Optimal. Leaf size=624 \[ \frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3} \]
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Rubi [A]
time = 0.49, antiderivative size = 624, normalized size of antiderivative = 1.00, number of steps
used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 481,
541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt [4]{a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (3 a d+5 b c)}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (3 a d+5 b c)}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {a \sqrt {x}}{2 b \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {\sqrt {x} (a d+b c)}{2 b \left (c+d x^2\right ) (b c-a d)^2}+\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (3 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (5 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} \sqrt [4]{d} (b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 477
Rule 481
Rule 536
Rule 541
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^{7/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {x^8}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {a c+(-4 b c-3 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 b (b c-a d)}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {8 a b c^2-12 b c (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 b c (b c-a d)^2}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {(a (5 b c+3 a d)) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}+\frac {(c (3 b c+5 a d)) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \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 )}{8 \sqrt {b} (b c-a d)^3}-\frac {\left (\sqrt {a} (5 b c+3 a d)\right ) \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 )}{8 \sqrt {b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 a d)\right ) \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 )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 a d)\right ) \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 )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\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 )}{8 \sqrt {d} (b c-a d)^3}+\frac {\left (\sqrt {c} (3 b c+5 a d)\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 )}{8 \sqrt {d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 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} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 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} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 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]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 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]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{a} (5 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{a} (5 b c+3 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}+\frac {\left (\sqrt [4]{c} (3 b c+5 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} \sqrt [4]{d} (b c-a d)^3}-\frac {\left (\sqrt [4]{c} (3 b c+5 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} \sqrt [4]{d} (b c-a d)^3}\\ &=\frac {(b c+a d) \sqrt {x}}{2 b (b c-a d)^2 \left (c+d x^2\right )}+\frac {a \sqrt {x}}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} \sqrt [4]{d} (b c-a d)^3}+\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{a} (5 b c+3 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} \sqrt [4]{b} (b c-a d)^3}-\frac {\sqrt [4]{c} (3 b c+5 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]{d} (b c-a d)^3}+\frac {\sqrt [4]{c} (3 b c+5 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]{d} (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 1.31, size = 344, normalized size = 0.55 \begin {gather*} \frac {1}{8} \left (\frac {4 \sqrt {x} \left (2 a c+b c x^2+a d x^2\right )}{(b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} \sqrt [4]{a} (5 b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{c} (3 b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{\sqrt [4]{d} (-b c+a d)^3}-\frac {\sqrt {2} \sqrt [4]{a} (5 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b} (b c-a d)^3}+\frac {\sqrt {2} \sqrt [4]{c} (3 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt [4]{d} (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 302, normalized size = 0.48
method | result | size |
derivativedivides | \(\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (3 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{3}}-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{\left (a d -b c \right )^{3}}\) | \(302\) |
default | \(\frac {2 a \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (3 a d +5 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{\left (a d -b c \right )^{3}}-\frac {2 c \left (\frac {\left (-\frac {a d}{4}+\frac {b c}{4}\right ) \sqrt {x}}{d \,x^{2}+c}+\frac {\left (5 a d +3 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c}\right )}{\left (a d -b c \right )^{3}}\) | \(302\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 617, normalized size = 0.99 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (5 \, b c + 3 \, a d\right )} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (5 \, b c + 3 \, a d\right )} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (5 \, b c + 3 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b c + 3 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} a}{16 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (\frac {2 \, \sqrt {2} {\left (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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 (3 \, b c + 5 \, a d\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}}}\right )} c}{16 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (b c + a d\right )} x^{\frac {5}{2}} + 2 \, a c \sqrt {x}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{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. 5375 vs.
\(2 (472) = 944\).
time = 32.22, size = 5375, normalized size = 8.61 \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 = 1.92, size = 912, normalized size = 1.46 \begin {gather*} -\frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + 5 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{3} d - 3 \, \sqrt {2} a b^{2} c^{2} d^{2} + 3 \, \sqrt {2} a^{2} b c d^{3} - \sqrt {2} a^{3} d^{4}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + 5 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{3} d - 3 \, \sqrt {2} a b^{2} c^{2} d^{2} + 3 \, \sqrt {2} a^{2} b c d^{3} - \sqrt {2} a^{3} d^{4}\right )}} - \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} + \frac {{\left (5 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} b^{4} c^{3} - 3 \, \sqrt {2} a b^{3} c^{2} d + 3 \, \sqrt {2} a^{2} b^{2} c d^{2} - \sqrt {2} a^{3} b d^{3}\right )}} + \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + 5 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{3} d - 3 \, \sqrt {2} a b^{2} c^{2} d^{2} + 3 \, \sqrt {2} a^{2} b c d^{3} - \sqrt {2} a^{3} d^{4}\right )}} - \frac {{\left (3 \, \left (c d^{3}\right )^{\frac {1}{4}} b c + 5 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{3} d - 3 \, \sqrt {2} a b^{2} c^{2} d^{2} + 3 \, \sqrt {2} a^{2} b c d^{3} - \sqrt {2} a^{3} d^{4}\right )}} + \frac {b c x^{\frac {5}{2}} + a d x^{\frac {5}{2}} + 2 \, a c \sqrt {x}}{2 \, {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.17, size = 2500, normalized size = 4.01 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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