Optimal. Leaf size=547 \[ -\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}+\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}-\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}-\frac{105 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{1575 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{7 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac{21 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.14344, antiderivative size = 547, normalized size of antiderivative = 1., number of steps used = 42, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {5865, 12, 5663, 5758, 5675, 5669, 5448, 3308, 2180, 2204, 2205} \[ -\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}+\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}-\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}-\frac{105 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{1575 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{7 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac{21 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5865
Rule 12
Rule 5663
Rule 5758
Rule 5675
Rule 5669
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (c e+d e x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^3 x^3 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 \operatorname{Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{\left (7 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{\left (21 b e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{32 d}+\frac{\left (35 b^2 e^3\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{64 d}\\ &=\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{\left (21 b e^3\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{64 d}-\frac{\left (105 b^2 e^3\right ) \operatorname{Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{128 d}-\frac{\left (105 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^4 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{512 d}\\ &=-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{2048 d}+\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{512 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{4096 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{4096 d}-\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{1024 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4096 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{8192 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{2048 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{a+b x}}+\frac{\sinh (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{4096 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8192 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2048 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32768 d}-\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16384 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8192 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{2048 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{65536 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{65536 d}+\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32768 d}-\frac{\left (105 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32768 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16384 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{4096 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{\left (105 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{\frac{4 a}{b}-\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32768 d}+\frac{\left (105 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{-\frac{4 a}{b}+\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{32768 d}+\frac{\left (105 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16384 d}-\frac{\left (105 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16384 d}+\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32768 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32768 d}+\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8192 d}-\frac{\left (315 b^4 e^3\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8192 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{105 b^{7/2} e^3 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}+\frac{105 b^{7/2} e^3 e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32768 d}+\frac{105 b^{7/2} e^3 e^{-\frac{4 a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}-\frac{105 b^{7/2} e^3 e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{32768 d}+\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16384 d}-\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16384 d}+\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4096 d}-\frac{\left (315 b^3 e^3\right ) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4096 d}\\ &=\frac{1575 b^3 e^3 (c+d x) \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}-\frac{105 b^3 e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}-\frac{525 b^2 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2048 d}-\frac{105 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{35 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{256 d}+\frac{21 b e^3 (c+d x) \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{64 d}-\frac{7 b e^3 (c+d x)^3 \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{32 d}-\frac{3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{32 d}+\frac{e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{4 d}-\frac{105 b^{7/2} e^3 e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}+\frac{105 b^{7/2} e^3 e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}+\frac{105 b^{7/2} e^3 e^{-\frac{4 a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{131072 d}-\frac{105 b^{7/2} e^3 e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2048 d}\\ \end{align*}
Mathematica [A] time = 0.313419, size = 225, normalized size = 0.41 \[ -\frac{b e^3 e^{-\frac{4 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (-\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+32 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \text{Gamma}\left (\frac{9}{2},-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left (e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{9}{2},\frac{4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-32 \sqrt{2} \text{Gamma}\left (\frac{9}{2},\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{8192 d \left (-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.196, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{3} \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 )}^{3}{\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 )}^{3}{\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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