Optimal. Leaf size=447 \[ \frac{8 b^2 p^2 q^2 \sqrt{f g-e h} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{8 b p q \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}-\frac{8 b^2 p^2 q^2 \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{16 b^2 p^2 q^2 \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}+\frac{16 b^2 p^2 q^2 \sqrt{f g-e h} \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}+\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h} \]
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Rubi [A] time = 2.21737, antiderivative size = 447, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 16, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {2398, 2411, 2346, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 50, 2445} \[ \frac{8 b^2 p^2 q^2 \sqrt{f g-e h} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{8 b p q \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}-\frac{8 b^2 p^2 q^2 \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{16 b^2 p^2 q^2 \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}+\frac{16 b^2 p^2 q^2 \sqrt{f g-e h} \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}+\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h} \]
Antiderivative was successfully verified.
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Rule 2398
Rule 2411
Rule 2346
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rule 50
Rule 2445
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{\sqrt{g+h x}} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{\sqrt{g+h x}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{(4 b f p q) \int \frac{\sqrt{g+h x} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 b (f g-e h) p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\operatorname{Subst}\left (\frac{\left (8 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}}{x} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (4 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int -\frac{2 \sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{\sqrt{f g-e h} x} \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{\left (8 b^2 \sqrt{f g-e h} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{x} \, dx,x,e+f x\right )}{\sqrt{f} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{\left (16 b^2 \sqrt{f} \sqrt{f g-e h} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{e h+f \left (-g+x^2\right )} \, dx,x,\sqrt{g+h x}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{f g-e h}{h}+\frac{f x^2}{h}} \, dx,x,\sqrt{g+h x}\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}-\operatorname{Subst}\left (\frac{\left (16 b^2 \sqrt{f} \sqrt{f g-e h} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{-f g+e h+f x^2} \, dx,x,\sqrt{g+h x}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{8 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\operatorname{Subst}\left (\frac{\left (16 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}} \, dx,x,\sqrt{g+h x}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{8 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}-\operatorname{Subst}\left (\frac{\left (16 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}}\right )}{1-\frac{f x^2}{f g-e h}} \, dx,x,\sqrt{g+h x}\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{8 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}+\operatorname{Subst}\left (\frac{\left (16 b^2 \sqrt{f g-e h} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 \sqrt{g+h x}}{h}-\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{\sqrt{f} h}-\frac{8 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{\sqrt{f} h}-\frac{8 b p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac{8 b \sqrt{f g-e h} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt{f} h}+\frac{2 \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h}+\frac{16 b^2 \sqrt{f g-e h} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}+\frac{8 b^2 \sqrt{f g-e h} p^2 q^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{\sqrt{f} h}\\ \end{align*}
Mathematica [C] time = 1.60494, size = 646, normalized size = 1.45 \[ \frac{2 \left (b^2 h p^2 q^2 (e+f x) \sqrt{\frac{f (g+h x)}{f g-e h}} \, _4F_3\left (\frac{1}{2},1,1,1;2,2,2;\frac{h (e+f x)}{e h-f g}\right )-b^2 h p^2 q^2 (e+f x) \log (e+f x) \sqrt{\frac{f (g+h x)}{f g-e h}} \, _3F_2\left (\frac{1}{2},1,1;2,2;\frac{h (e+f x)}{e h-f g}\right )+a^2 f g+a^2 f h x+2 a b f g \log \left (c \left (d (e+f x)^p\right )^q\right )+2 a b f h x \log \left (c \left (d (e+f x)^p\right )^q\right )+4 a b \sqrt{f} p q \sqrt{g+h x} \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )-4 a b f g p q-4 a b f h p q x+4 b^2 \sqrt{f} p q \sqrt{g+h x} \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 f g \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-4 b^2 f g p q \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 f h x \log ^2\left (c \left (d (e+f x)^p\right )^q\right )-4 b^2 f h p q x \log \left (c \left (d (e+f x)^p\right )^q\right )-b^2 f g p^2 q^2 \log ^2(e+f x) \sqrt{\frac{f (g+h x)}{f g-e h}}+b^2 e h p^2 q^2 \log ^2(e+f x) \sqrt{\frac{f (g+h x)}{f g-e h}}-4 b^2 \sqrt{f} p^2 q^2 \sqrt{g+h x} \sqrt{f g-e h} \log (e+f x) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )+4 b^2 f g p^2 q^2 \log (e+f x)+4 b^2 f h p^2 q^2 x \log (e+f x)\right )}{f h \sqrt{g+h x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.667, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2}{\frac{1}{\sqrt{hx+g}}}}\, dx \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{\sqrt{h x + g} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, \sqrt{h x + g} a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt{h x + g} a^{2}}{h x + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\sqrt{g + h x}}\, dx \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 (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{\sqrt{h x + g}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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