Integrand size = 27, antiderivative size = 382 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {3}{16} a d x^2 \sqrt {a+b x^4}+\frac {4 a^2 e x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}+\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {4 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}} \]
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Time = 0.17 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1899, 1191, 1212, 226, 1210, 1262, 655, 201, 223, 212} \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {2 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (7 \sqrt {a} e+15 \sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}}-\frac {4 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {3 a^2 d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}+\frac {4 a^2 e x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{63} x \left (a+b x^4\right )^{3/2} \left (9 c+7 e x^2\right )+\frac {2}{105} a x \sqrt {a+b x^4} \left (15 c+7 e x^2\right )+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {3}{16} a d x^2 \sqrt {a+b x^4}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b} \]
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Rule 201
Rule 212
Rule 223
Rule 226
Rule 655
Rule 1191
Rule 1210
Rule 1212
Rule 1262
Rule 1899
Rubi steps \begin{align*} \text {integral}& = \int \left (\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}+x \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2}\right ) \, dx \\ & = \int \left (c+e x^2\right ) \left (a+b x^4\right )^{3/2} \, dx+\int x \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx \\ & = \frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{21} \int \left (18 a c+14 a e x^2\right ) \sqrt {a+b x^4} \, dx+\frac {1}{2} \text {Subst}\left (\int (d+f x) \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right ) \\ & = \frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}+\frac {1}{315} \int \frac {180 a^2 c+84 a^2 e x^2}{\sqrt {a+b x^4}} \, dx+\frac {1}{2} d \text {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^2\right ) \\ & = \frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}+\frac {1}{8} (3 a d) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^2\right )-\frac {\left (4 a^{5/2} e\right ) \int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx}{15 \sqrt {b}}+\frac {1}{105} \left (4 a^2 \left (15 c+\frac {7 \sqrt {a} e}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx \\ & = \frac {3}{16} a d x^2 \sqrt {a+b x^4}+\frac {4 a^2 e x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}-\frac {4 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{16} \left (3 a^2 d\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right ) \\ & = \frac {3}{16} a d x^2 \sqrt {a+b x^4}+\frac {4 a^2 e x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}-\frac {4 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}}+\frac {1}{16} \left (3 a^2 d\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right ) \\ & = \frac {3}{16} a d x^2 \sqrt {a+b x^4}+\frac {4 a^2 e x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {2}{105} a x \left (15 c+7 e x^2\right ) \sqrt {a+b x^4}+\frac {1}{8} d x^2 \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 c+7 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {f \left (a+b x^4\right )^{5/2}}{10 b}+\frac {3 a^2 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {4 a^{9/4} e \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {b} c+7 \sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.46 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {1}{240} \sqrt {a+b x^4} \left (\frac {24 f \left (a+b x^4\right )^2}{b}+15 d \left (5 a x^2+2 b x^6+\frac {3 a^{5/2} \sqrt {1+\frac {b x^4}{a}} \text {arcsinh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {b} \left (a+b x^4\right )}\right )+\frac {240 a c x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}+\frac {80 a e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{\sqrt {1+\frac {b x^4}{a}}}\right ) \]
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Result contains complex when optimal does not.
Time = 1.92 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {\left (504 b^{2} f \,x^{8}+560 b^{2} e \,x^{7}+630 b^{2} d \,x^{6}+720 b^{2} c \,x^{5}+1008 a b f \,x^{4}+1232 a b e \,x^{3}+1575 x^{2} a b d +2160 a b c x +504 a^{2} f \right ) \sqrt {b \,x^{4}+a}}{5040 b}+\frac {a^{2} \left (\frac {480 c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {224 i e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}+\frac {315 d \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{2 \sqrt {b}}\right )}{840}\) | \(287\) |
default | \(c \left (\frac {b \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {3 a x \sqrt {b \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+\frac {f \left (b \,x^{4}+a \right )^{\frac {5}{2}}}{10 b}+e \left (\frac {b \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+d \left (\frac {3 a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}+\frac {b \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {b \,x^{4}+a}}{16}\right )\) | \(309\) |
elliptic | \(\frac {b f \,x^{8} \sqrt {b \,x^{4}+a}}{10}+\frac {b e \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {b d \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {b c \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {a f \,x^{4} \sqrt {b \,x^{4}+a}}{5}+\frac {11 a e \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {5 a d \,x^{2} \sqrt {b \,x^{4}+a}}{16}+\frac {3 a c x \sqrt {b \,x^{4}+a}}{7}+\frac {a^{2} f \sqrt {b \,x^{4}+a}}{10 b}+\frac {4 a^{2} c \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a^{2} d \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}+\frac {4 i a^{\frac {5}{2}} e \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\) | \(347\) |
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none
Time = 0.13 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.56 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {2688 \, a^{2} \sqrt {b} e x \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 945 \, a^{2} \sqrt {b} d x \log \left (-2 \, b x^{4} - 2 \, \sqrt {b x^{4} + a} \sqrt {b} x^{2} - a\right ) + 384 \, {\left (15 \, a b c - 7 \, a^{2} e\right )} \sqrt {b} x \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + 2 \, {\left (504 \, b^{2} f x^{9} + 560 \, b^{2} e x^{8} + 630 \, b^{2} d x^{7} + 720 \, b^{2} c x^{6} + 1008 \, a b f x^{5} + 1232 \, a b e x^{4} + 1575 \, a b d x^{3} + 2160 \, a b c x^{2} + 504 \, a^{2} f x + 1344 \, a^{2} e\right )} \sqrt {b x^{4} + a}}{10080 \, b x} \]
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Time = 4.98 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.03 \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{\frac {3}{2}} d x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {a^{\frac {3}{2}} d x^{2}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {a} b c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {3 \sqrt {a} b d x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {3 a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {b}} + a f \left (\begin {cases} \frac {\sqrt {a} x^{4}}{4} & \text {for}\: b = 0 \\\frac {\left (a + b x^{4}\right )^{\frac {3}{2}}}{6 b} & \text {otherwise} \end {cases}\right ) + b f \left (\begin {cases} - \frac {a^{2} \sqrt {a + b x^{4}}}{15 b^{2}} + \frac {a x^{4} \sqrt {a + b x^{4}}}{30 b} + \frac {x^{8} \sqrt {a + b x^{4}}}{10} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases}\right ) + \frac {b^{2} d x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \]
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\[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} \,d x } \]
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\[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (f x^{3} + e x^{2} + d x + c\right )} \,d x } \]
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Timed out. \[ \int \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2} \, dx=\int {\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \]
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