Optimal. Leaf size=381 \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}-\frac{3 b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{3 b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (c^2 x^2+1\right )}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (c^2 x^2+1\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^3}+\frac{b^2 c^2}{12 d^3 \left (c^2 x^2+1\right )}-\frac{7 b^2 c^2 \log \left (c^2 x^2+1\right )}{6 d^3}+\frac{b^2 c^2 \log (x)}{d^3} \]
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Rubi [A] time = 0.802988, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 19, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.731, Rules used = {5747, 5755, 5720, 5461, 4182, 2531, 2282, 6589, 5687, 260, 5690, 261, 271, 192, 191, 5732, 12, 1251, 893} \[ \frac{3 b c^2 \text{PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}-\frac{3 b c^2 \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^3}-\frac{3 b^2 c^2 \text{PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{3 b^2 c^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (c^2 x^2+1\right )}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (c^2 x^2+1\right )^{3/2}}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac{6 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d^3}+\frac{b^2 c^2}{12 d^3 \left (c^2 x^2+1\right )}-\frac{7 b^2 c^2 \log \left (c^2 x^2+1\right )}{6 d^3}+\frac{b^2 c^2 \log (x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 5747
Rule 5755
Rule 5720
Rule 5461
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 5687
Rule 260
Rule 5690
Rule 261
Rule 271
Rule 192
Rule 191
Rule 5732
Rule 12
Rule 1251
Rule 893
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx &=-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\left (3 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^3} \, dx+\frac{(b c) \int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )^{5/2}} \, dx}{d^3}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{8 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{\left (b^2 c^2\right ) \int \frac{-3-12 c^2 x^2-8 c^4 x^4}{3 x \left (1+c^2 x^2\right )^2} \, dx}{d^3}+\frac{\left (3 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3}-\frac{\left (3 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac{8 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac{\left (b^2 c^2\right ) \int \frac{-3-12 c^2 x^2-8 c^4 x^4}{x \left (1+c^2 x^2\right )^2} \, dx}{3 d^3}+\frac{\left (b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^3}+\frac{\left (3 b c^3\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^3}-\frac{\left (b^2 c^4\right ) \int \frac{x}{\left (1+c^2 x^2\right )^2} \, dx}{2 d^3}-\frac{\left (3 c^2\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac{b^2 c^2}{4 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{-3-12 c^2 x-8 c^4 x^2}{x \left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^3}-\frac{\left (b^2 c^4\right ) \int \frac{x}{1+c^2 x^2} \, dx}{d^3}-\frac{\left (3 b^2 c^4\right ) \int \frac{x}{1+c^2 x^2} \, dx}{d^3}\\ &=\frac{b^2 c^2}{4 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}-\frac{2 b^2 c^2 \log \left (1+c^2 x^2\right )}{d^3}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}-\frac{\left (b^2 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{3}{x}-\frac{c^2}{\left (1+c^2 x\right )^2}-\frac{5 c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^3}\\ &=\frac{b^2 c^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{b^2 c^2 \log (x)}{d^3}-\frac{7 b^2 c^2 \log \left (1+c^2 x^2\right )}{6 d^3}+\frac{\left (6 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}-\frac{\left (6 b c^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b^2 c^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{b^2 c^2 \log (x)}{d^3}-\frac{7 b^2 c^2 \log \left (1+c^2 x^2\right )}{6 d^3}+\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{\left (3 b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}+\frac{\left (3 b^2 c^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^3}\\ &=\frac{b^2 c^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{b^2 c^2 \log (x)}{d^3}-\frac{7 b^2 c^2 \log \left (1+c^2 x^2\right )}{6 d^3}+\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{\left (3 b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{\left (3 b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac{b^2 c^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac{b c \left (a+b \sinh ^{-1}(c x)\right )}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac{5 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 b c^3 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^3 \sqrt{1+c^2 x^2}}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac{3 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac{6 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}+\frac{b^2 c^2 \log (x)}{d^3}-\frac{7 b^2 c^2 \log \left (1+c^2 x^2\right )}{6 d^3}+\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{3 b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^3}-\frac{3 b^2 c^2 \text{Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}+\frac{3 b^2 c^2 \text{Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d^3}\\ \end{align*}
Mathematica [C] time = 9.57152, size = 759, normalized size = 1.99 \[ \frac{2 a b \left (\frac{3}{2} c^3 \left (\frac{2 \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )}{c}\right )+\frac{3}{2} c^3 \left (\frac{2 \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )}{c}\right )-3 c^2 \left (\frac{1}{2} \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )-\frac{1}{2} \sinh ^{-1}(c x)^2+\sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )\right )-\frac{9 i c^3 \left (\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right )}{16 \left (c^2 x+i c\right )}-\frac{c^2 \left (-3 \sinh ^{-1}(c x)+(-c x+2 i) \sqrt{c^2 x^2+1}\right )}{48 (c x-i)^2}-\frac{9 i c^2 \left (\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right )}{16 (-1-i c x)}+\frac{c^2 \left (3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right )}{48 (c x+i)^2}-\frac{c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)}{2 x^2}\right )}{d^3}+\frac{b^2 c^2 \left (\frac{1}{24} \left (-36 \text{PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )+36 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )+\frac{2}{c^2 x^2+1}-56 \log \left (\sqrt{c^2 x^2+1}\right )-\frac{24 \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{12 \sinh ^{-1}(c x)^2}{c^2 x^2}-\frac{6 \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^2}-\frac{24 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c x}+\frac{56 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 c x \sinh ^{-1}(c x)}{\left (c^2 x^2+1\right )^{3/2}}+24 \log (c x)+48 \sinh ^{-1}(c x)^3+72 \sinh ^{-1}(c x)^2 \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )-72 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )-3 i \pi ^3\right )-3 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )-3 \sinh ^{-1}(c x) \text{PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )\right )}{d^3}-\frac{a^2 c^2}{d^3 \left (c^2 x^2+1\right )}-\frac{a^2 c^2}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac{3 a^2 c^2 \log \left (c^2 x^2+1\right )}{2 d^3}-\frac{3 a^2 c^2 \log (x)}{d^3}-\frac{a^2}{2 d^3 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.289, size = 1436, normalized size = 3.8 \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}{4} \, a^{2}{\left (\frac{6 \, c^{4} x^{4} + 9 \, c^{2} x^{2} + 2}{c^{4} d^{3} x^{6} + 2 \, c^{2} d^{3} x^{4} + d^{3} x^{2}} - \frac{6 \, c^{2} \log \left (c^{2} x^{2} + 1\right )}{d^{3}} + \frac{12 \, c^{2} \log \left (x\right )}{d^{3}}\right )} + \int \frac{b^{2} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} x^{3}} + \frac{2 \, a b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}\,{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{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{c^{6} d^{3} x^{9} + 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} + d^{3} x^{3}}, x\right ) \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 \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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