Optimal. Leaf size=302 \[ -\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{2 b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{15 e^2 \sqrt{-c^2 x^2}}-\frac{b x \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (c^2 d-9 e\right ) \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}} \]
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Rubi [A] time = 0.433657, antiderivative size = 302, 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, 6302, 12, 573, 154, 157, 63, 217, 203, 93, 204} \[ -\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{2 b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{15 e^2 \sqrt{-c^2 x^2}}-\frac{b x \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left (c^2 d-9 e\right ) \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6302
Rule 12
Rule 573
Rule 154
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 204
Rubi steps
\begin{align*} \int x^3 \sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{(b c x) \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}{\sqrt{-c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{(b c x) \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 \sqrt{-c^2 x^2}}\\ &=-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{(b c x) \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 \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{(b x) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x} \left (4 c^2 d^2-\frac{1}{2} \left (c^2 d-9 e\right ) e x\right )}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{60 c e^2 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (c^2 d-9 e\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{(b x) \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^3 e^2 \sqrt{-c^2 x^2}}\\ &=\frac{b \left (c^2 d-9 e\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (b c d^3 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 \sqrt{-c^2 x^2}}+\frac{\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{240 c^3 e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (c^2 d-9 e\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}+\frac{\left (2 b c d^3 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 \sqrt{-c^2 x^2}}-\frac{\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (c^2 d-9 e\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{2 b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{15 e^2 \sqrt{-c^2 x^2}}-\frac{\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt{-c^2 x^2}}\\ &=\frac{b \left (c^2 d-9 e\right ) x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{d \left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2}+\frac{\left (d+e x^2\right )^{5/2} \left (a+b \text{csch}^{-1}(c x)\right )}{5 e^2}-\frac{b \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt{-c^2 x^2}}-\frac{2 b c d^{5/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{15 e^2 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.740532, size = 337, normalized size = 1.12 \[ \frac{\sqrt{d+e x^2} \left (8 a c^3 \left (-2 d^2+d e x^2+3 e^2 x^4\right )+8 b c^3 \text{csch}^{-1}(c x) \left (-2 d^2+d e x^2+3 e^2 x^4\right )+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 \left (7 d+6 e x^2\right )-9 e\right )\right )}{120 c^3 e^2}+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (16 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right )-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right )\right )}{120 c^6 e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.414, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b{\rm arccsch} \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 = 9.54428, size = 3614, normalized size = 11.97 \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{arcsch}\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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