Optimal. Leaf size=401 \[ -\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {x^{3/2} \left (\frac {3 a^2 d}{c}+42 a b-\frac {77 b^2 c}{d}\right )}{48 c d^2}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]
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Rubi [A] time = 0.34, antiderivative size = 401, 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, 457, 321, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {x^{3/2} \left (\frac {3 a^2 d}{c}+42 a b-\frac {77 b^2 c}{d}\right )}{48 c d^2}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 321
Rule 329
Rule 457
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 )^3} \, dx &=\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {x^{5/2} \left (\frac {1}{2} \left (-8 a^2 d^2+7 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac {x^{5/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac {\sqrt {x}}{c+d x^2} \, dx}{32 c d^3}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c d^3}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c d^{7/2}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c d^{7/2}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \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 )}{64 c d^4}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \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 )}{64 c d^4}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{5/4} d^{15/4}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}\\ &=\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac {(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{5/4} d^{15/4}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 363, normalized size = 0.91 \begin {gather*} \frac {\frac {24 d^{3/4} x^{3/2} \left (3 a^2 d^2-22 a b c d+19 b^2 c^2\right )}{c \left (c+d x^2\right )}-\frac {3 \sqrt {2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {3 \sqrt {2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {6 \sqrt {2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac {6 \sqrt {2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac {96 d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 d^{3/4} x^{3/2}}{384 d^{15/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.02, size = 254, normalized size = 0.63 \begin {gather*} \frac {\left (-3 a^2 d^2-42 a b c d+77 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 )}{32 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (-3 a^2 d^2-42 a b c d+77 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 )}{32 \sqrt {2} c^{5/4} d^{15/4}}+\frac {x^{3/2} \left (-3 a^2 c d^2+9 a^2 d^3 x^2-42 a b c^2 d-66 a b c d^2 x^2+77 b^2 c^3+121 b^2 c^2 d x^2+32 b^2 c d^2 x^4\right )}{48 c d^3 \left (c+d x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 1813, normalized size = 4.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 427, normalized size = 1.06 \begin {gather*} \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {19 \, b^{2} c^{2} d x^{\frac {7}{2}} - 22 \, a b c d^{2} x^{\frac {7}{2}} + 3 \, a^{2} d^{3} x^{\frac {7}{2}} + 15 \, b^{2} c^{3} x^{\frac {3}{2}} - 14 \, a b c^{2} d x^{\frac {3}{2}} - a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \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 )}{64 \, c^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \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 )}{64 \, c^{2} d^{6}} + \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \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 )}{128 \, c^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \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 )}{128 \, c^{2} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 562, normalized size = 1.40 \begin {gather*} \frac {3 a^{2} x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} c}-\frac {11 a b \,x^{\frac {7}{2}}}{8 \left (d \,x^{2}+c \right )^{2} d}+\frac {19 b^{2} c \,x^{\frac {7}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d^{2}}-\frac {a^{2} x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d}-\frac {7 a b c \,x^{\frac {3}{2}}}{8 \left (d \,x^{2}+c \right )^{2} d^{2}}+\frac {15 b^{2} c^{2} x^{\frac {3}{2}}}{16 \left (d \,x^{2}+c \right )^{2} d^{3}}+\frac {2 b^{2} 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 )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,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 )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,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 )}{128 \left (\frac {c}{d}\right )^{\frac {1}{4}} c \,d^{2}}+\frac {21 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}+\frac {21 \sqrt {2}\, a b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{32 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}+\frac {21 \sqrt {2}\, a b \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 )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{3}}-\frac {77 \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}-\frac {77 \sqrt {2}\, b^{2} c \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{64 \left (\frac {c}{d}\right )^{\frac {1}{4}} d^{4}}-\frac {77 \sqrt {2}\, b^{2} 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 )}{128 \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 = 2.54, size = 306, normalized size = 0.76 \begin {gather*} \frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {{\left (19 \, b^{2} c^{2} d - 22 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (15 \, b^{2} c^{3} - 14 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} - \frac {{\left (77 \, b^{2} c^{2} - 42 \, 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 )}}{128 \, c d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 197, normalized size = 0.49 \begin {gather*} \frac {2\,b^2\,x^{3/2}}{3\,d^3}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{16}+\frac {7\,a\,b\,c\,d}{8}-\frac {15\,b^2\,c^2}{16}\right )-\frac {x^{7/2}\,\left (3\,a^2\,d^3-22\,a\,b\,c\,d^2+19\,b^2\,c^2\,d\right )}{16\,c}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{5/4}\,d^{15/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )\,1{}\mathrm {i}}{32\,{\left (-c\right )}^{5/4}\,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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