Optimal. Leaf size=984 \[ -\frac {b d x^3 \sqrt {c^2 d x^2+d} c^3}{9 g \sqrt {c^2 x^2+1}}+\frac {b d f x^2 \sqrt {c^2 d x^2+d} c^3}{4 g^2 \sqrt {c^2 x^2+1}}-\frac {d f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^2}-\frac {d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt {c^2 x^2+1}}-\frac {d f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} c}{g^3 \sqrt {c^2 x^2+1}}-\frac {b d x \sqrt {c^2 d x^2+d} c}{3 g \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g^3}+\frac {d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {c^2 x^2+1}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d}}{g^3}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt {c^2 x^2+1} c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.89, antiderivative size = 984, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 24, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5835, 5825, 5682, 5675, 30, 5717, 5823, 683, 5815, 6742, 261, 725, 206, 5859, 1654, 12, 5857, 8, 5831, 3322, 2264, 2190, 2279, 2391} \[ -\frac {b d x^3 \sqrt {c^2 d x^2+d} c^3}{9 g \sqrt {c^2 x^2+1}}+\frac {b d f x^2 \sqrt {c^2 d x^2+d} c^3}{4 g^2 \sqrt {c^2 x^2+1}}-\frac {d f x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^2}-\frac {d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^3 \sqrt {c^2 x^2+1}}-\frac {d f \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^2 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right ) x \sqrt {c^2 d x^2+d} c}{g^3 \sqrt {c^2 x^2+1}}-\frac {b d x \sqrt {c^2 d x^2+d} c}{3 g \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{g^3}+\frac {d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {a d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \tanh ^{-1}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {c^2 x^2+1}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x) \log \left (\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}-\frac {b d \left (c^2 f^2+g^2\right )^{3/2} \sqrt {c^2 d x^2+d} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {c^2 x^2+1}}+\frac {a d \left (c^2 f^2+g^2\right ) \sqrt {c^2 d x^2+d}}{g^3}+\frac {d \left (c^2 f^2+g^2\right ) \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^2 (f+g x) c}-\frac {d \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^4 (f+g x) \sqrt {c^2 x^2+1} c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 206
Rule 261
Rule 683
Rule 725
Rule 1654
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3322
Rule 5675
Rule 5682
Rule 5717
Rule 5815
Rule 5823
Rule 5825
Rule 5831
Rule 5835
Rule 5857
Rule 5859
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{g^2}+\frac {c^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g}+\frac {\left (c^2 f^2+g^2\right ) \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g \sqrt {1+c^2 x^2}}\\ &=-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \left (a+b \sinh ^{-1}(c x)\right )}{\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} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}}+\frac {\left (c^2 f^2+g^2\right ) \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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 {\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \tanh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {e^x x}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \tanh ^{-1}\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 ) \operatorname {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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 ) \operatorname {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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 ) \operatorname {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,\sinh ^{-1}(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 ) \operatorname {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,\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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 ) \operatorname {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^{\sinh ^{-1}(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 ) \operatorname {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^{\sinh ^{-1}(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} \sinh ^{-1}(c x)}{g^3}-\frac {c^2 d f x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^2}+\frac {d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g}-\frac {c d f \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \left (a+b \sinh ^{-1}(c x)\right )^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} \tanh ^{-1}\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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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} \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(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 {Li}_2\left (-\frac {e^{\sinh ^{-1}(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 {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^4 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 13.62, size = 2889, normalized size = 2.94 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{2} d x^{2} + a d + {\left (b c^{2} d x^{2} + b d\right )} \operatorname {arsinh}\left (c x\right )\right )} \sqrt {c^{2} d x^{2} + d}}{g x + f}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 1838, normalized size = 1.87 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2}}{f+g\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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