Optimal. Leaf size=1536 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.44762, antiderivative size = 1536, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 29, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.967, Rules used = {5835, 5825, 5682, 5675, 30, 5717, 5742, 5758, 266, 43, 5732, 12, 5823, 683, 5815, 6742, 261, 725, 206, 5859, 1654, 5857, 8, 5831, 3322, 2264, 2190, 2279, 2391} \[ -\frac{b d^2 x^5 \sqrt{c^2 d x^2+d} c^5}{25 g \sqrt{c^2 x^2+1}}+\frac{b d^2 f x^4 \sqrt{c^2 d x^2+d} c^5}{16 g^2 \sqrt{c^2 x^2+1}}-\frac{d^2 f x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^4}{4 g^2}-\frac{b d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{c^2 d x^2+d} c^3}{9 g^3 \sqrt{c^2 x^2+1}}-\frac{b d^2 x^3 \sqrt{c^2 d x^2+d} c^3}{45 g \sqrt{c^2 x^2+1}}+\frac{b d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{c^2 d x^2+d} c^3}{4 g^4 \sqrt{c^2 x^2+1}}+\frac{b d^2 f x^2 \sqrt{c^2 d x^2+d} c^3}{16 g^2 \sqrt{c^2 x^2+1}}-\frac{d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{2 g^4}-\frac{d^2 f x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right ) c^2}{8 g^2}-\frac{d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{4 b g^4 \sqrt{c^2 x^2+1}}-\frac{d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{2 b g^5 \sqrt{c^2 x^2+1}}+\frac{d^2 f \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2 c}{16 b g^2 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{c^2 d x^2+d} c}{g^5 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{c^2 d x^2+d} c}{3 g^3 \sqrt{c^2 x^2+1}}+\frac{2 b d^2 x \sqrt{c^2 d x^2+d} c}{15 g \sqrt{c^2 x^2+1}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^5}+\frac{d^2 \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{d^2 \left (c^2 f^2+2 g^2\right ) \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}-\frac{d^2 \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^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{c^2 x^2+1}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{c^2 x^2+1}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{c^2 x^2+1}}+\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{c^2 d x^2+d}}{g^5}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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^4 (f+g x) c}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^6 (f+g x) \sqrt{c^2 x^2+1} c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5835
Rule 5825
Rule 5682
Rule 5675
Rule 30
Rule 5717
Rule 5742
Rule 5758
Rule 266
Rule 43
Rule 5732
Rule 12
Rule 5823
Rule 683
Rule 5815
Rule 6742
Rule 261
Rule 725
Rule 206
Rule 5859
Rule 1654
Rule 5857
Rule 8
Rule 5831
Rule 3322
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{\left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (\frac{\left (-c^4 f^3-2 c^2 f g^2\right ) \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^4}+\frac{c^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^3}-\frac{c^4 f x^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^2}+\frac{c^4 x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g}+\frac{\left (c^2 f^2+g^2\right )^2 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=-\frac{\left (c^4 d^2 f \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}{g^2 \sqrt{1+c^2 x^2}}+\frac{\left (c^4 d^2 \sqrt{d+c^2 d x^2}\right ) \int x^3 \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{g \sqrt{1+c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2+g^2\right )^2 \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}{g^4 \sqrt{1+c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2+2 g^2\right ) \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^4 \sqrt{1+c^2 x^2}}+\frac{\left (c^2 d^2 \left (c^2 f^2+2 g^2\right ) \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^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{\left (c^4 d^2 f \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}{4 g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b c^5 d^2 f \sqrt{d+c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt{1+c^2 x^2}}-\frac{\left (b c^5 d^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt{1+c^2 x^2}}-\frac{\left (d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 \sqrt{1+c^2 x^2}}-\frac{\left (c^2 d^2 f \left (c^2 f^2+2 g^2\right ) \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^4 \sqrt{1+c^2 x^2}}+\frac{\left (b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 g^3 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}+\frac{\left (c^2 d^2 f \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{8 g^2 \sqrt{1+c^2 x^2}}+\frac{\left (b c^3 d^2 f \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt{1+c^2 x^2}}+\frac{\left (d^2 \left (c^2 f^2+g^2\right )^2 \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}{g^4 \sqrt{1+c^2 x^2}}\\ &=\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}+\frac{\left (d^2 \left (c^2 f^2+g^2\right )^2 \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}{g^4 \sqrt{1+c^2 x^2}}\\ &=\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}+\frac{\left (a d^2 \left (c^2 f^2+g^2\right )^2 \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^2 \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}+\frac{\left (a d^2 \left (c^2 f^2+g^2\right )^2 \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^8 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^2 \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}+\frac{\left (b c^2 d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{x \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{g^5 \sqrt{1+c^2 x^2}}+\frac{\left (a d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1+c^2 x^2}} \, dx}{g^6 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^3 \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{\left (b c d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt{1+c^2 x^2}}-\frac{\left (a d^2 \left (c^2 f^2+g^2\right )^3 \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^3 \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2}}{g^5 \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{a d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (2 b d^2 \left (c^2 f^2+g^2\right )^3 \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2}}{g^5 \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{a d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (2 b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^5 \sqrt{1+c^2 x^2}}-\frac{\left (2 b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^5 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2}}{g^5 \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{a d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2}}{g^5 \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{a d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{\left (b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}\\ &=\frac{a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2}}{g^5}+\frac{2 b c d^2 x \sqrt{d+c^2 d x^2}}{15 g \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2}}{g^5 \sqrt{1+c^2 x^2}}-\frac{b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2}}{3 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f x^2 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}+\frac{b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt{d+c^2 d x^2}}{4 g^4 \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 x^3 \sqrt{d+c^2 d x^2}}{45 g \sqrt{1+c^2 x^2}}-\frac{b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt{d+c^2 d x^2}}{9 g^3 \sqrt{1+c^2 x^2}}+\frac{b c^5 d^2 f x^4 \sqrt{d+c^2 d x^2}}{16 g^2 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 x^5 \sqrt{d+c^2 d x^2}}{25 g \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt{d+c^2 d x^2} \sinh ^{-1}(c x)}{g^5}-\frac{c^2 d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{4 g^2}-\frac{d^2 \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^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 g^3}+\frac{d^2 \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{1+c^2 x^2}}-\frac{c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{1+c^2 x^2}}-\frac{c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{1+c^2 x^2}}-\frac{d^2 \left (c^2 f^2+g^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c g^6 (f+g x) \sqrt{1+c^2 x^2}}+\frac{d^2 \left (c^2 f^2+g^2\right )^2 \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 g^4 (f+g x)}-\frac{a d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}+\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}-\frac{b d^2 \left (c^2 f^2+g^2\right )^{5/2} \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^6 \sqrt{1+c^2 x^2}}\\ \end{align*}
Mathematica [C] time = 25.4527, size = 7163, normalized size = 4.66 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.303, size = 3928, normalized size = 2.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a c^{4} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} + 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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