Optimal. Leaf size=364 \[ \frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} d^{9/4}} \]
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Rubi [A]
time = 0.21, antiderivative size = 364, 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, 468, 335,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (21 a^2 d^2+6 a b c d+5 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^{11/4} d^{9/4}}+\frac {\left (21 a^2 d^2+6 a b c d+5 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^{11/4} d^{9/4}}-\frac {\sqrt {x} (7 a d+9 b c) (b c-a d)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\sqrt {x} (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 210
Rule 217
Rule 335
Rule 468
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 )^3} \, dx &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-8 a^2 d^2+(b c-a d)^2\right )-4 b^2 c d x^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \int \frac {1}{\sqrt {x} \left (c+d x^2\right )} \, dx}{32 c^2 d^2}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 d^2}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2} d^2}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^{5/2} d^2}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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 )}{64 c^{5/2} d^{5/2}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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 )}{64 c^{5/2} d^{5/2}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \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 )}{64 \sqrt {2} c^{11/4} d^{9/4}}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {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^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 a^2 d^2\right ) \text {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^{11/4} d^{9/4}}\\ &=\frac {(b c-a d)^2 \sqrt {x}}{4 c d^2 \left (c+d x^2\right )^2}-\frac {(b c-a d) (9 b c+7 a d) \sqrt {x}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}-\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}+\frac {\left (5 b^2 c^2+6 a b c d+21 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^{11/4} d^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 225, normalized size = 0.62 \begin {gather*} \frac {-\frac {4 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (2 a b c d \left (3 c-d x^2\right )-a^2 d^2 \left (11 c+7 d x^2\right )+b^2 c^2 \left (5 c+9 d x^2\right )\right )}{\left (c+d x^2\right )^2}-\sqrt {2} \left (5 b^2 c^2+6 a b c d+21 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+6 a b c d+21 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 )}{64 c^{11/4} d^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 213, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{16 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) \sqrt {x}}{16 d^{2} c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}+6 a b c d +5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} d^{2}}\) | \(213\) |
default | \(\frac {\frac {\left (7 a^{2} d^{2}+2 a b c d -9 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{16 c^{2} d}+\frac {\left (11 a^{2} d^{2}-6 a b c d -5 b^{2} c^{2}\right ) \sqrt {x}}{16 d^{2} c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}+6 a b c d +5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c^{3} d^{2}}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 366, normalized size = 1.01 \begin {gather*} -\frac {{\left (9 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 7 \, a^{2} d^{3}\right )} x^{\frac {5}{2}} + {\left (5 \, b^{2} c^{3} + 6 \, a b c^{2} d - 11 \, a^{2} c d^{2}\right )} \sqrt {x}}{16 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, 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} + 6 \, a b c d + 21 \, 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} + 6 \, a b c d + 21 \, 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} + 6 \, a b c d + 21 \, 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}}}}{128 \, c^{2} 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. 1416 vs.
\(2 (284) = 568\).
time = 0.87, size = 1416, normalized size = 3.89 \begin {gather*} \frac {4 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{6} d^{4} \sqrt {-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}} + {\left (25 \, b^{4} c^{4} + 60 \, a b^{3} c^{3} d + 246 \, a^{2} b^{2} c^{2} d^{2} + 252 \, a^{3} b c d^{3} + 441 \, a^{4} d^{4}\right )} x} c^{8} d^{7} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {3}{4}} - {\left (5 \, b^{2} c^{10} d^{7} + 6 \, a b c^{9} d^{8} + 21 \, a^{2} c^{8} d^{9}\right )} \sqrt {x} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {3}{4}}}{625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}\right ) + {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {1}{4}} \log \left (c^{3} d^{2} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {1}{4}} + {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {1}{4}} \log \left (-c^{3} d^{2} \left (-\frac {625 \, b^{8} c^{8} + 3000 \, a b^{7} c^{7} d + 15900 \, a^{2} b^{6} c^{6} d^{2} + 42120 \, a^{3} b^{5} c^{5} d^{3} + 112806 \, a^{4} b^{4} c^{4} d^{4} + 176904 \, a^{5} b^{3} c^{3} d^{5} + 280476 \, a^{6} b^{2} c^{2} d^{6} + 222264 \, a^{7} b c d^{7} + 194481 \, a^{8} d^{8}}{c^{11} d^{9}}\right )^{\frac {1}{4}} + {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 21 \, a^{2} d^{2}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, b^{2} c^{3} + 6 \, a b c^{2} d - 11 \, a^{2} c d^{2} + {\left (9 \, b^{2} c^{2} d - 2 \, a b c d^{2} - 7 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} 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. 2258 vs.
\(2 (367) = 734\).
time = 144.21, size = 2258, normalized size = 6.20 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 416, normalized size = 1.14 \begin {gather*} \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \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 )}{64 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \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 )}{64 \, c^{3} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \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 )}{128 \, c^{3} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} + 6 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \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 )}{128 \, c^{3} d^{3}} - \frac {9 \, b^{2} c^{2} d x^{\frac {5}{2}} - 2 \, a b c d^{2} x^{\frac {5}{2}} - 7 \, a^{2} d^{3} x^{\frac {5}{2}} + 5 \, b^{2} c^{3} \sqrt {x} + 6 \, a b c^{2} d \sqrt {x} - 11 \, a^{2} c d^{2} \sqrt {x}}{16 \, {\left (d x^{2} + c\right )}^{2} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 1419, normalized size = 3.90 \begin {gather*} -\frac {\frac {\sqrt {x}\,\left (-11\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{16\,c\,d^2}-\frac {x^{5/2}\,\left (7\,a^2\,d^2+2\,a\,b\,c\,d-9\,b^2\,c^2\right )}{16\,c^2\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}-\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}-\frac {\left (\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}+\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}}{\frac {\left (\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}-\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}+\frac {\left (\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}+\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,1{}\mathrm {i}}{32\,{\left (-c\right )}^{11/4}\,d^{9/4}}-\frac {\mathrm {atan}\left (\frac {\frac {\left (-\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}+\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}-\frac {\left (\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}+\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}}{\frac {\left (-\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}+\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}+\frac {\left (\frac {\sqrt {x}\,\left (441\,a^4\,d^4+252\,a^3\,b\,c\,d^3+246\,a^2\,b^2\,c^2\,d^2+60\,a\,b^3\,c^3\,d+25\,b^4\,c^4\right )}{64\,c^4\,d}+\frac {\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,\left (21\,a^2\,d^3+6\,a\,b\,c\,d^2+5\,b^2\,c^2\,d\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{15/4}\,d^{9/4}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )\,1{}\mathrm {i}}{64\,{\left (-c\right )}^{11/4}\,d^{9/4}}}\right )\,\left (21\,a^2\,d^2+6\,a\,b\,c\,d+5\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{11/4}\,d^{9/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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