Optimal. Leaf size=346 \[ \frac {(11 b c-3 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 x^{7/2}}{7 d^2} \]
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Rubi [A] time = 0.32, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {463, 459, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac {(11 b c-3 a d) (b c-a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {2 b^2 x^{7/2}}{7 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 321
Rule 329
Rule 459
Rule 463
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {x^{5/2} \left (\frac {1}{2} \left (-4 a^2 d^2+7 (b c-a d)^2\right )-2 b^2 c d x^2\right )}{c+d x^2} \, dx}{2 c d^2}\\ &=\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((11 b c-3 a d) (b c-a d)) \int \frac {x^{5/2}}{c+d x^2} \, dx}{4 c d^2}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 a d) (b c-a d)) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{4 d^3}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 d^3}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^{7/2}}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 d^{7/2}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^4}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{8 d^4}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}+\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {((11 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}\\ &=-\frac {(11 b c-3 a d) (b c-a d) x^{3/2}}{6 c d^3}+\frac {2 b^2 x^{7/2}}{7 d^2}+\frac {(b c-a d)^2 x^{7/2}}{2 c d^2 \left (c+d x^2\right )}-\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {(11 b c-3 a d) (b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{c} d^{15/4}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 337, normalized size = 0.97 \begin {gather*} \frac {\frac {21 \sqrt {2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{\sqrt [4]{c}}-\frac {21 \sqrt {2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{\sqrt [4]{c}}-\frac {42 \sqrt {2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}+\frac {42 \sqrt {2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{c}}-448 b d^{3/4} x^{3/2} (b c-a d)-\frac {168 d^{3/4} x^{3/2} (b c-a d)^2}{c+d x^2}+96 b^2 d^{7/4} x^{7/2}}{336 d^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.74, size = 233, normalized size = 0.67 \begin {gather*} -\frac {\left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}-\frac {\left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} \sqrt [4]{c} d^{15/4}}+\frac {x^{3/2} \left (-21 a^2 d^2+98 a b c d+56 a b d^2 x^2-77 b^2 c^2-44 b^2 c d x^2+12 b^2 d^2 x^4\right )}{42 d^3 \left (c+d x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 1733, normalized size = 5.01
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 413, normalized size = 1.19 \begin {gather*} -\frac {b^{2} c^{2} x^{\frac {3}{2}} - 2 \, a b c d x^{\frac {3}{2}} + a^{2} d^{2} x^{\frac {3}{2}}}{2 \, {\left (d x^{2} + c\right )} d^{3}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c d^{6}} - \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{6}} + \frac {\sqrt {2} {\left (11 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c d^{6}} + \frac {2 \, {\left (3 \, b^{2} d^{12} x^{\frac {7}{2}} - 14 \, b^{2} c d^{11} x^{\frac {3}{2}} + 14 \, a b d^{12} x^{\frac {3}{2}}\right )}}{21 \, d^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 523, normalized size = 1.51 \begin {gather*} \frac {2 b^{2} x^{\frac {7}{2}}}{7 d^{2}}-\frac {a^{2} x^{\frac {3}{2}}}{2 \left (d \,x^{2}+c \right ) d}+\frac {a b c \,x^{\frac {3}{2}}}{\left (d \,x^{2}+c \right ) d^{2}}-\frac {b^{2} c^{2} x^{\frac {3}{2}}}{2 \left (d \,x^{2}+c \right ) d^{3}}+\frac {4 a b \,x^{\frac {3}{2}}}{3 d^{2}}-\frac {4 b^{2} c \,x^{\frac {3}{2}}}{3 d^{3}}+\frac {3 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}+\frac {3 \sqrt {2}\, a^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}+\frac {3 \sqrt {2}\, a^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{2}}-\frac {7 \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}-\frac {7 \sqrt {2}\, a b c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}-\frac {7 \sqrt {2}\, a b c \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{8 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}+\frac {11 \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}+\frac {11 \sqrt {2}\, b^{2} c^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}+\frac {11 \sqrt {2}\, b^{2} c^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{16 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 272, normalized size = 0.79 \begin {gather*} -\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{\frac {3}{2}}}{2 \, {\left (d^{4} x^{2} + c d^{3}\right )}} + \frac {{\left (11 \, b^{2} c^{2} - 14 \, a b c d + 3 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, d^{3}} + \frac {2 \, {\left (3 \, b^{2} d x^{\frac {7}{2}} - 14 \, {\left (b^{2} c - a b d\right )} x^{\frac {3}{2}}\right )}}{21 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 160, normalized size = 0.46 \begin {gather*} \frac {2\,b^2\,x^{7/2}}{7\,d^2}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{2}-a\,b\,c\,d+\frac {b^2\,c^2}{2}\right )}{d^4\,x^2+c\,d^3}-x^{3/2}\,\left (\frac {4\,b^2\,c}{3\,d^3}-\frac {4\,a\,b}{3\,d^2}\right )+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}}+\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-11\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{1/4}\,d^{15/4}} \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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