Optimal. Leaf size=901 \[ -\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 g^2 x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{24} d^2 f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{48} d^2 g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} d^2 f^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} d^2 g^2 x^3 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {2 d^2 f g \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac {5 d^2 f^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt {1+c^2 x^2}}-\frac {5 d^2 g^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{256 b c^3 \sqrt {1+c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.61, antiderivative size = 901, normalized size of antiderivative = 1.00, number of
steps used = 26, number of rules used = 15, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used
= {5845, 5838, 5786, 5785, 5783, 30, 14, 267, 5798, 200, 5808, 5806, 5812, 272, 45}
\begin {gather*} -\frac {b c^5 d^2 g^2 \sqrt {c^2 d x^2+d} x^8}{64 \sqrt {c^2 x^2+1}}-\frac {2 b c^5 d^2 f g \sqrt {c^2 d x^2+d} x^7}{49 \sqrt {c^2 x^2+1}}-\frac {17 b c^3 d^2 g^2 \sqrt {c^2 d x^2+d} x^6}{288 \sqrt {c^2 x^2+1}}-\frac {6 b c^3 d^2 f 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^2 \sqrt {c^2 d x^2+d} x^4}{96 \sqrt {c^2 x^2+1}}-\frac {59 b c d^2 g^2 \sqrt {c^2 d x^2+d} x^4}{768 \sqrt {c^2 x^2+1}}+\frac {5}{64} d^2 g^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x^3+\frac {1}{8} d^2 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}{48} d^2 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 {2 b c d^2 f g \sqrt {c^2 d x^2+d} x^3}{7 \sqrt {c^2 x^2+1}}-\frac {25 b c d^2 f^2 \sqrt {c^2 d x^2+d} x^2}{96 \sqrt {c^2 x^2+1}}-\frac {5 b d^2 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^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) x+\frac {5 d^2 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^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+\frac {5}{24} d^2 f^2 \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 f g \sqrt {c^2 d x^2+d} x}{7 c \sqrt {c^2 x^2+1}}+\frac {5 d^2 f^2 \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 {5 d^2 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 {2 d^2 f 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^2 \left (c^2 x^2+1\right )^{5/2} \sqrt {c^2 d x^2+d}}{36 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 45
Rule 200
Rule 267
Rule 272
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5806
Rule 5808
Rule 5812
Rule 5838
Rule 5845
Rubi steps
\begin {align*} \int (f+g x)^2 \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)^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 \sqrt {d+c^2 d x^2}\right ) \int \left (f^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+2 f g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g^2 x^2 \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^2 \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 (2 d^2 f 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 (d^2 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 {1}{6} d^2 f^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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 (2 b d^2 f 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 (5 d^2 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 (b c d^2 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 {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{24} d^2 f^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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 (2 b d^2 f 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 (5 d^2 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 (b c d^2 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 g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{48 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {\left (5 d^2 f^2 \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^2 \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^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {1+c^2 x^2}}+\frac {\left (5 d^2 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 (b c d^2 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 (5 b c d^2 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 g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{48 \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {5 d^2 f^2 \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 (5 d^2 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 (5 b d^2 g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{128 c \sqrt {1+c^2 x^2}}\\ &=-\frac {2 b d^2 f g x \sqrt {d+c^2 d x^2}}{7 c \sqrt {1+c^2 x^2}}-\frac {25 b c d^2 f^2 x^2 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 g^2 x^2 \sqrt {d+c^2 d x^2}}{256 c \sqrt {1+c^2 x^2}}-\frac {2 b c d^2 f g x^3 \sqrt {d+c^2 d x^2}}{7 \sqrt {1+c^2 x^2}}-\frac {5 b c^3 d^2 f^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 g^2 x^4 \sqrt {d+c^2 d x^2}}{768 \sqrt {1+c^2 x^2}}-\frac {6 b c^3 d^2 f g x^5 \sqrt {d+c^2 d x^2}}{35 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 g^2 x^6 \sqrt {d+c^2 d x^2}}{288 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 f g x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 g^2 x^8 \sqrt {d+c^2 d x^2}}{64 \sqrt {1+c^2 x^2}}-\frac {b d^2 f^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2}}{36 c}+\frac {5}{16} d^2 f^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {5 d^2 g^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{128 c^2}+\frac {5}{64} d^2 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^2 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}{48} d^2 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^2 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 {1}{8} d^2 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 {2 d^2 f 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 {5 d^2 f^2 \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 {5 d^2 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 = 1.80, size = 1047, normalized size = 1.16 \begin {gather*} \frac {d^2 \left (-5160960 b c^2 f g x \sqrt {d+c^2 d x^2}-5160960 b c^4 f g x^3 \sqrt {d+c^2 d x^2}-3096576 b c^6 f g x^5 \sqrt {d+c^2 d x^2}-737280 b c^8 f g x^7 \sqrt {d+c^2 d x^2}+5160960 a c f g \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+12418560 a c^3 f^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+705600 a c g^2 x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+15482880 a c^3 f g x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+9784320 a c^5 f^2 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+5550720 a c^3 g^2 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+15482880 a c^5 f g x^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+3010560 a c^7 f^2 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+6397440 a c^5 g^2 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+5160960 a c^7 f g x^6 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2257920 a c^7 g^2 x^7 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+352800 b \left (8 c^2 f^2-g^2\right ) \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2-141120 b \left (15 c^2 f^2-g^2\right ) \sqrt {d+c^2 d x^2} \cosh \left (2 \sinh ^{-1}(c x)\right )-211680 b c^2 f^2 \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )-35280 b g^2 \sqrt {d+c^2 d x^2} \cosh \left (4 \sinh ^{-1}(c x)\right )-15680 b c^2 f^2 \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )-15680 b g^2 \sqrt {d+c^2 d x^2} \cosh \left (6 \sinh ^{-1}(c x)\right )-2205 b g^2 \sqrt {d+c^2 d x^2} \cosh \left (8 \sinh ^{-1}(c x)\right )+5644800 a c^2 \sqrt {d} f^2 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-705600 a \sqrt {d} g^2 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+840 b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (6144 c f g \sqrt {1+c^2 x^2}+18432 c^3 f g x^2 \sqrt {1+c^2 x^2}+18432 c^5 f g x^4 \sqrt {1+c^2 x^2}+6144 c^7 f g x^6 \sqrt {1+c^2 x^2}+336 \left (15 c^2 f^2-g^2\right ) \sinh \left (2 \sinh ^{-1}(c x)\right )+168 \left (6 c^2 f^2+g^2\right ) \sinh \left (4 \sinh ^{-1}(c x)\right )+112 c^2 f^2 \sinh \left (6 \sinh ^{-1}(c x)\right )+112 g^2 \sinh \left (6 \sinh ^{-1}(c x)\right )+21 g^2 \sinh \left (8 \sinh ^{-1}(c x)\right )\right )\right )}{18063360 c^3 \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2334\) vs.
\(2(791)=1582\).
time = 4.40, size = 2335, normalized size = 2.59
method | result | size |
default | \(\text {Expression too large to display}\) | \(2335\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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