Optimal. Leaf size=316 \[ \frac {5 (b B-9 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.26, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1584, 457, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {5 (b B-9 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}+\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 290
Rule 297
Rule 325
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x^{3/2} \left (b+c x^2\right )^3} \, dx\\ &=-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}+\frac {\left (-\frac {b B}{2}+\frac {9 A c}{2}\right ) \int \frac {1}{x^{3/2} \left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}-\frac {(5 (b B-9 A c)) \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{32 b^2 c}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}+\frac {(5 (b B-9 A c)) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 b^3}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^3}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}-\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^3 \sqrt {c}}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^3 \sqrt {c}}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^3 c}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 b^3 c}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}+\frac {5 (b B-9 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}+\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}-\frac {(5 (b B-9 A c)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}\\ &=\frac {5 (b B-9 A c)}{16 b^3 c \sqrt {x}}-\frac {b B-A c}{4 b c \sqrt {x} \left (b+c x^2\right )^2}-\frac {b B-9 A c}{16 b^2 c \sqrt {x} \left (b+c x^2\right )}-\frac {5 (b B-9 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{13/4} c^{3/4}}+\frac {5 (b B-9 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}-\frac {5 (b B-9 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{13/4} c^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 147, normalized size = 0.47 \[ \frac {2 x^{3/2} (b B-A c) \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^4}-\frac {2 A c x^{3/2} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^4}-\frac {2 A}{b^3 \sqrt {x}}+\frac {A \sqrt [4]{c} \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )}{(-b)^{13/4}}+\frac {A b \sqrt [4]{c} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 988, normalized size = 3.13 \[ \frac {20 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (B^{6} b^{6} - 54 \, A B^{5} b^{5} c + 1215 \, A^{2} B^{4} b^{4} c^{2} - 14580 \, A^{3} B^{3} b^{3} c^{3} + 98415 \, A^{4} B^{2} b^{2} c^{4} - 354294 \, A^{5} B b c^{5} + 531441 \, A^{6} c^{6}\right )} x - {\left (B^{4} b^{11} c - 36 \, A B^{3} b^{10} c^{2} + 486 \, A^{2} B^{2} b^{9} c^{3} - 2916 \, A^{3} B b^{8} c^{4} + 6561 \, A^{4} b^{7} c^{5}\right )} \sqrt {-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}}} b^{3} c \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} + {\left (B^{3} b^{6} c - 27 \, A B^{2} b^{5} c^{2} + 243 \, A^{2} B b^{4} c^{3} - 729 \, A^{3} b^{3} c^{4}\right )} \sqrt {x} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}}}{B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}\right ) - 5 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (125 \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) + 5 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {1}{4}} \log \left (-125 \, b^{10} c^{2} \left (-\frac {B^{4} b^{4} - 36 \, A B^{3} b^{3} c + 486 \, A^{2} B^{2} b^{2} c^{2} - 2916 \, A^{3} B b c^{3} + 6561 \, A^{4} c^{4}}{b^{13} c^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} b^{3} - 27 \, A B^{2} b^{2} c + 243 \, A^{2} B b c^{2} - 729 \, A^{3} c^{3}\right )} \sqrt {x}\right ) + 4 \, {\left (5 \, {\left (B b c - 9 \, A c^{2}\right )} x^{4} - 32 \, A b^{2} + 9 \, {\left (B b^{2} - 9 \, A b c\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 300, normalized size = 0.95 \[ -\frac {2 \, A}{b^{3} \sqrt {x}} + \frac {5 \, B b c x^{\frac {7}{2}} - 13 \, A c^{2} x^{\frac {7}{2}} + 9 \, B b^{2} x^{\frac {3}{2}} - 17 \, A b c x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{4} c^{3}} - \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{3}} + \frac {5 \, \sqrt {2} {\left (\left (b c^{3}\right )^{\frac {3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{4} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 363, normalized size = 1.15 \[ -\frac {13 A \,c^{2} x^{\frac {7}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{3}}+\frac {5 B c \,x^{\frac {7}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{2}}-\frac {17 A c \,x^{\frac {3}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{2}}+\frac {9 B \,x^{\frac {3}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b}-\frac {45 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {45 \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {45 \sqrt {2}\, A \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {5 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{2} c}+\frac {5 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{2} c}+\frac {5 \sqrt {2}\, B \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{2} c}-\frac {2 A}{b^{3} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 255, normalized size = 0.81 \[ \frac {5 \, {\left (B b c - 9 \, A c^{2}\right )} x^{4} - 32 \, A b^{2} + 9 \, {\left (B b^{2} - 9 \, A b c\right )} x^{2}}{16 \, {\left (b^{3} c^{2} x^{\frac {9}{2}} + 2 \, b^{4} c x^{\frac {5}{2}} + b^{5} \sqrt {x}\right )}} + \frac {5 \, {\left (B b - 9 \, A c\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{128 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 133, normalized size = 0.42 \[ \frac {5\,\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (9\,A\,c-B\,b\right )}{32\,{\left (-b\right )}^{13/4}\,c^{3/4}}-\frac {\frac {2\,A}{b}+\frac {9\,x^2\,\left (9\,A\,c-B\,b\right )}{16\,b^2}+\frac {5\,c\,x^4\,\left (9\,A\,c-B\,b\right )}{16\,b^3}}{b^2\,\sqrt {x}+c^2\,x^{9/2}+2\,b\,c\,x^{5/2}}-\frac {5\,\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (9\,A\,c-B\,b\right )}{32\,{\left (-b\right )}^{13/4}\,c^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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