Optimal. Leaf size=312 \[ \frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}} \]
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Rubi [A]
time = 0.22, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {474, 470, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {(b c-a d) (3 a d+5 b c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}+\frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac {2 b^2 \sqrt {x}}{d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 470
Rule 474
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx &=\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}-\frac {\int \frac {\frac {1}{2} (b c-3 a d) (b c+a d)-2 b^2 c d x^2}{\sqrt {x} \left (c+d x^2\right )} \, dx}{2 c d^2}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (5 b c+3 a d)) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{4 c d^2}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (5 b c+3 a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c d^2}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (5 b c+3 a d)) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2} d^2}-\frac {((b c-a d) (5 b c+3 a d)) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^{3/2} d^2}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}-\frac {((b c-a d) (5 b c+3 a d)) \text {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 c^{3/2} d^{5/2}}-\frac {((b c-a d) (5 b c+3 a d)) \text {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 c^{3/2} d^{5/2}}+\frac {((b c-a d) (5 b c+3 a d)) \text {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} c^{7/4} d^{9/4}}+\frac {((b c-a d) (5 b c+3 a d)) \text {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} c^{7/4} d^{9/4}}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {((b c-a d) (5 b c+3 a d)) \text {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} c^{7/4} d^{9/4}}+\frac {((b c-a d) (5 b c+3 a d)) \text {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} c^{7/4} d^{9/4}}\\ &=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}+\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} d^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.59, size = 202, normalized size = 0.65 \begin {gather*} \frac {\frac {4 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (-2 a b c d+a^2 d^2+b^2 c \left (5 c+4 d x^2\right )\right )}{c+d x^2}+\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{8 c^{7/4} d^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 185, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\) | \(185\) |
default | \(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\) | \(185\) |
risch | \(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\sqrt {x}\, a^{2}}{2 c \left (d \,x^{2}+c \right )}-\frac {\sqrt {x}\, a b}{d \left (d \,x^{2}+c \right )}+\frac {\sqrt {x}\, b^{2} c}{2 d^{2} \left (d \,x^{2}+c \right )}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a^{2}}{8 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) a b}{4 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right ) b^{2}}{8 d^{2}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a^{2}}{8 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) a b}{4 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right ) b^{2}}{8 d^{2}}+\frac {3 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a^{2}}{16 c^{2}}+\frac {\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) a b}{8 d c}-\frac {5 \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right ) b^{2}}{16 d^{2}}\) | \(496\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 327, normalized size = 1.05 \begin {gather*} \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, c d^{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. 1341 vs.
\(2 (234) = 468\).
time = 0.92, size = 1341, normalized size = 4.30 \begin {gather*} \frac {4 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{4} d^{4} \sqrt {-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}} + {\left (25 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d - 26 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + 9 \, a^{4} d^{4}\right )} x} c^{5} d^{7} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {3}{4}} + {\left (5 \, b^{2} c^{7} d^{7} - 2 \, a b c^{6} d^{8} - 3 \, a^{2} c^{5} d^{9}\right )} \sqrt {x} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {3}{4}}}{625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}\right ) + {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {1}{4}} \log \left (c^{2} d^{2} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {1}{4}} - {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - {\left (c d^{3} x^{2} + c^{2} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {1}{4}} \log \left (-c^{2} d^{2} \left (-\frac {625 \, b^{8} c^{8} - 1000 \, a b^{7} c^{7} d - 900 \, a^{2} b^{6} c^{6} d^{2} + 1640 \, a^{3} b^{5} c^{5} d^{3} + 646 \, a^{4} b^{4} c^{4} d^{4} - 984 \, a^{5} b^{3} c^{3} d^{5} - 324 \, a^{6} b^{2} c^{2} d^{6} + 216 \, a^{7} b c d^{7} + 81 \, a^{8} d^{8}}{c^{7} d^{9}}\right )^{\frac {1}{4}} - {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \sqrt {x}\right ) + 4 \, {\left (4 \, b^{2} c d x^{2} + 5 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}}{8 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1248 vs.
\(2 (298) = 596\).
time = 25.56, size = 1248, normalized size = 4.00 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}\right ) & \text {for}\: c = 0 \wedge d = 0 \\\frac {- \frac {2 a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 a b}{3 x^{\frac {3}{2}}} + 2 b^{2} \sqrt {x}}{d^{2}} & \text {for}\: c = 0 \\\frac {2 a^{2} \sqrt {x} + \frac {4 a b x^{\frac {5}{2}}}{5} + \frac {2 b^{2} x^{\frac {9}{2}}}{9}}{c^{2}} & \text {for}\: d = 0 \\\frac {4 a^{2} c d^{2} \sqrt {x}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {3 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {3 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {6 a^{2} c d^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {3 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {3 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {6 a^{2} d^{3} x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {8 a b c^{2} d \sqrt {x}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {2 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {2 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {4 a b c^{2} d \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {2 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {2 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {4 a b c d^{2} x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {20 b^{2} c^{3} \sqrt {x}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {5 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {5 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {10 b^{2} c^{3} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {16 b^{2} c^{2} d x^{\frac {5}{2}}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} + \frac {5 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {5 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {c}{d}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} - \frac {10 b^{2} c^{2} d x^{2} \sqrt [4]{- \frac {c}{d}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {c}{d}}} \right )}}{8 c^{3} d^{2} + 8 c^{2} d^{3} x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 388, normalized size = 1.24 \begin {gather*} \frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{2} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{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^{2} d^{3}} + \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} c d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 1267, normalized size = 4.06 \begin {gather*} \frac {2\,b^2\,\sqrt {x}}{d^2}+\frac {\sqrt {x}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c\,\left (d^3\,x^2+c\,d^2\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}-\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}+\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}+\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}-\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}-\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}+\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{4\,{\left (-c\right )}^{7/4}\,d^{9/4}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}-\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}+\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}+\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}}{\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}-\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}-\frac {\left (\frac {\sqrt {x}\,\left (9\,a^4\,d^4+12\,a^3\,b\,c\,d^3-26\,a^2\,b^2\,c^2\,d^2-20\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{c^2\,d}+\frac {\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,\left (24\,a^2\,d^3+16\,a\,b\,c\,d^2-40\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-c\right )}^{7/4}\,d^{9/4}}}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d+5\,b\,c\right )}{4\,{\left (-c\right )}^{7/4}\,d^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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