Integrand size = 30, antiderivative size = 984 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^4 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^2 (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}} \]
[Out]
Time = 1.37 (sec) , antiderivative size = 984, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5845, 5840, 5785, 5783, 30, 5798, 5839, 697, 5835, 6874, 267, 739, 212, 5856, 1668, 12, 5855, 8, 5843, 3403, 2296, 2221, 2317, 2438} \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=-\frac {b d x^3 \sqrt {c^2 d x^2+d} c^3}{9 g \sqrt {c^2 x^2+1}}+\frac {b d f x^2 \sqrt {c^2 d x^2+d} c^3}{4 g^2 \sqrt {c^2 x^2+1}}-\frac {d f x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x)) c^2}{2 g^2}-\frac {d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2 c}{2 b g^3 \sqrt {c^2 x^2+1}}-\frac {d f \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2 c}{4 b g^2 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} c}{g^3 \sqrt {c^2 x^2+1}}-\frac {b d x \sqrt {c^2 d x^2+d} c}{3 g \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d} \text {arcsinh}(c x)}{g^3}+\frac {d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 g}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {c^2 x^2+1}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {arcsinh}(c x) \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {arcsinh}(c x) \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d}}{g^3}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 b g^4 (f+g x) \sqrt {c^2 x^2+1} c} \]
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 212
Rule 267
Rule 697
Rule 739
Rule 1668
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3403
Rule 5783
Rule 5785
Rule 5798
Rule 5835
Rule 5839
Rule 5840
Rule 5843
Rule 5845
Rule 5855
Rule 5856
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \left (-\frac {c^2 f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{g^2}+\frac {c^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{g}+\frac {\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{g^2 (f+g x)}\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (c^2 d f \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (c^2 d \sqrt {d+c^2 d x^2}\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{g \sqrt {1+c^2 x^2}} \\ & = -\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {\left (d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (-g+2 c^2 f x+c^2 g x^2\right ) (a+b \text {arcsinh}(c x))^2}{(f+g x)^2} \, dx}{2 b c \sqrt {1+c^2 x^2}}-\frac {\left (c^2 d f \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b c^3 d f \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 g \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (\frac {c^2 x}{g}+\frac {1+\frac {c^2 f^2}{g^2}}{f+g x}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {a \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 (f+g x) \sqrt {1+c^2 x^2}}+\frac {b \left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \text {arcsinh}(c x)}{g^2 (f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (a d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (c^2 f^2+g^2+c^2 f g x+c^2 g^2 x^2\right ) \text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (a d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {c^2 g^2 \left (c^2 f^2+g^2\right )}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{c^2 g^4 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \left (\frac {c^2 g x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 f^2+g^2\right ) \text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}}\right ) \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}+\frac {\left (b c^2 d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{g \sqrt {1+c^2 x^2}}+\frac {\left (a d \left (1+\frac {c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \int \frac {\text {arcsinh}(c x)}{(f+g x) \sqrt {1+c^2 x^2}} \, dx}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {\left (b c d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{g \sqrt {1+c^2 x^2}}-\frac {\left (a d \left (1+\frac {c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c^2 f^2+g^2-x^2} \, dx,x,\frac {g-c^2 f x}{\sqrt {1+c^2 x^2}}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \sinh (x)} \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {\left (2 b d \left (1+\frac {c^2 f^2}{g^2}\right ) \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {\left (2 b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{g \sqrt {1+c^2 x^2}}-\frac {\left (2 b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\text {arcsinh}(c x)\right )}{g \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{g^2 \sqrt {1+c^2 x^2}}+\frac {\left (b d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{g^2 \sqrt {1+c^2 x^2}} \\ & = \frac {a d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}{g^3}-\frac {b c d x \sqrt {d+c^2 d x^2}}{3 g \sqrt {1+c^2 x^2}}-\frac {b c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2}}{g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d+c^2 d x^2}}{4 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d+c^2 d x^2}}{9 g \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d \left (c^2 f^2+g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^3 \sqrt {1+c^2 x^2}}-\frac {d \left (1+\frac {c^2 f^2}{g^2}\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x) \sqrt {1+c^2 x^2}}+\frac {d \left (1+\frac {c^2 f^2}{g^2}\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c (f+g x)}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 12.24 (sec) , antiderivative size = 2869, normalized size of antiderivative = 2.92 \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]
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Time = 0.86 (sec) , antiderivative size = 1557, normalized size of antiderivative = 1.58
method | result | size |
default | \(\text {Expression too large to display}\) | \(1557\) |
parts | \(\text {Expression too large to display}\) | \(1557\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{f+g\,x} \,d x \]
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