Optimal. Leaf size=536 \[ -\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{5/4} (b c-a d)^2} \]
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Rubi [A]
time = 0.41, antiderivative size = 536, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {477, 483, 598,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{5/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}-\frac {d x^{3/2}}{2 c \left (c+d x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 477
Rule 483
Rule 598
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (4 b c-a d-b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {4 b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {d (-5 b c+a d) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^2}-\frac {(d (5 b c-a d)) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b^{3/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^2}+\frac {b^{3/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^2}+\frac {\left (\sqrt {d} (5 b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)^2}-\frac {\left (\sqrt {d} (5 b c-a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^2}+\frac {b \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^2}+\frac {b^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {(5 b c-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 (b c-a d)^2}-\frac {(5 b c-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 (b c-a d)^2}-\frac {\left (\sqrt [4]{d} (5 b c-a d)\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 )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\left (\sqrt [4]{d} (5 b c-a d)\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 )}{8 \sqrt {2} c^{5/4} (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\left (\sqrt [4]{d} (5 b c-a d)\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 )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {\left (\sqrt [4]{d} (5 b c-a d)\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 )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}\\ &=-\frac {d x^{3/2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{5/4} (b c-a d)^2}+\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)^2}-\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}+\frac {\sqrt [4]{d} (5 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} c^{5/4} (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 1.01, size = 273, normalized size = 0.51 \begin {gather*} \frac {\frac {4 d (-b c+a d) x^{3/2}}{c \left (c+d x^2\right )}-\frac {4 \sqrt {2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{5/4}}-\frac {4 \sqrt {2} b^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}+\frac {\sqrt {2} \sqrt [4]{d} (5 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{5/4}}}{8 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 270, normalized size = 0.50
method | result | size |
derivativedivides | \(\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -5 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}\) | \(270\) |
default | \(\frac {b \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -5 b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{2}}\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 450, normalized size = 0.84 \begin {gather*} \frac {b^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {d x^{\frac {3}{2}}}{2 \, {\left (b c^{3} - a c^{2} d + {\left (b c^{2} d - a c d^{2}\right )} x^{2}\right )}} - \frac {{\left (5 \, b c d - a 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 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )}} \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. 3457 vs.
\(2 (393) = 786\).
time = 12.24, size = 3457, normalized size = 6.45 \begin {gather*} \text {Too large to display} \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 = 1.60, size = 701, normalized size = 1.31 \begin {gather*} -\frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} + \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} - \frac {{\left (5 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{4} d^{2} - 2 \, \sqrt {2} a b c^{3} d^{3} + \sqrt {2} a^{2} c^{2} d^{4}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c^{2} - 2 \, \sqrt {2} a^{2} b^{2} c d + \sqrt {2} a^{3} b d^{2}\right )}} - \frac {d x^{\frac {3}{2}}}{2 \, {\left (b c^{2} - a c d\right )} {\left (d x^{2} + c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 2500, normalized size = 4.66 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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