Optimal. Leaf size=337 \[ -b d^3 \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}(c x)}\right )-\frac{1}{18} b c d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{24} b c d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2+\frac{1}{108} b^2 d^3 \left (c^2 x^2+1\right )^3 \]
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Rubi [A] time = 0.68407, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 13, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5744, 5659, 3716, 2190, 2531, 2282, 6589, 5682, 5675, 30, 5684, 14, 261} \[ b d^3 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{2} b^2 d^3 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{18} b c d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{24} b c d^3 x \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d^3 \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2+\frac{1}{108} b^2 d^3 \left (c^2 x^2+1\right )^3 \]
Warning: Unable to verify antiderivative.
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Rule 5744
Rule 5659
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 5682
Rule 5675
Rule 30
Rule 5684
Rule 14
Rule 261
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx &=\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d \int \frac{\left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{3} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^2 \int \frac{\left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{18} \left (5 b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{1}{2} \left (b c d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{18} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right )^2 \, dx\\ &=\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x} \, dx-\frac{1}{24} \left (5 b c d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac{1}{8} \left (3 b c d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac{1}{72} \left (5 b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx+\frac{1}{8} \left (b^2 c^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx\\ &=\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+d^3 \operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )-\frac{1}{48} \left (5 b c d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{16} \left (3 b c d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{2} \left (b c d^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx+\frac{1}{72} \left (5 b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{48} \left (5 b^2 c^2 d^3\right ) \int x \, dx+\frac{1}{8} \left (b^2 c^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac{1}{16} \left (3 b^2 c^2 d^3\right ) \int x \, dx+\frac{1}{2} \left (b^2 c^2 d^3\right ) \int x \, dx\\ &=\frac{71}{144} b^2 c^2 d^3 x^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}-\left (2 d^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^2}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-\left (2 b d^3\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\left (b^2 d^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \left (b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )\\ &=\frac{71}{144} b^2 c^2 d^3 x^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{1}{108} b^2 d^3 \left (1+c^2 x^2\right )^3-\frac{19}{24} b c d^3 x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{7}{36} b c d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{1}{18} b c d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{19}{48} d^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{2} d^3 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{4} d^3 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{1}{6} d^3 \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+b d^3 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} b^2 d^3 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )\\ \end{align*}
Mathematica [A] time = 0.658456, size = 416, normalized size = 1.23 \[ \frac{d^3 \left (-3456 a b \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+3456 b^2 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-1728 b^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+576 a^2 c^6 x^6+2592 a^2 c^4 x^4+5184 a^2 c^2 x^2+3456 a^2 \log (c x)-192 a b c^5 x^5 \sqrt{c^2 x^2+1}-1056 a b c^3 x^3 \sqrt{c^2 x^2+1}-3600 a b c x \sqrt{c^2 x^2+1}+1152 a b c^6 x^6 \sinh ^{-1}(c x)+5184 a b c^4 x^4 \sinh ^{-1}(c x)+10368 a b c^2 x^2 \sinh ^{-1}(c x)+3456 a b \sinh ^{-1}(c x)^2+3600 a b \sinh ^{-1}(c x)+6912 a b \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-1152 b^2 \sinh ^{-1}(c x)^3-1566 b^2 \sinh ^{-1}(c x) \sinh \left (2 \sinh ^{-1}(c x)\right )-108 b^2 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )-6 b^2 \sinh ^{-1}(c x) \sinh \left (6 \sinh ^{-1}(c x)\right )+3456 b^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+783 b^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+1566 b^2 \sinh ^{-1}(c x)^2 \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+216 b^2 \sinh ^{-1}(c x)^2 \cosh \left (4 \sinh ^{-1}(c x)\right )+b^2 \cosh \left (6 \sinh ^{-1}(c x)\right )+18 b^2 \sinh ^{-1}(c x)^2 \cosh \left (6 \sinh ^{-1}(c x)\right )\right )}{3456} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.199, size = 706, normalized size = 2.1 \begin{align*} \text{result too large to display} \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}{6} \, a^{2} c^{6} d^{3} x^{6} + \frac{3}{4} \, a^{2} c^{4} d^{3} x^{4} + \frac{3}{2} \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} \log \left (x\right ) + \int b^{2} c^{6} d^{3} x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \, a b c^{6} d^{3} x^{5} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 3 \, b^{2} c^{4} d^{3} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \, a b c^{4} d^{3} x^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 3 \, b^{2} c^{2} d^{3} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 6 \, a b c^{2} d^{3} x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + \frac{b^{2} d^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{x} + \frac{2 \, a b d^{3} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{2} c^{6} d^{3} x^{6} + 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} +{\left (b^{2} c^{6} d^{3} x^{6} + 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} + b^{2} d^{3}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b c^{6} d^{3} x^{6} + 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} + a b d^{3}\right )} \operatorname{arsinh}\left (c x\right )}{x}, x\right ) \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*} d^{3} \left (\int \frac{a^{2}}{x}\, dx + \int 3 a^{2} c^{2} x\, dx + \int 3 a^{2} c^{4} x^{3}\, dx + \int a^{2} c^{6} x^{5}\, dx + \int \frac{b^{2} \operatorname{asinh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac{2 a b \operatorname{asinh}{\left (c x \right )}}{x}\, dx + \int 3 b^{2} c^{2} x \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int 3 b^{2} c^{4} x^{3} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{6} x^{5} \operatorname{asinh}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname{asinh}{\left (c x \right )}\, dx + \int 6 a b c^{4} x^{3} \operatorname{asinh}{\left (c x \right )}\, dx + \int 2 a b c^{6} x^{5} \operatorname{asinh}{\left (c x \right )}\, dx\right ) \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 (c^{2} d x^{2} + d\right )}^{3}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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