Optimal. Leaf size=1228 \[ -\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} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac {d^2 g^3 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{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} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.14, antiderivative size = 1228, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 261, 5717, 194, 5744, 5742, 5758, 266, 43, 5732, 12, 373} \[ -\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} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) 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} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {c^2 x^2+1}}-\frac {15 d^2 f g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}-\frac {d^2 g^3 \left (c^2 x^2+1\right )^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{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} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 43
Rule 194
Rule 261
Rule 266
Rule 373
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rule 5732
Rule 5742
Rule 5744
Rule 5758
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int (f+g x)^3 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+3 f^2 g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+3 f g^2 x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g^3 x^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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 ) \operatorname {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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {\left (5 d^2 f^3 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(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 \left (a+b \sinh ^{-1}(c x)\right )}{\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 ) \operatorname {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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {15 d^2 f g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac {3 d^2 f^2 g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac {d^2 g^3 \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^4}+\frac {d^2 g^3 \left (1+c^2 x^2\right )^4 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^4}+\frac {5 d^2 f^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {15 d^2 f g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 7.08, size = 1899, normalized size = 1.55 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g^{3} x^{7} + 3 \, a c^{4} d^{2} f g^{2} x^{6} + 3 \, a d^{2} f^{2} g x + a d^{2} f^{3} + {\left (3 \, a c^{4} d^{2} f^{2} g + 2 \, a c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a c^{4} d^{2} f^{3} + 6 \, a c^{2} d^{2} f g^{2}\right )} x^{4} + {\left (6 \, a c^{2} d^{2} f^{2} g + a d^{2} g^{3}\right )} x^{3} + {\left (2 \, a c^{2} d^{2} f^{3} + 3 \, a d^{2} f g^{2}\right )} x^{2} + {\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} + {\left (3 \, b c^{4} d^{2} f^{2} g + 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b c^{4} d^{2} f^{3} + 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} + {\left (6 \, b c^{2} d^{2} f^{2} g + b d^{2} g^{3}\right )} x^{3} + {\left (2 \, b c^{2} d^{2} f^{3} + 3 \, b d^{2} f g^{2}\right )} x^{2}\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.01, size = 1954, normalized size = 1.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \]
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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