Optimal. Leaf size=298 \[ -\frac {(3 A c+5 b B) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(3 A c+5 b B) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}-\frac {(3 A c+5 b B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(3 A c+5 b B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}-\frac {\sqrt {x} (3 A c+5 b B)}{16 b c^2 \left (b+c x^2\right )}-\frac {x^{5/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.24, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1584, 457, 288, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {(3 A c+5 b B) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(3 A c+5 b B) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}-\frac {(3 A c+5 b B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(3 A c+5 b B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}-\frac {\sqrt {x} (3 A c+5 b B)}{16 b c^2 \left (b+c x^2\right )}-\frac {x^{5/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 288
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{15/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}+\frac {\left (\frac {5 b B}{2}+\frac {3 A c}{2}\right ) \int \frac {x^{3/2}}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}+\frac {(5 b B+3 A c) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 b c^2}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}+\frac {(5 b B+3 A c) \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b c^2}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}+\frac {(5 b B+3 A c) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{3/2} c^2}+\frac {(5 b B+3 A c) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^{3/2} c^2}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}+\frac {(5 b B+3 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^{3/2} c^{5/2}}+\frac {(5 b B+3 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^{3/2} c^{5/2}}-\frac {(5 b B+3 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^{7/4} c^{9/4}}-\frac {(5 b B+3 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^{7/4} c^{9/4}}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}-\frac {(5 b B+3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(5 b B+3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(5 b B+3 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^{7/4} c^{9/4}}-\frac {(5 b B+3 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^{7/4} c^{9/4}}\\ &=-\frac {(b B-A c) x^{5/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(5 b B+3 A c) \sqrt {x}}{16 b c^2 \left (b+c x^2\right )}-\frac {(5 b B+3 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(5 b B+3 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}-\frac {(5 b B+3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(5 b B+3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{7/4} c^{9/4}}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 389, normalized size = 1.31 \begin {gather*} \frac {-\frac {2 \sqrt {2} (3 A c+5 b B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{b^{7/4}}+\frac {2 \sqrt {2} (3 A c+5 b B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{b^{7/4}}-\frac {3 \sqrt {2} A c \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{7/4}}+\frac {3 \sqrt {2} A c \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{7/4}}+\frac {8 A c^{5/4} \sqrt {x}}{b^2+b c x^2}-\frac {32 A c^{5/4} \sqrt {x}}{\left (b+c x^2\right )^2}-\frac {5 \sqrt {2} B \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{3/4}}+\frac {5 \sqrt {2} B \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{b^{3/4}}-\frac {72 B \sqrt [4]{c} \sqrt {x}}{b+c x^2}+\frac {32 b B \sqrt [4]{c} \sqrt {x}}{\left (b+c x^2\right )^2}}{128 c^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.94, size = 191, normalized size = 0.64 \begin {gather*} -\frac {(3 A c+5 b B) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}+\frac {(3 A c+5 b B) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{7/4} c^{9/4}}+\frac {-3 A b c \sqrt {x}+A c^2 x^{5/2}-5 b^2 B \sqrt {x}-9 b B c x^{5/2}}{16 b c^2 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 806, normalized size = 2.70 \begin {gather*} \frac {4 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {b^{4} c^{4} \sqrt {-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}} + {\left (25 \, B^{2} b^{2} + 30 \, A B b c + 9 \, A^{2} c^{2}\right )} x} b^{5} c^{7} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {3}{4}} - {\left (5 \, B b^{6} c^{7} + 3 \, A b^{5} c^{8}\right )} \sqrt {x} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {3}{4}}}{625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}\right ) + {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {1}{4}} \log \left (b^{2} c^{2} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B b + 3 \, A c\right )} \sqrt {x}\right ) - {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {1}{4}} \log \left (-b^{2} c^{2} \left (-\frac {625 \, B^{4} b^{4} + 1500 \, A B^{3} b^{3} c + 1350 \, A^{2} B^{2} b^{2} c^{2} + 540 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{9}}\right )^{\frac {1}{4}} + {\left (5 \, B b + 3 \, A c\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, B b^{2} + 3 \, A b c + {\left (9 \, B b c - A c^{2}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 298, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac {1}{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^{2} c^{3}} + \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac {1}{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^{2} c^{3}} + \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac {1}{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^{2} c^{3}} - \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \left (b c^{3}\right )^{\frac {1}{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^{2} c^{3}} - \frac {9 \, B b c x^{\frac {5}{2}} - A c^{2} x^{\frac {5}{2}} + 5 \, B b^{2} \sqrt {x} + 3 \, A b c \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 334, normalized size = 1.12 \begin {gather*} \frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 b^{2} c}+\frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 b^{2} c}+\frac {3 \left (\frac {b}{c}\right )^{\frac {1}{4}} \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 b^{2} c}+\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 b \,c^{2}}+\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 b \,c^{2}}+\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \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 b \,c^{2}}+\frac {\frac {\left (A c -9 b B \right ) x^{\frac {5}{2}}}{16 b c}-\frac {\left (3 A c +5 b B \right ) \sqrt {x}}{16 c^{2}}}{\left (c \,x^{2}+b \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 280, normalized size = 0.94 \begin {gather*} -\frac {{\left (9 \, B b c - A c^{2}\right )} x^{\frac {5}{2}} + {\left (5 \, B b^{2} + 3 \, A b c\right )} \sqrt {x}}{16 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, B b + 3 \, A c\right )} \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 {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (5 \, B b + 3 \, A c\right )} \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 {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (5 \, B b + 3 \, A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, B b + 3 \, A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}}{128 \, b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 799, normalized size = 2.68 \begin {gather*} -\frac {\frac {\sqrt {x}\,\left (3\,A\,c+5\,B\,b\right )}{16\,c^2}-\frac {x^{5/2}\,\left (A\,c-9\,B\,b\right )}{16\,b\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}-\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}+\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}+\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}}{\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}-\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}-\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}+\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}}\right )\,\left (3\,A\,c+5\,B\,b\right )\,1{}\mathrm {i}}{32\,{\left (-b\right )}^{7/4}\,c^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}-\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}+\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}+\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}}{\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}-\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}-\frac {\left (3\,A\,c+5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^2+30\,A\,B\,b\,c+25\,B^2\,b^2\right )}{64\,b^2\,c}+\frac {\left (3\,A\,c^2+5\,B\,b\,c\right )\,\left (3\,A\,c+5\,B\,b\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{64\,{\left (-b\right )}^{7/4}\,c^{9/4}}}\right )\,\left (3\,A\,c+5\,B\,b\right )}{32\,{\left (-b\right )}^{7/4}\,c^{9/4}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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