Optimal. Leaf size=514 \[ \frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 c \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{a^2 x^2+1}}-\frac{45 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{a^2 x^2+1}}+\frac{225}{512} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.764722, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {5684, 5682, 5675, 5663, 5758, 5669, 5448, 12, 3308, 2180, 2204, 2205, 5717, 5779} \[ \frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{Erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 c \left (a^2 x^2+1\right )^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{a^2 x^2+1}}-\frac{45 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{a^2 x^2+1}}+\frac{225}{512} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5684
Rule 5682
Rule 5675
Rule 5663
Rule 5758
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5717
Rule 5779
Rubi steps
\begin{align*} \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2} \, dx &=\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} (3 c) \int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2} \, dx-\frac{\left (5 a c \sqrt{c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^{3/2} \, dx}{8 \sqrt{1+a^2 x^2}}\\ &=-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \sqrt{\sinh ^{-1}(a x)} \, dx}{64 \sqrt{1+a^2 x^2}}+\frac{\left (3 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\sinh ^{-1}(a x)^{5/2}}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{1+a^2 x^2}}-\frac{\left (15 a c \sqrt{c+a^2 c x^2}\right ) \int x \sinh ^{-1}(a x)^{3/2} \, dx}{16 \sqrt{1+a^2 x^2}}\\ &=\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \int \sqrt{1+a^2 x^2} \sqrt{\sinh ^{-1}(a x)} \, dx}{256 \sqrt{1+a^2 x^2}}-\frac{\left (15 a c \sqrt{c+a^2 c x^2}\right ) \int \frac{x \left (1+a^2 x^2\right )}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{512 \sqrt{1+a^2 x^2}}+\frac{\left (45 a^2 c \sqrt{c+a^2 c x^2}\right ) \int \frac{x^2 \sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{64 \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{512 \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{128 \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a \sqrt{1+a^2 x^2}}-\frac{\left (45 a c \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{1024 \sqrt{1+a^2 x^2}}-\frac{\left (45 a c \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{256 \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{512 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8192 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8192 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{2048 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{512 a \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{4096 a \sqrt{1+a^2 x^2}}+\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}-\frac{\left (15 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4096 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{1024 a \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{15 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{1+a^2 x^2}}+\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{4096 a \sqrt{1+a^2 x^2}}-\frac{15 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{1+a^2 x^2}}-\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{4096 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{2048 a \sqrt{1+a^2 x^2}}+\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{512 a \sqrt{1+a^2 x^2}}-\frac{\left (45 c \sqrt{c+a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{512 a \sqrt{1+a^2 x^2}}\\ &=\frac{225}{512} c x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left (1+a^2 x^2\right ) \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}-\frac{45 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{1+a^2 x^2}}-\frac{15 a c x^2 \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{1+a^2 x^2}}-\frac{5 c \left (1+a^2 x^2\right )^{3/2} \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{32 a}+\frac{3}{8} c x \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac{1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3 c \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{1+a^2 x^2}}+\frac{15 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erf}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{1+a^2 x^2}}+\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}-\frac{15 c \sqrt{\pi } \sqrt{c+a^2 c x^2} \text{erfi}\left (2 \sqrt{\sinh ^{-1}(a x)}\right )}{16384 a \sqrt{1+a^2 x^2}}-\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{256 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.344886, size = 201, normalized size = 0.39 \[ \frac{c \sqrt{a^2 c x^2+c} \left (-7 \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},4 \sinh ^{-1}(a x)\right )-7 \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 \sinh ^{-1}(a x)\right )+420 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )-420 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )+1536 \sinh ^{-1}(a x)^4+3584 \sinh \left (2 \sinh ^{-1}(a x)\right ) \sinh ^{-1}(a x)^3+3360 \sinh \left (2 \sinh ^{-1}(a x)\right ) \sinh ^{-1}(a x)-4480 \sinh ^{-1}(a x)^2 \cosh \left (2 \sinh ^{-1}(a x)\right )\right )}{14336 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.164, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{{\frac{5}{2}}}\, dx \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*} \int{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arsinh}\left (a x\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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