Optimal. Leaf size=835 \[ \frac{e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} (c+d x)^5}{5 d}+\frac{7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^5}{100 d}-\frac{7 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^4}{50 d}-\frac{21 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^4}{1000 d}-\frac{7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^3}{45 d}+\frac{14 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^2}{75 d}+\frac{119 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^2}{1125 d}+\frac{14 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)}{15 d}-\frac{28 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{105 b^{7/2} e^4 e^{a/b} \sqrt{\pi } \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{21 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{119 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{18000 d}+\frac{21 b^{7/2} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{105 b^{7/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{21 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{119 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{18000 d}+\frac{21 b^{7/2} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{1813 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.2885, antiderivative size = 835, normalized size of antiderivative = 1., number of steps used = 77, 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} \[ \frac{e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} (c+d x)^5}{5 d}+\frac{7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^5}{100 d}-\frac{7 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^4}{50 d}-\frac{21 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^4}{1000 d}-\frac{7 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)^3}{45 d}+\frac{14 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} (c+d x)^2}{75 d}+\frac{119 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^2}{1125 d}+\frac{14 b^2 e^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} (c+d x)}{15 d}-\frac{28 b e^4 \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{105 b^{7/2} e^4 e^{a/b} \sqrt{\pi } \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{21 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{119 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{18000 d}+\frac{21 b^{7/2} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{105 b^{7/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{21 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{119 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{18000 d}+\frac{21 b^{7/2} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{1813 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d} \]
Antiderivative was successfully verified.
[In]
[Out]
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)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 \operatorname{Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac{\left (7 b e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{10 d}\\ &=-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (14 b e^4\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 )}{25 d}+\frac{\left (7 b^2 e^4\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{20 d}\\ &=\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac{\left (28 b e^4\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 )}{75 d}-\frac{\left (7 b^2 e^4\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{15 d}-\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^5 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{200 d}\\ &=-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (14 b^2 e^4\right ) \operatorname{Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{15 d}+\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{250 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x^3 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{30 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{2000 d}\\ &=\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}-\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{125 d}-\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{45 d}-\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{5 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^4(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{500 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{180 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \left (\frac{\cosh (x)}{8 \sqrt{a+b x}}-\frac{3 \cosh (3 x)}{16 \sqrt{a+b x}}+\frac{\cosh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{500 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{250 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{180 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{90 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{10 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (7 b^3 e^4\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 )}{250 d}+\frac{\left (7 b^3 e^4\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 )}{90 d}+\frac{\left (7 b^3 e^4\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 )}{10 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16000 d}-\frac{\left (63 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}-\frac{\left (7 b^4 e^4\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 )}{500 d}-\frac{\left (7 b^4 e^4\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 )}{180 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (7 b^3 e^4\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 )}{500 d}+\frac{\left (7 b^3 e^4\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 )}{500 d}+\frac{\left (7 b^3 e^4\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 )}{180 d}+\frac{\left (7 b^3 e^4\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 )}{180 d}+\frac{\left (7 b^3 e^4\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 )}{20 d}+\frac{\left (7 b^3 e^4\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 )}{20 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}+\frac{\left (21 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32000 d}-\frac{\left (63 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}-\frac{\left (63 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64000 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2000 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{720 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{720 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}+\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32000 d}+\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16000 d}+\frac{\left (21 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16000 d}-\frac{\left (63 b^3 e^4\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 )}{32000 d}-\frac{\left (63 b^3 e^4\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 )}{32000 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{250 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{250 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{90 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{90 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{10 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{10 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4000 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}+\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}-\frac{\left (7 b^4 e^4\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1440 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{116221 b^{7/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{288000 d}-\frac{21 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{21 b^{7/2} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{116221 b^{7/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{288000 d}-\frac{21 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{21 b^{7/2} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}-\frac{\left (7 b^3 e^4\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 )}{2000 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2000 d}-\frac{\left (7 b^3 e^4\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 )}{2000 d}-\frac{\left (7 b^3 e^4\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 )}{720 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{720 d}+\frac{\left (7 b^3 e^4\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{720 d}-\frac{\left (7 b^3 e^4\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 )}{720 d}\\ &=-\frac{1813 b^3 e^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}+\frac{119 b^3 e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}-\frac{21 b^3 e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1000 d}+\frac{14 b^2 e^4 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac{7 b^2 e^4 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{45 d}+\frac{7 b^2 e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{100 d}-\frac{28 b e^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}+\frac{14 b e^4 (c+d x)^2 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{75 d}-\frac{7 b e^4 (c+d x)^4 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{50 d}+\frac{e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{5 d}+\frac{105 b^{7/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{119 b^{7/2} e^4 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 )}{18000 d}-\frac{21 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{21 b^{7/2} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{105 b^{7/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{256 d}-\frac{119 b^{7/2} e^4 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 )}{18000 d}-\frac{21 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}+\frac{21 b^{7/2} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{64000 d}\\ \end{align*}
Mathematica [A] time = 0.501047, size = 343, normalized size = 0.41 \[ \frac{b e^4 e^{-\frac{5 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (506250 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{9}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+81 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-3125 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+506250 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )-3125 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{9}{2},\frac{3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+81 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \text{Gamma}\left (\frac{9}{2},\frac{5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{8100000 d \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.329, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{4} \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 )}^{4}{\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 )}^{4}{\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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