Optimal. Leaf size=261 \[ -\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}-\frac {(b B+3 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}-\frac {(b B+3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}} \]
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Rubi [A]
time = 0.13, 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 = {1598, 468,
335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {(3 A c+b B) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(3 A c+b B) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}-\frac {(3 A c+b B) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(3 A c+b B) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}-\frac {\sqrt {x} (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 335
Rule 468
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{\sqrt {x} \left (b+c x^2\right )^2} \, dx\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}+\frac {\left (\frac {b B}{2}+\frac {3 A c}{2}\right ) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{2 b c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}+\frac {\left (\frac {b B}{2}+\frac {3 A c}{2}\right ) \text {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{b c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}+\frac {(b B+3 A c) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^{3/2} c}+\frac {(b B+3 A c) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{4 b^{3/2} c}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}+\frac {(b B+3 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 b^{3/2} c^{3/2}}+\frac {(b B+3 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 b^{3/2} c^{3/2}}-\frac {(b B+3 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^{7/4} c^{5/4}}-\frac {(b B+3 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^{7/4} c^{5/4}}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}-\frac {(b B+3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 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^{7/4} c^{5/4}}-\frac {(b B+3 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^{7/4} c^{5/4}}\\ &=-\frac {(b B-A c) \sqrt {x}}{2 b c \left (b+c x^2\right )}-\frac {(b B+3 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{7/4} c^{5/4}}-\frac {(b B+3 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}+\frac {(b B+3 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} b^{7/4} c^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 151, normalized size = 0.58 \begin {gather*} \frac {\frac {4 b^{3/4} \sqrt [4]{c} (-b B+A c) \sqrt {x}}{b+c x^2}-\sqrt {2} (b B+3 A c) \tan ^{-1}\left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+\sqrt {2} (b B+3 A c) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{8 b^{7/4} c^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 146, normalized size = 0.56
method | result | size |
derivativedivides | \(\frac {\left (A c -B b \right ) \sqrt {x}}{2 b c \left (c \,x^{2}+b \right )}+\frac {\left (3 A c +B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} c}\) | \(146\) |
default | \(\frac {\left (A c -B b \right ) \sqrt {x}}{2 b c \left (c \,x^{2}+b \right )}+\frac {\left (3 A c +B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b^{2} c}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 241, normalized size = 0.92 \begin {gather*} -\frac {{\left (B b - A c\right )} \sqrt {x}}{2 \, {\left (b c^{2} x^{2} + b^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (B b + 3 \, 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 (B b + 3 \, 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 (B b + 3 \, 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 (B b + 3 \, 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 \, b c} \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. 717 vs.
\(2 (185) = 370\).
time = 2.06, size = 717, normalized size = 2.75 \begin {gather*} \frac {4 \, {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {b^{4} c^{2} \sqrt {-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}} + {\left (B^{2} b^{2} + 6 \, A B b c + 9 \, A^{2} c^{2}\right )} x} b^{5} c^{4} \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {3}{4}} - {\left (B b^{6} c^{4} + 3 \, A b^{5} c^{5}\right )} \sqrt {x} \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {3}{4}}}{B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}\right ) + {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (b^{2} c \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {1}{4}} + {\left (B b + 3 \, A c\right )} \sqrt {x}\right ) - {\left (b c^{2} x^{2} + b^{2} c\right )} \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {1}{4}} \log \left (-b^{2} c \left (-\frac {B^{4} b^{4} + 12 \, A B^{3} b^{3} c + 54 \, A^{2} B^{2} b^{2} c^{2} + 108 \, A^{3} B b c^{3} + 81 \, A^{4} c^{4}}{b^{7} c^{5}}\right )^{\frac {1}{4}} + {\left (B b + 3 \, A c\right )} \sqrt {x}\right ) - 4 \, {\left (B b - A c\right )} \sqrt {x}}{8 \, {\left (b c^{2} x^{2} + b^{2} c\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 = 273, normalized size = 1.05 \begin {gather*} \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \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^{2} c^{2}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \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^{2} c^{2}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \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^{2} c^{2}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b + 3 \, \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^{2} c^{2}} - \frac {B b \sqrt {x} - A c \sqrt {x}}{2 \, {\left (c x^{2} + b\right )} b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 750, normalized size = 2.87 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}-\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}+\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}+\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}}{\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}-\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}-\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}+\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}}\right )\,\left (3\,A\,c+B\,b\right )}{4\,{\left (-b\right )}^{7/4}\,c^{5/4}}+\frac {\sqrt {x}\,\left (A\,c-B\,b\right )}{2\,b\,c\,\left (c\,x^2+b\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}-\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}+\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}+\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )\,1{}\mathrm {i}}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}}{\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}-\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}-\frac {\left (3\,A\,c+B\,b\right )\,\left (\frac {\sqrt {x}\,\left (9\,A^2\,c^3+6\,A\,B\,b\,c^2+B^2\,b^2\,c\right )}{b^2}+\frac {\left (3\,A\,c+B\,b\right )\,\left (24\,A\,c^3+8\,B\,b\,c^2\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}\right )}{8\,{\left (-b\right )}^{7/4}\,c^{5/4}}}\right )\,\left (3\,A\,c+B\,b\right )\,1{}\mathrm {i}}{4\,{\left (-b\right )}^{7/4}\,c^{5/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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