d)-e} \left (15 c^4 d^2-10 c^2 d e-9 e^2\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right )+16 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )\right )}{120 c^7 e^2 (c x-1) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.626, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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(-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 = 7.63659, size = 3636, normalized size = 11.05 \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{arsech}\left (c x\right ) + a\right )} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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