Optimal. Leaf size=332 \[ \frac {c^{5/4} (9 b B-13 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{17/4}}+\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{17/4}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \]
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Rubi [A] time = 0.29, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1584, 457, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {c^{5/4} (9 b B-13 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{17/4}}+\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{17/4}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 325
Rule 329
Rule 457
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1584
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^{3/2} \left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x^{11/2} \left (b+c x^2\right )^2} \, dx\\ &=-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {\left (-\frac {9 b B}{2}+\frac {13 A c}{2}\right ) \int \frac {1}{x^{11/2} \left (b+c x^2\right )} \, dx}{2 b c}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {(9 b B-13 A c) \int \frac {1}{x^{7/2} \left (b+c x^2\right )} \, dx}{4 b^2}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac {(c (9 b B-13 A c)) \int \frac {1}{x^{3/2} \left (b+c x^2\right )} \, dx}{4 b^3}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {\left (c^2 (9 b B-13 A c)\right ) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{4 b^4}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {\left (c^2 (9 b B-13 A c)\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{2 b^4}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac {\left (c^{3/2} (9 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 )}{4 b^4}+\frac {\left (c^{3/2} (9 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 )}{4 b^4}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {(c (9 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 )}{8 b^4}+\frac {(c (9 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 )}{8 b^4}+\frac {\left (c^{5/4} (9 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 )}{8 \sqrt {2} b^{17/4}}+\frac {\left (c^{5/4} (9 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 )}{8 \sqrt {2} b^{17/4}}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}+\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}+\frac {\left (c^{5/4} (9 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 )}{4 \sqrt {2} b^{17/4}}-\frac {\left (c^{5/4} (9 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 )}{4 \sqrt {2} b^{17/4}}\\ &=\frac {9 b B-13 A c}{18 b^2 c x^{9/2}}-\frac {9 b B-13 A c}{10 b^3 x^{5/2}}+\frac {c (9 b B-13 A c)}{2 b^4 \sqrt {x}}-\frac {b B-A c}{2 b c x^{9/2} \left (b+c x^2\right )}-\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{17/4}}+\frac {c^{5/4} (9 b B-13 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{17/4}}+\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}-\frac {c^{5/4} (9 b B-13 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{17/4}}\\ \end {align*}
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Mathematica [C] time = 0.59, size = 176, normalized size = 0.53 \begin {gather*} \frac {2 c^2 x^{3/2} (b B-A c) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^5}+\frac {2 c (2 b B-3 A c)}{b^4 \sqrt {x}}-\frac {2 (b B-2 A c)}{5 b^3 x^{5/2}}-\frac {2 A}{9 b^2 x^{9/2}}+\frac {c^{5/4} (2 b B-3 A c) \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )}{(-b)^{17/4}}+\frac {c^{5/4} (3 A c-2 b B) \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 = 0.73, size = 224, normalized size = 0.67 \begin {gather*} -\frac {\left (9 b B c^{5/4}-13 A c^{9/4}\right ) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{4 \sqrt {2} b^{17/4}}-\frac {\left (9 b B c^{5/4}-13 A c^{9/4}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{4 \sqrt {2} b^{17/4}}+\frac {-20 A b^3+52 A b^2 c x^2-468 A b c^2 x^4-585 A c^3 x^6-36 b^3 B x^2+324 b^2 B c x^4+405 b B c^2 x^6}{90 b^4 x^{9/2} \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1024, normalized size = 3.08 \begin {gather*} \frac {180 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (531441 \, B^{6} b^{6} c^{8} - 4605822 \, A B^{5} b^{5} c^{9} + 16632135 \, A^{2} B^{4} b^{4} c^{10} - 32032260 \, A^{3} B^{3} b^{3} c^{11} + 34701615 \, A^{4} B^{2} b^{2} c^{12} - 20049822 \, A^{5} B b c^{13} + 4826809 \, A^{6} c^{14}\right )} x - {\left (6561 \, B^{4} b^{13} c^{5} - 37908 \, A B^{3} b^{12} c^{6} + 82134 \, A^{2} B^{2} b^{11} c^{7} - 79092 \, A^{3} B b^{10} c^{8} + 28561 \, A^{4} b^{9} c^{9}\right )} \sqrt {-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}}} b^{4} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {1}{4}} + {\left (729 \, B^{3} b^{7} c^{4} - 3159 \, A B^{2} b^{6} c^{5} + 4563 \, A^{2} B b^{5} c^{6} - 2197 \, A^{3} b^{4} c^{7}\right )} \sqrt {x} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {1}{4}}}{6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}\right ) - 45 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {1}{4}} \log \left (b^{13} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {3}{4}} - {\left (729 \, B^{3} b^{3} c^{4} - 3159 \, A B^{2} b^{2} c^{5} + 4563 \, A^{2} B b c^{6} - 2197 \, A^{3} c^{7}\right )} \sqrt {x}\right ) + 45 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {1}{4}} \log \left (-b^{13} \left (-\frac {6561 \, B^{4} b^{4} c^{5} - 37908 \, A B^{3} b^{3} c^{6} + 82134 \, A^{2} B^{2} b^{2} c^{7} - 79092 \, A^{3} B b c^{8} + 28561 \, A^{4} c^{9}}{b^{17}}\right )^{\frac {3}{4}} - {\left (729 \, B^{3} b^{3} c^{4} - 3159 \, A B^{2} b^{2} c^{5} + 4563 \, A^{2} B b c^{6} - 2197 \, A^{3} c^{7}\right )} \sqrt {x}\right ) + 4 \, {\left (45 \, {\left (9 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 36 \, {\left (9 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} - 20 \, A b^{3} - 4 \, {\left (9 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x}}{360 \, {\left (b^{4} c x^{7} + b^{5} x^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 328, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} {\left (9 \, \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 )}{8 \, b^{5} c} + \frac {\sqrt {2} {\left (9 \, \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 )}{8 \, b^{5} c} - \frac {\sqrt {2} {\left (9 \, \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 )}{16 \, b^{5} c} + \frac {\sqrt {2} {\left (9 \, \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 )}{16 \, b^{5} c} + \frac {B b c^{2} x^{\frac {3}{2}} - A c^{3} x^{\frac {3}{2}}}{2 \, {\left (c x^{2} + b\right )} b^{4}} + \frac {2 \, {\left (90 \, B b c x^{4} - 135 \, A c^{2} x^{4} - 9 \, B b^{2} x^{2} + 18 \, A b c x^{2} - 5 \, A b^{2}\right )}}{45 \, b^{4} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 372, normalized size = 1.12 \begin {gather*} -\frac {A \,c^{3} x^{\frac {3}{2}}}{2 \left (c \,x^{2}+b \right ) b^{4}}+\frac {B \,c^{2} x^{\frac {3}{2}}}{2 \left (c \,x^{2}+b \right ) b^{3}}-\frac {13 \sqrt {2}\, A \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}-\frac {13 \sqrt {2}\, A \,c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}-\frac {13 \sqrt {2}\, A \,c^{2} \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 )}{16 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{4}}+\frac {9 \sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {9 \sqrt {2}\, B c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}+\frac {9 \sqrt {2}\, B 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 )}{16 \left (\frac {b}{c}\right )^{\frac {1}{4}} b^{3}}-\frac {6 A \,c^{2}}{b^{4} \sqrt {x}}+\frac {4 B c}{b^{3} \sqrt {x}}+\frac {4 A c}{5 b^{3} x^{\frac {5}{2}}}-\frac {2 B}{5 b^{2} x^{\frac {5}{2}}}-\frac {2 A}{9 b^{2} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 276, normalized size = 0.83 \begin {gather*} \frac {45 \, {\left (9 \, B b c^{2} - 13 \, A c^{3}\right )} x^{6} + 36 \, {\left (9 \, B b^{2} c - 13 \, A b c^{2}\right )} x^{4} - 20 \, A b^{3} - 4 \, {\left (9 \, B b^{3} - 13 \, A b^{2} c\right )} x^{2}}{90 \, {\left (b^{4} c x^{\frac {13}{2}} + b^{5} x^{\frac {9}{2}}\right )}} + \frac {{\left (9 \, B b c^{2} - 13 \, A c^{3}\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 )}}{16 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 142, normalized size = 0.43 \begin {gather*} \frac {{\left (-c\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-9\,B\,b\right )}{4\,b^{17/4}}-\frac {\frac {2\,A}{9\,b}-\frac {2\,x^2\,\left (13\,A\,c-9\,B\,b\right )}{45\,b^2}+\frac {c^2\,x^6\,\left (13\,A\,c-9\,B\,b\right )}{2\,b^4}+\frac {2\,c\,x^4\,\left (13\,A\,c-9\,B\,b\right )}{5\,b^3}}{b\,x^{9/2}+c\,x^{13/2}}-\frac {{\left (-c\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-c\right )}^{1/4}\,\sqrt {x}}{b^{1/4}}\right )\,\left (13\,A\,c-9\,B\,b\right )}{4\,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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