Optimal. Leaf size=343 \[ -\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}} \]
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Rubi [A]
time = 0.19, 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 = {1598, 468,
294, 327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {9 \sqrt [4]{b} (13 b B-5 A c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}-\frac {9 \sqrt {x} (13 b B-5 A c)}{16 c^4}+\frac {9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac {x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac {x^{13/2} (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 210
Rule 217
Rule 294
Rule 327
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{23/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^{11/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}+\frac {\left (\frac {13 b B}{2}-\frac {5 A c}{2}\right ) \int \frac {x^{11/2}}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 (13 b B-5 A c)) \int \frac {x^{7/2}}{b+c x^2} \, dx}{32 b c^2}\\ &=\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {(9 (13 b B-5 A c)) \int \frac {x^{3/2}}{b+c x^2} \, dx}{32 c^3}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 b (13 b B-5 A c)) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 b (13 b B-5 A c)) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^4}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \text {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 c^{9/2}}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \text {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 c^{9/2}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \text {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} c^{17/4}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \text {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} c^{17/4}}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \text {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} c^{17/4}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \text {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} c^{17/4}}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 204, normalized size = 0.59 \begin {gather*} \frac {\frac {4 \sqrt [4]{c} \sqrt {x} \left (-585 b^3 B+b c^2 x^2 \left (405 A-416 B x^2\right )+9 b^2 c \left (25 A-117 B x^2\right )+32 c^3 x^4 \left (5 A+B x^2\right )\right )}{\left (b+c x^2\right )^2}-45 \sqrt {2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} (13 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{320 c^{17/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 191, normalized size = 0.56
method | result | size |
derivativedivides | \(\frac {\frac {2 B \,x^{\frac {5}{2}} c}{5}+2 A c \sqrt {x}-6 B b \sqrt {x}}{c^{4}}-\frac {2 b \left (\frac {\left (-\frac {17}{32} A \,c^{2}+\frac {25}{32} b B c \right ) x^{\frac {5}{2}}-\frac {b \left (13 A c -21 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {9 \left (5 A c -13 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{c^{4}}\) | \(191\) |
default | \(\frac {\frac {2 B \,x^{\frac {5}{2}} c}{5}+2 A c \sqrt {x}-6 B b \sqrt {x}}{c^{4}}-\frac {2 b \left (\frac {\left (-\frac {17}{32} A \,c^{2}+\frac {25}{32} b B c \right ) x^{\frac {5}{2}}-\frac {b \left (13 A c -21 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {9 \left (5 A c -13 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{c^{4}}\) | \(191\) |
risch | \(\frac {2 \left (B c \,x^{2}+5 A c -15 B b \right ) \sqrt {x}}{5 c^{4}}+\frac {17 b \,x^{\frac {5}{2}} A}{16 c^{2} \left (c \,x^{2}+b \right )^{2}}-\frac {25 b^{2} x^{\frac {5}{2}} B}{16 c^{3} \left (c \,x^{2}+b \right )^{2}}+\frac {13 b^{2} A \sqrt {x}}{16 c^{3} \left (c \,x^{2}+b \right )^{2}}-\frac {21 b^{3} B \sqrt {x}}{16 c^{4} \left (c \,x^{2}+b \right )^{2}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 c^{3}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 c^{3}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{128 c^{3}}+\frac {117 b \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 c^{4}}+\frac {117 b \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 c^{4}}+\frac {117 b \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{128 c^{4}}\) | \(376\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 306, normalized size = 0.89 \begin {gather*} -\frac {{\left (25 \, B b^{2} c - 17 \, A b c^{2}\right )} x^{\frac {5}{2}} + {\left (21 \, B b^{3} - 13 \, A b^{2} c\right )} \sqrt {x}}{16 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac {9 \, {\left (\frac {2 \, \sqrt {2} {\left (13 \, B b - 5 \, 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 (13 \, B b - 5 \, 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 (13 \, B b - 5 \, 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 (13 \, B b - 5 \, 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}}}\right )} b}{128 \, c^{4}} + \frac {2 \, {\left (B c x^{\frac {5}{2}} - 5 \, {\left (3 \, B b - A c\right )} \sqrt {x}\right )}}{5 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 817 vs.
\(2 (255) = 510\).
time = 2.47, size = 817, normalized size = 2.38 \begin {gather*} -\frac {180 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{8} \sqrt {-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}} + {\left (169 \, B^{2} b^{2} - 130 \, A B b c + 25 \, A^{2} c^{2}\right )} x} c^{13} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {3}{4}} + {\left (13 \, B b c^{13} - 5 \, A c^{14}\right )} \sqrt {x} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {3}{4}}}{28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}\right ) + 45 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \log \left (9 \, c^{4} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} - 9 \, {\left (13 \, B b - 5 \, A c\right )} \sqrt {x}\right ) - 45 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \log \left (-9 \, c^{4} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} - 9 \, {\left (13 \, B b - 5 \, A c\right )} \sqrt {x}\right ) - 4 \, {\left (32 \, B c^{3} x^{6} - 32 \, {\left (13 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} - 585 \, B b^{3} + 225 \, A b^{2} c - 81 \, {\left (13 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 0.47, size = 321, normalized size = 0.94 \begin {gather*} \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \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 \, c^{5}} + \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \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 \, c^{5}} + \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \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 \, c^{5}} - \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \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 \, c^{5}} - \frac {25 \, B b^{2} c x^{\frac {5}{2}} - 17 \, A b c^{2} x^{\frac {5}{2}} + 21 \, B b^{3} \sqrt {x} - 13 \, A b^{2} c \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} c^{4}} + \frac {2 \, {\left (B c^{12} x^{\frac {5}{2}} - 15 \, B b c^{11} \sqrt {x} + 5 \, A c^{12} \sqrt {x}\right )}}{5 \, c^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 865, normalized size = 2.52 \begin {gather*} \frac {x^{5/2}\,\left (\frac {17\,A\,b\,c^2}{16}-\frac {25\,B\,b^2\,c}{16}\right )-\sqrt {x}\,\left (\frac {21\,B\,b^3}{16}-\frac {13\,A\,b^2\,c}{16}\right )}{b^2\,c^4+2\,b\,c^5\,x^2+c^6\,x^4}+\sqrt {x}\,\left (\frac {2\,A}{c^3}-\frac {6\,B\,b}{c^4}\right )+\frac {2\,B\,x^{5/2}}{5\,c^3}+\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}-\frac {81\,{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )\,9{}\mathrm {i}}{64\,c^{17/4}}+\frac {{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}+\frac {81\,{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )\,9{}\mathrm {i}}{64\,c^{17/4}}}{\frac {9\,{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}-\frac {81\,{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )}{64\,c^{17/4}}-\frac {9\,{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}+\frac {81\,{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )}{64\,c^{17/4}}}\right )\,\left (5\,A\,c-13\,B\,b\right )\,9{}\mathrm {i}}{32\,c^{17/4}}+\frac {9\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {\frac {9\,{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}-\frac {{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )\,81{}\mathrm {i}}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )}{64\,c^{17/4}}+\frac {9\,{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}+\frac {{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )\,81{}\mathrm {i}}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )}{64\,c^{17/4}}}{\frac {{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}-\frac {{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )\,81{}\mathrm {i}}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )\,9{}\mathrm {i}}{64\,c^{17/4}}-\frac {{\left (-b\right )}^{1/4}\,\left (\frac {81\,\sqrt {x}\,\left (25\,A^2\,b^2\,c^2-130\,A\,B\,b^3\,c+169\,B^2\,b^4\right )}{64\,c^5}+\frac {{\left (-b\right )}^{1/4}\,\left (5\,A\,c-13\,B\,b\right )\,\left (13\,B\,b^3-5\,A\,b^2\,c\right )\,81{}\mathrm {i}}{64\,c^{21/4}}\right )\,\left (5\,A\,c-13\,B\,b\right )\,9{}\mathrm {i}}{64\,c^{17/4}}}\right )\,\left (5\,A\,c-13\,B\,b\right )}{32\,c^{17/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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