Optimal. Leaf size=481 \[ -\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}-\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}+\frac{175 b^3 e^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{7 b e^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d} \]
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Rubi [A] time = 1.67977, antiderivative size = 481, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {5865, 12, 5663, 5758, 5717, 5653, 5657, 3307, 2180, 2205, 2204, 5669, 5448} \[ -\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}-\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}+\frac{175 b^3 e^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{7 b e^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5663
Rule 5758
Rule 5717
Rule 5653
Rule 5657
Rule 3307
Rule 2180
Rule 2205
Rule 2204
Rule 5669
Rule 5448
Rubi steps
\begin{align*} \int (c e+d e x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^2 x^2 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (7 b e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac{\left (7 b e^2\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{9 d}+\frac{\left (35 b^2 e^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{36 d}\\ &=\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^2 e^2\right ) \operatorname{Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{18 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{108 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{12 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{432 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{432 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{216 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{24 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{24 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}+\frac{\cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{432 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{432 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{432 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{48 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{48 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1728 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1728 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{216 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{24 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3456 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3456 d}-\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3456 d}+\frac{\left (35 b^4 e^2\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3456 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{175 b^{7/2} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{216 d}-\frac{175 b^{7/2} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{216 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{1728 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{1728 d}-\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{1728 d}+\frac{\left (35 b^3 e^2\right ) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{1728 d}\\ &=\frac{175 b^3 e^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}-\frac{35 b^2 e^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac{35 b^2 e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac{7 b e^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac{105 b^{7/2} e^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 b^{7/2} e^2 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}-\frac{105 b^{7/2} e^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{128 d}+\frac{35 b^{7/2} e^2 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3456 d}\\ \end{align*}
Mathematica [A] time = 0.328138, size = 238, normalized size = 0.49 \[ \frac{b e^2 e^{-\frac{3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (-243 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{9}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-243 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{9}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{1944 d \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{2} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{7}{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 (d e x + c e\right )}^{2}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{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 (d e x + c e\right )}^{2}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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