Optimal. Leaf size=601 \[ \frac {3 \sqrt {\pi } b^{3/2} e^4 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {3 \sqrt {3 \pi } b^{3/2} e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^4 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{200 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{3/2} e^4 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}+\frac {3 \sqrt {\pi } b^{3/2} e^4 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {3 \sqrt {3 \pi } b^{3/2} e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^4 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{200 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{3/2} e^4 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac {3 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {4 b e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.66, antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps used = 43, 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, 5657, 3307, 2180, 2205, 2204, 5669, 5448} \[ \frac {3 \sqrt {\pi } b^{3/2} e^4 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {3 \sqrt {3 \pi } b^{3/2} e^4 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^4 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{200 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{3/2} e^4 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}+\frac {3 \sqrt {\pi } b^{3/2} e^4 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {3 \sqrt {3 \pi } b^{3/2} e^4 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}-\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e^4 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{200 d}+\frac {3 \sqrt {\frac {\pi }{5}} b^{3/2} e^4 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3200 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac {3 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^4 \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {2 b e^4 \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {4 b e^4 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5657
Rule 5663
Rule 5669
Rule 5717
Rule 5758
Rule 5865
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^4 x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int x^4 \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac {\left (3 b 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 )}{10 d}\\ &=-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {\left (6 b 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 )}{25 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{100 d}\\ &=\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}-\frac {\left (4 b 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 )}{25 d}+\frac {\left (3 b^2 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 )}{100 d}-\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {\left (3 b^2 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 )}{100 d}-\frac {\left (b^2 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 )}{25 d}+\frac {\left (2 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {\left (2 b 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 )}{25 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{800 d}-\frac {\left (9 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}-\frac {\left (b^2 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 )}{25 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {\left (b 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 )}{25 d}+\frac {\left (b 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 )}{25 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}+\frac {\left (3 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{1600 d}-\frac {\left (9 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}-\frac {\left (9 b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3200 d}+\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{100 d}-\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{100 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {\left (3 b 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 )}{1600 d}+\frac {\left (3 b 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 )}{1600 d}+\frac {\left (3 b 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 )}{800 d}+\frac {\left (3 b 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 )}{800 d}-\frac {\left (9 b 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 )}{1600 d}-\frac {\left (9 b 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 )}{1600 d}+\frac {\left (2 b 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 )}{25 d}+\frac {\left (2 b 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 )}{25 d}-\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}+\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}-\frac {\left (b^2 e^4\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{200 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {67 b^{3/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1600 d}-\frac {3 b^{3/2} 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 )}{3200 d}+\frac {3 b^{3/2} 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 )}{3200 d}+\frac {67 b^{3/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1600 d}-\frac {3 b^{3/2} 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 )}{3200 d}+\frac {3 b^{3/2} 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 )}{3200 d}-\frac {\left (b 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 )}{100 d}+\frac {\left (b 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 )}{100 d}+\frac {\left (b 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 )}{100 d}-\frac {\left (b 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 )}{100 d}\\ &=-\frac {4 b e^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}+\frac {2 b e^4 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac {3 b e^4 (c+d x)^4 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac {e^4 (c+d x)^5 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{5 d}+\frac {3 b^{3/2} e^4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {b^{3/2} 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 )}{200 d}-\frac {3 b^{3/2} 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 )}{3200 d}+\frac {3 b^{3/2} 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 )}{3200 d}+\frac {3 b^{3/2} e^4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {b^{3/2} 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 )}{200 d}-\frac {3 b^{3/2} 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 )}{3200 d}+\frac {3 b^{3/2} 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 )}{3200 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.48, size = 343, normalized size = 0.57 \[ -\frac {b e^4 e^{-\frac {5 a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (2250 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+9 \sqrt {5} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-125 \sqrt {3} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+2250 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )-125 \sqrt {3} e^{\frac {8 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+9 \sqrt {5} e^{\frac {10 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {5 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{36000 d \sqrt {-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{4} \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{4} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^4\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{4} \left (\int a c^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a d^{4} x^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b c^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 4 a c d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 6 a c^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int 4 a c^{3} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b d^{4} x^{4} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 4 b c d^{3} x^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 6 b c^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 4 b c^{3} d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________