Optimal. Leaf size=620 \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\sqrt{2} b e^{5/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{7/2}}+\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{5 d^{5/4} h^{7/2}}-\frac{\sqrt{2} b e^{3/4} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{7/2}}+\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{3 d^{3/4} h^{7/2}}-\frac{8 b e f p}{5 d h^3 \sqrt{h x}} \]
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Rubi [A] time = 0.799025, antiderivative size = 620, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {2467, 2476, 2455, 325, 297, 1162, 617, 204, 1165, 628, 211} \[ -\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\sqrt{2} b e^{5/4} f p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{7/2}}+\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{5 d^{5/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{5 d^{5/4} h^{7/2}}-\frac{\sqrt{2} b e^{3/4} g p \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{7/2}}+\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}+\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x\right )}{3 d^{3/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}+1\right )}{3 d^{3/4} h^{7/2}}-\frac{8 b e f p}{5 d h^3 \sqrt{h x}} \]
Antiderivative was successfully verified.
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Rule 2467
Rule 2476
Rule 2455
Rule 325
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 211
Rubi steps
\begin{align*} \int \frac{(f+g x) \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{(h x)^{7/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\left (f+\frac{g x^2}{h}\right ) \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{x^6} \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{f \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{x^6}+\frac{g \left (a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )\right )}{h x^4}\right ) \, dx,x,\sqrt{h x}\right )}{h}\\ &=\frac{(2 g) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^4} \, dx,x,\sqrt{h x}\right )}{h^2}+\frac{(2 f) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e x^4}{h^2}\right )^p\right )}{x^6} \, dx,x,\sqrt{h x}\right )}{h}\\ &=-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}+\frac{(8 b e g p) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 h^4}+\frac{(8 b e f p) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (d+\frac{e x^4}{h^2}\right )} \, dx,x,\sqrt{h x}\right )}{5 h^3}\\ &=-\frac{8 b e f p}{5 d h^3 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\left (8 b e^2 f p\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^5}+\frac{(4 b e g p) \operatorname{Subst}\left (\int \frac{\sqrt{d} h-\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 \sqrt{d} h^5}+\frac{(4 b e g p) \operatorname{Subst}\left (\int \frac{\sqrt{d} h+\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{3 \sqrt{d} h^5}\\ &=-\frac{8 b e f p}{5 d h^3 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}+\frac{\left (4 b e^{3/2} f p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h-\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^5}-\frac{\left (4 b e^{3/2} f p\right ) \operatorname{Subst}\left (\int \frac{\sqrt{d} h+\sqrt{e} x^2}{d+\frac{e x^4}{h^2}} \, dx,x,\sqrt{h x}\right )}{5 d h^5}-\frac{\left (\sqrt{2} b e^{3/4} g p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}+2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}-\frac{\left (\sqrt{2} b e^{3/4} g p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}-2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}+\frac{\left (2 b \sqrt{e} g p\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{3 \sqrt{d} h^3}+\frac{\left (2 b \sqrt{e} g p\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{3 \sqrt{d} h^3}\\ &=-\frac{8 b e f p}{5 d h^3 \sqrt{h x}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}+\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}-\frac{\left (\sqrt{2} b e^{5/4} f p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}+2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}-\frac{\left (\sqrt{2} b e^{5/4} f p\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h}}{\sqrt [4]{e}}-2 x}{-\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}-x^2} \, dx,x,\sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}+\frac{\left (2 \sqrt{2} b e^{3/4} g p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}-\frac{\left (2 \sqrt{2} b e^{3/4} g p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}-\frac{(2 b e f p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{5 d h^3}-\frac{(2 b e f p) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{d} h}{\sqrt{e}}+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{h} x}{\sqrt [4]{e}}+x^2} \, dx,x,\sqrt{h x}\right )}{5 d h^3}\\ &=-\frac{8 b e f p}{5 d h^3 \sqrt{h x}}-\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}-\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}+\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}+\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}-\frac{\left (2 \sqrt{2} b e^{5/4} f p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}+\frac{\left (2 \sqrt{2} b e^{5/4} f p\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}\\ &=-\frac{8 b e f p}{5 d h^3 \sqrt{h x}}+\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}-\frac{2 \sqrt{2} b e^{5/4} f p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{5 d^{5/4} h^{7/2}}+\frac{2 \sqrt{2} b e^{3/4} g p \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{e} \sqrt{h x}}{\sqrt [4]{d} \sqrt{h}}\right )}{3 d^{3/4} h^{7/2}}-\frac{2 f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 h (h x)^{5/2}}-\frac{2 g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 h^2 (h x)^{3/2}}-\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}-\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}+\frac{\sqrt{2} b e^{5/4} f p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{5 d^{5/4} h^{7/2}}+\frac{\sqrt{2} b e^{3/4} g p \log \left (\sqrt{d} \sqrt{h}+\sqrt{e} \sqrt{h} x+\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{h x}\right )}{3 d^{3/4} h^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.182868, size = 309, normalized size = 0.5 \[ \frac{2 x^{7/2} \left (-\frac{f \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{5 x^{5/2}}-\frac{g \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{3 x^{3/2}}-\frac{1}{6} b g p \left (\frac{\frac{\sqrt{2} e^{3/4} \log \left (-\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )}{\sqrt [4]{d}}-\frac{\sqrt{2} e^{3/4} \log \left (\sqrt{2} \sqrt [4]{d} \sqrt [4]{e} \sqrt{x}+\sqrt{d}+\sqrt{e} x\right )}{\sqrt [4]{d}}}{\sqrt{d}}+\frac{2 \left (\frac{\sqrt{2} e^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )}{\sqrt [4]{d}}-\frac{\sqrt{2} e^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}+1\right )}{\sqrt [4]{d}}\right )}{\sqrt{d}}\right )-\frac{4 b e f p \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};-\frac{e x^2}{d}\right )}{5 d \sqrt{x}}\right )}{(h x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.239, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx+f \right ) \left ( a+b\ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) \left ( hx \right ) ^{-{\frac{7}{2}}}}\, 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 [B] time = 2.08976, size = 2877, normalized size = 4.64 \begin{align*} \text{result too large to display} \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 [A] time = 1.62067, size = 647, normalized size = 1.04 \begin{align*} \frac{\frac{2 \,{\left (5 \, \sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} - 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f p e^{\frac{9}{4}}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} e^{\left (-\frac{1}{4}\right )} + 2 \, \sqrt{h x}\right )} e^{\frac{1}{4}}}{2 \, \left (d h^{2}\right )^{\frac{1}{4}}}\right ) e^{\left (-1\right )}}{d^{2} h^{2}} + \frac{2 \,{\left (5 \, \sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} - 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f p e^{\frac{9}{4}}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} e^{\left (-\frac{1}{4}\right )} - 2 \, \sqrt{h x}\right )} e^{\frac{1}{4}}}{2 \, \left (d h^{2}\right )^{\frac{1}{4}}}\right ) e^{\left (-1\right )}}{d^{2} h^{2}} + \frac{{\left (5 \, \sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} + 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f p e^{\frac{9}{4}}\right )} e^{\left (-1\right )} \log \left (\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} \sqrt{h x} e^{\left (-\frac{1}{4}\right )} + h x + \sqrt{d h^{2}} e^{\left (-\frac{1}{2}\right )}\right )}{d^{2} h^{2}} - \frac{{\left (5 \, \sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} b d g h p e^{\frac{7}{4}} + 3 \, \sqrt{2} \left (d h^{2}\right )^{\frac{3}{4}} b f p e^{\frac{9}{4}}\right )} e^{\left (-1\right )} \log \left (-\sqrt{2} \left (d h^{2}\right )^{\frac{1}{4}} \sqrt{h x} e^{\left (-\frac{1}{4}\right )} + h x + \sqrt{d h^{2}} e^{\left (-\frac{1}{2}\right )}\right )}{d^{2} h^{2}}}{15 \, h^{3}} - \frac{2 \,{\left (12 \, b f h^{3} p x^{2} e + 5 \, b d g h^{3} p x \log \left (h^{2} x^{2} e + d h^{2}\right ) - 5 \, b d g h^{3} p x \log \left (h^{2}\right ) + 3 \, b d f h^{3} p \log \left (h^{2} x^{2} e + d h^{2}\right ) - 3 \, b d f h^{3} p \log \left (h^{2}\right ) + 5 \, b d g h^{3} x \log \left (c\right ) + 5 \, a d g h^{3} x + 3 \, b d f h^{3} \log \left (c\right ) + 3 \, a d f h^{3}\right )}}{15 \, \sqrt{h x} d h^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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