Optimal. Leaf size=701 \[ \frac {15 \sqrt {\pi } b^{5/2} e^4 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {\sqrt {3 \pi } b^{5/2} e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1280 d}-\frac {\sqrt {\frac {\pi }{3}} b^{5/2} e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{240 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{5/2} e^4 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6400 d}-\frac {15 \sqrt {\pi } b^{5/2} e^4 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {\sqrt {3 \pi } b^{5/2} e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1280 d}+\frac {\sqrt {\frac {\pi }{3}} b^{5/2} e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{240 d}-\frac {3 \sqrt {\frac {\pi }{5}} b^{5/2} e^4 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6400 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac {b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {4 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d} \]
[Out]
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Rubi [A] time = 2.20, antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5865, 12, 5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ \frac {15 \sqrt {\pi } b^{5/2} e^4 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {\sqrt {3 \pi } b^{5/2} e^4 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1280 d}-\frac {\sqrt {\frac {\pi }{3}} b^{5/2} e^4 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{240 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{5/2} e^4 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6400 d}-\frac {15 \sqrt {\pi } b^{5/2} e^4 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {\sqrt {3 \pi } b^{5/2} e^4 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1280 d}+\frac {\sqrt {\frac {\pi }{3}} b^{5/2} e^4 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{240 d}-\frac {3 \sqrt {\frac {\pi }{5}} b^{5/2} e^4 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6400 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac {b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {4 b e^4 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 3312
Rule 5653
Rule 5663
Rule 5717
Rule 5758
Rule 5779
Rule 5865
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \frac {x^5 \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (2 b e^4\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{5 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int x^4 \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{20 d}\\ &=\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}-\frac {\left (4 b e^4\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{15 d}-\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int x^2 \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{5 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{200 d}\\ &=-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (2 b^2 e^4\right ) \operatorname {Subst}\left (\int \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{5 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh ^5(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{30 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (3 i b^3 e^4\right ) \operatorname {Subst}\left (\int \left (\frac {5 i \sinh (x)}{8 \sqrt {a+b x}}-\frac {5 i \sinh (3 x)}{16 \sqrt {a+b x}}+\frac {i \sinh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{30 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{5 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (i b^3 e^4\right ) \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {a+b x}}-\frac {i \sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{30 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{320 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{5 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6400 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6400 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1280 d}+\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1280 d}+\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}-\frac {\left (3 b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{640 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{120 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{40 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{10 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{10 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {\left (3 b^2 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 )}{3200 d}-\frac {\left (3 b^2 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 )}{3200 d}-\frac {\left (3 b^2 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 )}{640 d}+\frac {\left (3 b^2 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 )}{640 d}+\frac {\left (3 b^2 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 )}{320 d}-\frac {\left (3 b^2 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 )}{320 d}+\frac {\left (b^2 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 )}{5 d}-\frac {\left (b^2 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 )}{5 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{240 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{240 d}+\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{80 d}-\frac {\left (b^3 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{80 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {67 b^{5/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{640 d}-\frac {b^{5/2} 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 )}{1280 d}+\frac {3 b^{5/2} 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 )}{6400 d}-\frac {67 b^{5/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{640 d}+\frac {b^{5/2} 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 )}{1280 d}-\frac {3 b^{5/2} 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 )}{6400 d}-\frac {\left (b^2 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 )}{120 d}+\frac {\left (b^2 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 )}{120 d}+\frac {\left (b^2 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 )}{40 d}-\frac {\left (b^2 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 )}{40 d}\\ &=\frac {2 b^2 e^4 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{5 d}-\frac {b^2 e^4 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac {3 b^2 e^4 (c+d x)^5 \sqrt {a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{15 d}-\frac {b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{10 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{5 d}+\frac {15 b^{5/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {b^{5/2} 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 )}{240 d}-\frac {b^{5/2} 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 )}{1280 d}+\frac {3 b^{5/2} 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 )}{6400 d}-\frac {15 b^{5/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {b^{5/2} 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 )}{240 d}+\frac {b^{5/2} 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 )}{1280 d}-\frac {3 b^{5/2} 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 )}{6400 d}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 342, normalized size = 0.49 \[ -\frac {e^4 e^{-\frac {5 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (-33750 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {7}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+27 \sqrt {5} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {7}{2},-\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-625 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {7}{2},-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+33750 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {7}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )+625 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {7}{2},\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-27 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {7}{2},\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{540000 d \left (-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{4} \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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