Integrand size = 16, antiderivative size = 389 \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{64 d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {15 b^{5/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 b^{5/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \]
[Out]
Time = 0.85 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5830, 6873, 6874, 5433, 5432, 5406, 2236, 2235, 5407} \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 \sqrt {\pi } b^{5/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {15 b^2 \cosh (2 \text {arcsinh}(c+d x)) \sqrt {a+b \text {arcsinh}(c+d x)}}{64 d^2}+\frac {5 b c \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}-\frac {5 b \sinh (2 \text {arcsinh}(c+d x)) (a+b \text {arcsinh}(c+d x))^{3/2}}{16 d^2}+\frac {\cosh (2 \text {arcsinh}(c+d x)) (a+b \text {arcsinh}(c+d x))^{5/2}}{4 d^2} \]
[In]
[Out]
Rule 2235
Rule 2236
Rule 5406
Rule 5407
Rule 5432
Rule 5433
Rule 5830
Rule 5859
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) (a+b \text {arcsinh}(x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int (a+b x)^{5/2} \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right ) \, dx,x,\text {arcsinh}(c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int x^6 \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int x^6 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {2 \text {Subst}\left (\int \left (c x^6 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^6 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {\text {Subst}\left (\int x^6 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int x^6 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b d^2} \\ & = -\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {5 \text {Subst}\left (\int x^4 \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^2}-\frac {(5 c) \text {Subst}\left (\int x^4 \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{d^2} \\ & = \frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2}-\frac {(15 b) \text {Subst}\left (\int x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{16 d^2}-\frac {(15 b c) \text {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{2 d^2} \\ & = -\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{64 d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{64 d^2}-\frac {\left (15 b^2 c\right ) \text {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{4 d^2} \\ & = -\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{64 d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{128 d^2}-\frac {\left (15 b^2\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{128 d^2}-\frac {\left (15 b^2 c\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d^2}+\frac {\left (15 b^2 c\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{8 d^2} \\ & = -\frac {15 b^2 c (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{2 d^2}-\frac {c (c+d x) (a+b \text {arcsinh}(c+d x))^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))}{64 d^2}+\frac {(a+b \text {arcsinh}(c+d x))^{5/2} \cosh (2 \text {arcsinh}(c+d x))}{4 d^2}-\frac {15 b^{5/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 b^{5/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {5 b (a+b \text {arcsinh}(c+d x))^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{16 d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(939\) vs. \(2(389)=778\).
Time = 8.10 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.41 \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\frac {-1920 b^2 c^2 \sqrt {a+b \text {arcsinh}(c+d x)}-1920 b^2 c d x \sqrt {a+b \text {arcsinh}(c+d x)}+1280 a b c \sqrt {1+c^2+2 c d x+d^2 x^2} \sqrt {a+b \text {arcsinh}(c+d x)}-1024 a b c^2 \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-1024 a b c d x \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+1280 b^2 c \sqrt {1+c^2+2 c d x+d^2 x^2} \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-512 b^2 c^2 \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}-512 b^2 c d x \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}+128 a^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+120 b^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+256 a b \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))+128 b^2 \text {arcsinh}(c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)} \cosh (2 \text {arcsinh}(c+d x))-128 a^2 \sqrt {b} c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+480 b^{5/2} c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-15 b^{5/2} \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {256 a^2 b c e^{a/b} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {256 a^2 b c e^{-\frac {a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}+128 a^2 \sqrt {b} c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-480 b^{5/2} c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+32 \sqrt {b} \left (4 a^2-15 b^2\right ) c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+15 b^{5/2} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-15 b^{5/2} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-160 a b \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))-160 b^2 \text {arcsinh}(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)} \sinh (2 \text {arcsinh}(c+d x))}{512 d^2} \]
[In]
[Out]
\[\int x \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
[In]
[Out]
\[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x \,d x } \]
[In]
[Out]
Timed out. \[ \int x (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
[In]
[Out]