Optimal. Leaf size=289 \[ \frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac {(5 b B-A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}-\frac {(5 b B-A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}+\frac {(5 b B-A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{3/4} c^{9/4}}-\frac {(5 b B-A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{3/4} c^{9/4}} \]
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Rubi [A]
time = 0.15, antiderivative size = 289, 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 = {1598, 468,
327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {(5 b B-A c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}-\frac {(5 b B-A c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}+\frac {(5 b B-A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{3/4} c^{9/4}}-\frac {(5 b B-A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{3/4} c^{9/4}}+\frac {\sqrt {x} (5 b B-A c)}{2 b c^2}-\frac {x^{5/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 327
Rule 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{11/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac {\left (\frac {5 b B}{2}-\frac {A c}{2}\right ) \int \frac {x^{3/2}}{b+c x^2} \, dx}{2 b c}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac {(5 b B-A c) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{4 c^2}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac {(5 b B-A c) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{2 c^2}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac {(5 b B-A c) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {b} c^2}-\frac {(5 b B-A c) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {b} c^2}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}-\frac {(5 b B-A c) \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 )}{8 \sqrt {b} c^{5/2}}-\frac {(5 b B-A c) \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 )}{8 \sqrt {b} c^{5/2}}+\frac {(5 b B-A c) \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 )}{8 \sqrt {2} b^{3/4} c^{9/4}}+\frac {(5 b B-A c) \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 )}{8 \sqrt {2} b^{3/4} c^{9/4}}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac {(5 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^{3/4} c^{9/4}}-\frac {(5 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^{3/4} c^{9/4}}-\frac {(5 b B-A c) \text {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^{3/4} c^{9/4}}+\frac {(5 b B-A c) \text {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^{3/4} c^{9/4}}\\ &=\frac {(5 b B-A c) \sqrt {x}}{2 b c^2}-\frac {(b B-A c) x^{5/2}}{2 b c \left (b+c x^2\right )}+\frac {(5 b B-A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}-\frac {(5 b B-A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{3/4} c^{9/4}}+\frac {(5 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^{3/4} c^{9/4}}-\frac {(5 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^{3/4} c^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 161, normalized size = 0.56 \begin {gather*} \frac {\frac {4 \sqrt [4]{c} \sqrt {x} \left (5 b B-A c+4 B c x^2\right )}{b+c x^2}+\frac {\sqrt {2} (5 b B-A c) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{b^{3/4}}-\frac {\sqrt {2} (5 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{b^{3/4}}}{8 c^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 152, normalized size = 0.53
method | result | size |
derivativedivides | \(\frac {2 B \sqrt {x}}{c^{2}}+\frac {\frac {2 \left (-\frac {A c}{4}+\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (A c -5 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b}}{c^{2}}\) | \(152\) |
default | \(\frac {2 B \sqrt {x}}{c^{2}}+\frac {\frac {2 \left (-\frac {A c}{4}+\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (A c -5 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b}}{c^{2}}\) | \(152\) |
risch | \(\frac {2 B \sqrt {x}}{c^{2}}-\frac {\sqrt {x}\, A}{2 c \left (c \,x^{2}+b \right )}+\frac {\sqrt {x}\, B b}{2 c^{2} \left (c \,x^{2}+b \right )}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 c b}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 c b}+\frac {\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{16 c b}-\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{8 c^{2}}-\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{8 c^{2}}-\frac {5 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )}{16 c^{2}}\) | \(323\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 250, normalized size = 0.87 \begin {gather*} \frac {{\left (B b - A c\right )} \sqrt {x}}{2 \, {\left (c^{3} x^{2} + b c^{2}\right )}} + \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, B b - 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 - 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 - 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 - 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}}}}{16 \, c^{2}} \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. 725 vs.
\(2 (209) = 418\).
time = 1.92, size = 725, normalized size = 2.51 \begin {gather*} \frac {4 \, {\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {b^{2} c^{4} \sqrt {-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}} + {\left (25 \, B^{2} b^{2} - 10 \, A B b c + A^{2} c^{2}\right )} x} b^{2} c^{7} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {3}{4}} + {\left (5 \, B b^{3} c^{7} - A b^{2} c^{8}\right )} \sqrt {x} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {3}{4}}}{625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}\right ) + {\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (b c^{2} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B b - A c\right )} \sqrt {x}\right ) - {\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {1}{4}} \log \left (-b c^{2} \left (-\frac {625 \, B^{4} b^{4} - 500 \, A B^{3} b^{3} c + 150 \, A^{2} B^{2} b^{2} c^{2} - 20 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{3} c^{9}}\right )^{\frac {1}{4}} - {\left (5 \, B b - A c\right )} \sqrt {x}\right ) + 4 \, {\left (4 \, B c x^{2} + 5 \, B b - A c\right )} \sqrt {x}}{8 \, {\left (c^{3} x^{2} + b c^{2}\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 = 0.45, size = 283, normalized size = 0.98 \begin {gather*} \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - \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 )}{8 \, b c^{3}} - \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - \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 )}{8 \, b c^{3}} - \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - \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 )}{16 \, b c^{3}} + \frac {\sqrt {2} {\left (5 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - \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 )}{16 \, b c^{3}} + \frac {B b \sqrt {x} - A c \sqrt {x}}{2 \, {\left (c x^{2} + b\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 744, normalized size = 2.57 \begin {gather*} \frac {2\,B\,\sqrt {x}}{c^2}-\frac {\sqrt {x}\,\left (\frac {A\,c}{2}-\frac {B\,b}{2}\right )}{c^3\,x^2+b\,c^2}+\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}-\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}+\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}+\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}}{\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}-\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}-\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}+\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}}\right )\,\left (A\,c-5\,B\,b\right )\,1{}\mathrm {i}}{4\,{\left (-b\right )}^{3/4}\,c^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}-\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}+\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}+\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}}{\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}-\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}-\frac {\left (A\,c-5\,B\,b\right )\,\left (\frac {\sqrt {x}\,\left (A^2\,c^2-10\,A\,B\,b\,c+25\,B^2\,b^2\right )}{c}+\frac {\left (A\,c-5\,B\,b\right )\,\left (8\,A\,b\,c^2-40\,B\,b^2\,c\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{3/4}\,c^{9/4}}}\right )\,\left (A\,c-5\,B\,b\right )}{4\,{\left (-b\right )}^{3/4}\,c^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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