Optimal. Leaf size=403 \[ \frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {4 a^{9/4} f \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+b x^4}}+\frac {2 a^{7/4} \left (15 \sqrt {b} d+7 \sqrt {a} f\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{105 b^{3/4} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.23, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {1847, 1266,
829, 858, 223, 212, 272, 65, 214, 1191, 1212, 226, 1210} \begin {gather*} \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} f+15 \sqrt {b} d\right ) F\left (2 \text {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} f \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 \text {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 {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}+\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \sqrt {a+b x^4} \left (8 c+3 e x^2\right )+\frac {1}{24} \left (a+b x^4\right )^{3/2} \left (4 c+3 e x^2\right )+\frac {2}{105} a x \sqrt {a+b x^4} \left (15 d+7 f x^2\right )+\frac {1}{63} x \left (a+b x^4\right )^{3/2} \left (9 d+7 f x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 226
Rule 272
Rule 829
Rule 858
Rule 1191
Rule 1210
Rule 1212
Rule 1266
Rule 1847
Rubi steps
\begin {align*} \int \frac {\left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^{3/2}}{x} \, dx &=\int \left (\frac {\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 \frac {\left (c+e x^2\right ) \left (a+b x^4\right )^{3/2}}{x} \, dx+\int \left (d+f x^2\right ) \left (a+b x^4\right )^{3/2} \, dx\\ &=\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{21} \int \left (18 a d+14 a f x^2\right ) \sqrt {a+b x^4} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {(c+e x) \left (a+b x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{315} \int \frac {180 a^2 d+84 a^2 f x^2}{\sqrt {a+b x^4}} \, dx+\frac {\text {Subst}\left (\int \frac {(4 a b c+3 a b e x) \sqrt {a+b x^2}}{x} \, dx,x,x^2\right )}{8 b}\\ &=\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {\text {Subst}\left (\int \frac {8 a^2 b^2 c+3 a^2 b^2 e x}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )}{16 b^2}-\frac {\left (4 a^{5/2} f\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 d+\frac {7 \sqrt {a} f}{\sqrt {b}}\right )\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac {4 a^{9/4} f \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} d+7 \sqrt {a} f\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}{2} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,x^2\right )+\frac {1}{16} \left (3 a^2 e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^2\right )\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}-\frac {4 a^{9/4} f \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} d+7 \sqrt {a} f\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}{4} \left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^4\right )+\frac {1}{16} \left (3 a^2 e\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^2}{\sqrt {a+b x^4}}\right )\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {4 a^{9/4} f \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} d+7 \sqrt {a} f\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 {\left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^4}\right )}{2 b}\\ &=\frac {4 a^2 f x \sqrt {a+b x^4}}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}+\frac {1}{16} a \left (8 c+3 e x^2\right ) \sqrt {a+b x^4}+\frac {2}{105} a x \left (15 d+7 f x^2\right ) \sqrt {a+b x^4}+\frac {1}{24} \left (4 c+3 e x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {1}{63} x \left (9 d+7 f x^2\right ) \left (a+b x^4\right )^{3/2}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )-\frac {4 a^{9/4} f \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} d+7 \sqrt {a} f\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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.46, size = 319, normalized size = 0.79 \begin {gather*} \frac {\sqrt {a+b x^4} \left (10 b x^4 (84 c+x (72 d+7 x (9 e+8 f x)))+a (3360 c+x (2160 d+7 x (225 e+176 f x)))\right )}{5040}+\frac {3 a^2 e \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a+b x^4}}\right )}{16 \sqrt {b}}-\frac {1}{2} a^{3/2} c \tanh ^{-1}\left (\frac {\sqrt {a+b x^4}}{\sqrt {a}}\right )+\frac {4 i a^2 f \sqrt {1+\frac {b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )\right )}{15 \left (\frac {i \sqrt {b}}{\sqrt {a}}\right )^{3/2} \sqrt {a+b x^4}}-\frac {4 i a^2 d \sqrt {1+\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.37, size = 352, normalized size = 0.87
method | result | size |
elliptic | \(\frac {b f \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {b e \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {b d \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {b c \,x^{4} \sqrt {b \,x^{4}+a}}{6}+\frac {11 a f \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {5 a e \,x^{2} \sqrt {b \,x^{4}+a}}{16}+\frac {3 a d x \sqrt {b \,x^{4}+a}}{7}+\frac {2 a c \sqrt {b \,x^{4}+a}}{3}+\frac {4 a^{2} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {3 a^{2} e \ln \left (2 x^{2} \sqrt {b}+2 \sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}+\frac {4 i a^{\frac {5}{2}} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}-\frac {a^{\frac {3}{2}} c \arctanh \left (\frac {\sqrt {a}}{\sqrt {b \,x^{4}+a}}\right )}{2}\) | \(346\) |
default | \(f \left (\frac {b \,x^{7} \sqrt {b \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {b \,x^{4}+a}}{45}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )+e \left (\frac {b \,x^{6} \sqrt {b \,x^{4}+a}}{8}+\frac {5 a \,x^{2} \sqrt {b \,x^{4}+a}}{16}+\frac {3 a^{2} \ln \left (x^{2} \sqrt {b}+\sqrt {b \,x^{4}+a}\right )}{16 \sqrt {b}}\right )+d \left (\frac {b \,x^{5} \sqrt {b \,x^{4}+a}}{7}+\frac {3 a x \sqrt {b \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+c \left (\frac {b \,x^{4} \sqrt {b \,x^{4}+a}}{6}+\frac {2 a \sqrt {b \,x^{4}+a}}{3}-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{4}+a}}{x^{2}}\right )}{2}\right )\) | \(352\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.25, size = 59, normalized size = 0.15 \begin {gather*} {\rm integral}\left (\frac {{\left (b f x^{7} + b e x^{6} + b d x^{5} + b c x^{4} + a f x^{3} + a e x^{2} + a d x + a c\right )} \sqrt {b x^{4} + a}}{x}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.00, size = 405, normalized size = 1.00 \begin {gather*} - \frac {a^{\frac {3}{2}} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2} + \frac {a^{\frac {3}{2}} d 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}} e x^{2} \sqrt {1 + \frac {b x^{4}}{a}}}{4} + \frac {a^{\frac {3}{2}} e x^{2}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {a^{\frac {3}{2}} f 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 d 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 e x^{6}}{16 \sqrt {1 + \frac {b x^{4}}{a}}} + \frac {\sqrt {a} b f 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 {a^{2} c}{2 \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{4}} + 1}} + \frac {3 a^{2} e \operatorname {asinh}{\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {a \sqrt {b} c x^{2}}{2 \sqrt {\frac {a}{b x^{4}} + 1}} + b c \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 ) + \frac {b^{2} e x^{10}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{4}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{3/2}\,\left (f\,x^3+e\,x^2+d\,x+c\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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