Integrand size = 30, antiderivative size = 1228 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.81 (sec) , antiderivative size = 1228, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 267, 5798, 200, 5808, 5806, 5812, 272, 45, 5804, 12, 380} \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b c^5 d^2 g^3 \sqrt {c^2 d x^2+d} x^9}{81 \sqrt {c^2 x^2+1}}-\frac {3 b c^5 d^2 f g^2 \sqrt {c^2 d x^2+d} x^8}{64 \sqrt {c^2 x^2+1}}-\frac {19 b c^3 d^2 g^3 \sqrt {c^2 d x^2+d} x^7}{441 \sqrt {c^2 x^2+1}}-\frac {3 b c^5 d^2 f^2 g \sqrt {c^2 d x^2+d} x^7}{49 \sqrt {c^2 x^2+1}}-\frac {17 b c^3 d^2 f g^2 \sqrt {c^2 d x^2+d} x^6}{96 \sqrt {c^2 x^2+1}}-\frac {b c d^2 g^3 \sqrt {c^2 d x^2+d} x^5}{21 \sqrt {c^2 x^2+1}}-\frac {9 b c^3 d^2 f^2 g \sqrt {c^2 d x^2+d} x^5}{35 \sqrt {c^2 x^2+1}}-\frac {5 b c^3 d^2 f^3 \sqrt {c^2 d x^2+d} x^4}{96 \sqrt {c^2 x^2+1}}-\frac {59 b c d^2 f g^2 \sqrt {c^2 d x^2+d} x^4}{256 \sqrt {c^2 x^2+1}}+\frac {15}{64} d^2 f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3+\frac {3}{8} d^2 f g^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3+\frac {5}{16} d^2 f g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x^3-\frac {b d^2 g^3 \sqrt {c^2 d x^2+d} x^3}{189 c \sqrt {c^2 x^2+1}}-\frac {3 b c d^2 f^2 g \sqrt {c^2 d x^2+d} x^3}{7 \sqrt {c^2 x^2+1}}-\frac {25 b c d^2 f^3 \sqrt {c^2 d x^2+d} x^2}{96 \sqrt {c^2 x^2+1}}-\frac {15 b d^2 f g^2 \sqrt {c^2 d x^2+d} x^2}{256 c \sqrt {c^2 x^2+1}}+\frac {5}{16} d^2 f^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x}{128 c^2}+\frac {1}{6} d^2 f^3 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {5}{24} d^2 f^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) x+\frac {2 b d^2 g^3 \sqrt {c^2 d x^2+d} x}{63 c^3 \sqrt {c^2 x^2+1}}-\frac {3 b d^2 f^2 g \sqrt {c^2 d x^2+d} x}{7 c \sqrt {c^2 x^2+1}}+\frac {5 d^2 f^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {c^2 x^2+1}}+\frac {d^2 g^3 \left (c^2 x^2+1\right )^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{9 c^4}-\frac {d^2 g^3 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {3 d^2 f^2 g \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {b d^2 f^3 \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c} \]
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 45
Rule 200
Rule 267
Rule 272
Rule 380
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5804
Rule 5806
Rule 5808
Rule 5812
Rule 5838
Rule 5845
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+3 f^2 g x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+3 f g^2 x^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+g^3 x^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right )^2 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (3 b d^2 f^2 g \sqrt {d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int x \left (1+c^2 x\right )^2 \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (x+2 c^2 x^2+c^4 x^3\right ) \, dx,x,x^2\right )}{16 \sqrt {1+c^2 x^2}}-\frac {\left (15 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{64 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{16 \sqrt {1+c^2 x^2}}-\frac {\left (b d^2 g^3 \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {\left (15 d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{128 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (15 b d^2 f g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1+c^2 x^2}} \\ & = -\frac {3 b d^2 f^2 g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}+\frac {2 b d^2 g^3 x \sqrt {d+c^2 d x^2}}{63 c^3 \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {15 b d^2 f g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {b d^2 g^3 x^3 \sqrt {d+c^2 d x^2}}{189 c \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d+c^2 d x^2}}{256 \sqrt {1+c^2 x^2}}-\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {b c d^2 g^3 x^5 \sqrt {d+c^2 d x^2}}{21 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d+c^2 d x^2}}{441 \sqrt {1+c^2 x^2}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d+c^2 d x^2}}{81 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^3 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3}{8} d^2 f g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{256 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 810, normalized size of antiderivative = 0.66 \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^3 \left (1+c^2 x^2\right ) \left (-20160 a \sqrt {1+c^2 x^2} \left (-256 g^3+c^2 g \left (3456 f^2+945 f g x+128 g^2 x^2\right )+16 c^8 x^5 \left (84 f^3+216 f^2 g x+189 f g^2 x^2+56 g^3 x^3\right )+8 c^6 x^3 \left (546 f^3+1296 f^2 g x+1071 f g^2 x^2+304 g^3 x^3\right )+6 c^4 x \left (924 f^3+1728 f^2 g x+1239 f g^2 x^2+320 g^3 x^3\right )\right )+b \left (-315 c g^2 (7539 f+16384 g x)+30240 c^5 x^2 \left (1848 f^3+2304 f^2 g x+1239 f g^2 x^2+256 g^3 x^3\right )+3360 c^3 \left (6279 f^3+20736 f^2 g x+2835 f g^2 x^2+256 g^3 x^3\right )+2304 c^7 x^4 \left (9555 f^3+18144 f^2 g x+12495 f g^2 x^2+3040 g^3 x^3\right )+640 c^9 x^6 \left (7056 f^3+15552 f^2 g x+11907 f g^2 x^2+3136 g^3 x^3\right )\right )\right )+3175200 b c d^3 f \left (8 c^2 f^2-3 g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+6350400 a c d^{5/2} f \left (8 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+2520 b d^3 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (27648 c^2 f^2 g \sqrt {1+c^2 x^2}-2048 g^3 \sqrt {1+c^2 x^2}+82944 c^4 f^2 g x^2 \sqrt {1+c^2 x^2}+1024 c^2 g^3 x^2 \sqrt {1+c^2 x^2}+82944 c^6 f^2 g x^4 \sqrt {1+c^2 x^2}+15360 c^4 g^3 x^4 \sqrt {1+c^2 x^2}+27648 c^8 f^2 g x^6 \sqrt {1+c^2 x^2}+19456 c^6 g^3 x^6 \sqrt {1+c^2 x^2}+7168 c^8 g^3 x^8 \sqrt {1+c^2 x^2}+3024 c f \left (5 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+1512 c f \left (2 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+336 c^3 f^3 \sinh (6 \text {arcsinh}(c x))+1008 c f g^2 \sinh (6 \text {arcsinh}(c x))+189 c f g^2 \sinh (8 \text {arcsinh}(c x))\right )}{162570240 c^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2883\) vs. \(2(1080)=2160\).
Time = 1.05 (sec) , antiderivative size = 2884, normalized size of antiderivative = 2.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(2884\) |
parts | \(\text {Expression too large to display}\) | \(2884\) |
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\[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{3} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \]
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Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]
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