Optimal. Leaf size=343 \[ -\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}+\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.28, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \begin {gather*} -\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} b^{17/4}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2} \end {gather*}
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^{5/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x^{7/2} \left (b+c x^2\right )^3} \, dx\\ &=-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}+\frac {\left (-\frac {5 b B}{2}+\frac {13 A c}{2}\right ) \int \frac {1}{x^{7/2} \left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 (5 b B-13 A c)) \int \frac {1}{x^{7/2} \left (b+c x^2\right )} \, dx}{32 b^2 c}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {(9 (5 b B-13 A c)) \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{32 b^3}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 c (5 b B-13 A c)) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{32 b^4}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 c (5 b B-13 A c)) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 b^4}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {\left (9 \sqrt {c} (5 b B-13 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4}-\frac {\left (9 \sqrt {c} (5 b B-13 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 b^4}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {(9 (5 b B-13 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^4}-\frac {(9 (5 b B-13 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^4}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \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^{17/4}}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \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^{17/4}}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}-\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \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^{17/4}}+\frac {\left (9 \sqrt [4]{c} (5 b B-13 A c)\right ) \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^{17/4}}\\ &=\frac {9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac {9 (5 b B-13 A c)}{16 b^4 \sqrt {x}}-\frac {b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2}-\frac {5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{17/4}}-\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}+\frac {9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} b^{17/4}}\\ \end {align*}
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Mathematica [C] time = 0.52, size = 189, normalized size = 0.55 \begin {gather*} -\frac {2 c x^{3/2} (b B-2 A c) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^5}+\frac {2 c x^{3/2} (A c-b B) \, _2F_1\left (\frac {3}{4},3;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^5}+\frac {6 A c-2 b B}{b^4 \sqrt {x}}-\frac {2 A}{5 b^3 x^{5/2}}+\frac {\sqrt [4]{c} (3 A c-b B) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}}+\frac {\sqrt [4]{c} (b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 224, normalized size = 0.65 \begin {gather*} \frac {9 \left (5 b B \sqrt [4]{c}-13 A c^{5/4}\right ) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{32 \sqrt {2} b^{17/4}}+\frac {9 \left (5 b B \sqrt [4]{c}-13 A c^{5/4}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{17/4}}+\frac {-32 A b^3+416 A b^2 c x^2+1053 A b c^2 x^4+585 A c^3 x^6-160 b^3 B x^2-405 b^2 B c x^4-225 b B c^2 x^6}{80 b^4 x^{5/2} \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 1043, normalized size = 3.04 \begin {gather*} -\frac {180 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (15625 \, B^{6} b^{6} c^{2} - 243750 \, A B^{5} b^{5} c^{3} + 1584375 \, A^{2} B^{4} b^{4} c^{4} - 5492500 \, A^{3} B^{3} b^{3} c^{5} + 10710375 \, A^{4} B^{2} b^{2} c^{6} - 11138790 \, A^{5} B b c^{7} + 4826809 \, A^{6} c^{8}\right )} x - {\left (625 \, B^{4} b^{13} c - 6500 \, A B^{3} b^{12} c^{2} + 25350 \, A^{2} B^{2} b^{11} c^{3} - 43940 \, A^{3} B b^{10} c^{4} + 28561 \, A^{4} b^{9} c^{5}\right )} \sqrt {-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}}} b^{4} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} + {\left (125 \, B^{3} b^{7} c - 975 \, A B^{2} b^{6} c^{2} + 2535 \, A^{2} B b^{5} c^{3} - 2197 \, A^{3} b^{4} c^{4}\right )} \sqrt {x} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}}}{625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}\right ) - 45 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (729 \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) + 45 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {1}{4}} \log \left (-729 \, b^{13} \left (-\frac {625 \, B^{4} b^{4} c - 6500 \, A B^{3} b^{3} c^{2} + 25350 \, A^{2} B^{2} b^{2} c^{3} - 43940 \, A^{3} B b c^{4} + 28561 \, A^{4} c^{5}}{b^{17}}\right )^{\frac {3}{4}} - 729 \, {\left (125 \, B^{3} b^{3} c - 975 \, A B^{2} b^{2} c^{2} + 2535 \, A^{2} B b c^{3} - 2197 \, A^{3} c^{4}\right )} \sqrt {x}\right ) + 4 \, {\left (45 \, {\left (5 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 81 \, {\left (5 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} + 32 \, A b^{3} + 32 \, {\left (5 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 326, normalized size = 0.95 \begin {gather*} -\frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \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^{5} c^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \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^{5} c^{2}} + \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \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^{5} c^{2}} - \frac {9 \, \sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {3}{4}} B b - 13 \, \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^{5} c^{2}} - \frac {13 \, B b c^{2} x^{\frac {7}{2}} - 21 \, A c^{3} x^{\frac {7}{2}} + 17 \, B b^{2} c x^{\frac {3}{2}} - 25 \, A b c^{2} x^{\frac {3}{2}}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} - \frac {2 \, {\left (5 \, B b x^{2} - 15 \, A c x^{2} + A b\right )}}{5 \, b^{4} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 381, normalized size = 1.11 \begin {gather*} \frac {21 A \,c^{3} x^{\frac {7}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{4}}-\frac {13 B \,c^{2} x^{\frac {7}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{3}}+\frac {25 A \,c^{2} x^{\frac {3}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{3}}-\frac {17 B c \,x^{\frac {3}{2}}}{16 \left (c \,x^{2}+b \right )^{2} b^{2}}+\frac {117 \sqrt {2}\, A c \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^{4}}+\frac {117 \sqrt {2}\, A c \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^{4}}+\frac {117 \sqrt {2}\, A c \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^{4}}-\frac {45 \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^{3}}-\frac {45 \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^{3}}-\frac {45 \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^{3}}+\frac {6 A c}{b^{4} \sqrt {x}}-\frac {2 B}{b^{3} \sqrt {x}}-\frac {2 A}{5 b^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 285, normalized size = 0.83 \begin {gather*} -\frac {45 \, {\left (5 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 81 \, {\left (5 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} + 32 \, A b^{3} + 32 \, {\left (5 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}}{80 \, {\left (b^{4} c^{2} x^{\frac {13}{2}} + 2 \, b^{5} c x^{\frac {9}{2}} + b^{6} x^{\frac {5}{2}}\right )}} - \frac {9 \, {\left (5 \, B b c - 13 \, A c^{2}\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^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 152, normalized size = 0.44 \begin {gather*} \frac {\frac {2\,x^2\,\left (13\,A\,c-5\,B\,b\right )}{5\,b^2}-\frac {2\,A}{5\,b}+\frac {9\,c^2\,x^6\,\left (13\,A\,c-5\,B\,b\right )}{16\,b^4}+\frac {81\,c\,x^4\,\left (13\,A\,c-5\,B\,b\right )}{80\,b^3}}{b^2\,x^{5/2}+c^2\,x^{13/2}+2\,b\,c\,x^{9/2}}+\frac {9\,{\left (-c\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-5\,B\,b\right )}{32\,b^{17/4}}-\frac {9\,{\left (-c\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-5\,B\,b\right )}{32\,b^{17/4}} \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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