Optimal. Leaf size=1385 \[ -\frac {b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} x^9}{81 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} x^7}{441 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} x^6}{96 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 g^3 \sqrt {d-c^2 d x^2} x^5}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {9 b c^3 d^2 f^2 g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d^2 f^3 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {c x-1} \sqrt {c x+1}}-\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} x^4}{256 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {3}{8} d^2 f g^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {b d^2 g^3 \sqrt {d-c^2 d x^2} x^3}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d^2 f^2 g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 g^3 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{9 c^2}-\frac {25 b c d^2 f^3 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15 b d^2 f g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{16} d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 f^2 g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d^2 g^3 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {3 d^2 f^2 g (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.52, antiderivative size = 1385, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 20, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.645, Rules used = {5836, 5822, 5685, 5683, 5676, 30, 14, 261, 5718, 194, 5745, 5743, 5759, 266, 43, 100, 12, 74, 5733, 373} \[ -\frac {b c^5 d^2 g^3 \sqrt {d-c^2 d x^2} x^9}{81 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 d^2 f g^2 \sqrt {d-c^2 d x^2} x^8}{64 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 g^3 \sqrt {d-c^2 d x^2} x^7}{441 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c^5 d^2 f^2 g \sqrt {d-c^2 d x^2} x^7}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {17 b c^3 d^2 f g^2 \sqrt {d-c^2 d x^2} x^6}{96 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 g^3 \sqrt {d-c^2 d x^2} x^5}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {9 b c^3 d^2 f^2 g \sqrt {d-c^2 d x^2} x^5}{35 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d^2 f^3 \sqrt {d-c^2 d x^2} x^4}{96 \sqrt {c x-1} \sqrt {c x+1}}-\frac {59 b c d^2 f g^2 \sqrt {d-c^2 d x^2} x^4}{256 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15}{64} d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {3}{8} d^2 f g^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {5}{16} d^2 f g^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3+\frac {b d^2 g^3 \sqrt {d-c^2 d x^2} x^3}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 b c d^2 f^2 g \sqrt {d-c^2 d x^2} x^3}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 g^3 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{9 c^2}-\frac {25 b c d^2 f^3 \sqrt {d-c^2 d x^2} x^2}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {15 b d^2 f g^2 \sqrt {d-c^2 d x^2} x^2}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{16} d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x}{128 c^2}+\frac {1}{6} d^2 f^3 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {5}{24} d^2 f^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x+\frac {2 b d^2 g^3 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 f^2 g \sqrt {d-c^2 d x^2} x}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d^2 g^3 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {3 d^2 f^2 g (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 30
Rule 43
Rule 74
Rule 100
Rule 194
Rule 261
Rule 266
Rule 373
Rule 5676
Rule 5683
Rule 5685
Rule 5718
Rule 5733
Rule 5743
Rule 5745
Rule 5759
Rule 5822
Rule 5836
Rubi steps
\begin {align*} \int (f+g x)^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} (f+g x)^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (f^3 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+3 f^2 g x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+3 f g^2 x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+g^3 x^3 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d^2 f^2 g \sqrt {d-c^2 d x^2}\right ) \int x (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int x^3 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{6} d^2 f^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 f^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}-\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}-\frac {\left (b c d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{24} d^2 f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 f^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\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 x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}+\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}+\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 x} \sqrt {1+c x}}-\frac {\left (b d^2 g^3 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {15}{64} d^2 f g^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 f^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {\left (5 d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\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 x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 f^3 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}+\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 x} \sqrt {1+c x}}-\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 x} \sqrt {1+c x}}\\ &=\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-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 \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 f^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 d^2 f g^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\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 x} \sqrt {1+c x}}\\ &=\frac {3 b d^2 f^2 g x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 g^3 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {25 b c d^2 f^3 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 b d^2 f g^2 x^2 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c d^2 f^2 g x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 g^3 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 f^3 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 f g^2 x^4 \sqrt {d-c^2 d x^2}}{256 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {9 b c^3 d^2 f^2 g x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 g^3 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 f g^2 x^6 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f^2 g x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 g^3 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 b c^5 d^2 f g^2 x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 g^3 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 f^3 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 f^3 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {15 d^2 f g^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-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 \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^3 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{16} d^2 f g^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^3 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{8} d^2 f g^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 f^2 g (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {2 d^2 g^3 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d^2 g^3 x^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {5 d^2 f^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {15 d^2 f g^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 7.98, size = 1802, normalized size = 1.30 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} g^{3} x^{7} + 3 \, a c^{4} d^{2} f g^{2} x^{6} + 3 \, a d^{2} f^{2} g x + a d^{2} f^{3} + {\left (3 \, a c^{4} d^{2} f^{2} g - 2 \, a c^{2} d^{2} g^{3}\right )} x^{5} + {\left (a c^{4} d^{2} f^{3} - 6 \, a c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, a c^{2} d^{2} f^{2} g - a d^{2} g^{3}\right )} x^{3} - {\left (2 \, a c^{2} d^{2} f^{3} - 3 \, a d^{2} f g^{2}\right )} x^{2} + {\left (b c^{4} d^{2} g^{3} x^{7} + 3 \, b c^{4} d^{2} f g^{2} x^{6} + 3 \, b d^{2} f^{2} g x + b d^{2} f^{3} + {\left (3 \, b c^{4} d^{2} f^{2} g - 2 \, b c^{2} d^{2} g^{3}\right )} x^{5} + {\left (b c^{4} d^{2} f^{3} - 6 \, b c^{2} d^{2} f g^{2}\right )} x^{4} - {\left (6 \, b c^{2} d^{2} f^{2} g - b d^{2} g^{3}\right )} x^{3} - {\left (2 \, b c^{2} d^{2} f^{3} - 3 \, b d^{2} f g^{2}\right )} x^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
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 = 1.23, size = 2116, normalized size = 1.53 \[ \text {result too large to display} \]
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 \[ \frac {1}{48} \, {\left (8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x + 10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x + 15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c}\right )} a f^{3} + \frac {1}{128} \, {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{2}} - \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{2} d} + \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{2}} + \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{2}} + \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{3}}\right )} a f g^{2} - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a g^{3} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a f^{2} g}{7 \, c^{2} d} + \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b g^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b f g^{2} x^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b f^{2} g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b f^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
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 \[ \int {\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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