Optimal. Leaf size=261 \[ \frac {(A c+3 b B) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(A c+3 b B) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(A c+3 b B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(A c+3 b B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}-\frac {x^{3/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1584, 457, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac {(A c+3 b B) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(A c+3 b B) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(A c+3 b B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(A c+3 b B) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}-\frac {x^{3/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
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 )^2} \, dx &=\int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}+\frac {\left (\frac {3 b B}{2}+\frac {A c}{2}\right ) \int \frac {\sqrt {x}}{b+c x^2} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}+\frac {\left (\frac {3 b B}{2}+\frac {A c}{2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b c}\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}-\frac {(3 b B+A c) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b c^{3/2}}+\frac {(3 b B+A c) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b c^{3/2}}\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}+\frac {(3 b B+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 c^2}+\frac {(3 b B+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 c^2}+\frac {(3 b B+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 )}{8 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(3 b B+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 )}{8 \sqrt {2} b^{5/4} c^{7/4}}\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}+\frac {(3 b B+A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(3 b B+A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(3 b B+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 )}{4 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(3 b B+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 )}{4 \sqrt {2} b^{5/4} c^{7/4}}\\ &=-\frac {(b B-A c) x^{3/2}}{2 b c \left (b+c x^2\right )}-\frac {(3 b B+A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(3 b B+A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{5/4} c^{7/4}}+\frac {(3 b B+A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}-\frac {(3 b B+A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{5/4} c^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 95, normalized size = 0.36 \[ \frac {2 x^{3/2} (A c-b B) \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-\frac {c x^2}{b}\right )}{3 b^2 c}+\frac {B \left (\tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{-b}}\right )+\tanh ^{-1}\left (\frac {b \sqrt [4]{c} \sqrt {x}}{(-b)^{5/4}}\right )\right )}{\sqrt [4]{-b} c^{7/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 912, normalized size = 3.49 \[ -\frac {4 \, {\left (B b - A c\right )} x^{\frac {3}{2}} + 4 \, {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {{\left (729 \, B^{6} b^{6} + 1458 \, A B^{5} b^{5} c + 1215 \, A^{2} B^{4} b^{4} c^{2} + 540 \, A^{3} B^{3} b^{3} c^{3} + 135 \, A^{4} B^{2} b^{2} c^{4} + 18 \, A^{5} B b c^{5} + A^{6} c^{6}\right )} x - {\left (81 \, B^{4} b^{7} c^{3} + 108 \, A B^{3} b^{6} c^{4} + 54 \, A^{2} B^{2} b^{5} c^{5} + 12 \, A^{3} B b^{4} c^{6} + A^{4} b^{3} c^{7}\right )} \sqrt {-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}}} b c^{2} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {1}{4}} - {\left (27 \, B^{3} b^{4} c^{2} + 27 \, A B^{2} b^{3} c^{3} + 9 \, A^{2} B b^{2} c^{4} + A^{3} b c^{5}\right )} \sqrt {x} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {1}{4}}}{81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}\right ) - {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {1}{4}} \log \left (b^{4} c^{5} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} b^{3} + 27 \, A B^{2} b^{2} c + 9 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right ) + {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {1}{4}} \log \left (-b^{4} c^{5} \left (-\frac {81 \, B^{4} b^{4} + 108 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 12 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{7}}\right )^{\frac {3}{4}} + {\left (27 \, B^{3} b^{3} + 27 \, A B^{2} b^{2} c + 9 \, A^{2} B b c^{2} + A^{3} c^{3}\right )} \sqrt {x}\right )}{8 \, {\left (b c^{2} x^{2} + b^{2} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 273, normalized size = 1.05 \[ -\frac {B b x^{\frac {3}{2}} - A c x^{\frac {3}{2}}}{2 \, {\left (c x^{2} + b\right )} b c} + \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{2} c^{4}} + \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{2} c^{4}} - \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{2} c^{4}} + \frac {\sqrt {2} {\left (3 \, \left (b c^{3}\right )^{\frac {3}{4}} B b + \left (b c^{3}\right )^{\frac {3}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{2} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 305, normalized size = 1.17 \[ \frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b c}+\frac {\sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} b c}+\frac {\sqrt {2}\, A \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{16 \left (\frac {b}{c}\right )^{\frac {1}{4}} b c}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {3 \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {b}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {3 \sqrt {2}\, B \ln \left (\frac {x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{16 \left (\frac {b}{c}\right )^{\frac {1}{4}} c^{2}}+\frac {\left (A c -b B \right ) x^{\frac {3}{2}}}{2 \left (c \,x^{2}+b \right ) b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.11, size = 217, normalized size = 0.83 \[ -\frac {{\left (B b - A c\right )} x^{\frac {3}{2}}}{2 \, {\left (b c^{2} x^{2} + b^{2} c\right )}} + \frac {{\left (3 \, B b + A c\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {\sqrt {b} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {1}{4}} c^{\frac {3}{4}}}\right )}}{16 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 91, normalized size = 0.35 \[ \frac {\mathrm {atanh}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (A\,c+3\,B\,b\right )}{4\,{\left (-b\right )}^{5/4}\,c^{7/4}}-\frac {\mathrm {atan}\left (\frac {c^{1/4}\,\sqrt {x}}{{\left (-b\right )}^{1/4}}\right )\,\left (A\,c+3\,B\,b\right )}{4\,{\left (-b\right )}^{5/4}\,c^{7/4}}+\frac {x^{3/2}\,\left (A\,c-B\,b\right )}{2\,b\,c\,\left (c\,x^2+b\right )} \]
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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