Optimal. Leaf size=628 \[ \frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3} \]
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Rubi [A]
time = 0.58, antiderivative size = 628, 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, 425,
541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (3 b c-11 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (3 b c-11 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 b c-3 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {d^{7/4} (11 b c-3 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{7/4} (b c-a d)^3}+\frac {b \sqrt {x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d \sqrt {x} (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 425
Rule 477
Rule 536
Rule 541
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-3 b c+4 a d-7 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-4 \left (3 b^2 c^2-8 a b c d+3 a^2 d^2\right )-12 b d (b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{8 a c (b c-a d)^2}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 a d)\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (3 b c-11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} (b c-a d)^3}+\frac {\left (b^2 (3 b c-11 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 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^{3/2} (b c-a d)^3}+\frac {\left (d^2 (11 b c-3 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^{3/2} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^{3/2} (3 b c-11 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 a^{3/2} (b c-a d)^3}+\frac {\left (b^{3/2} (3 b c-11 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 a^{3/2} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 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} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 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} a^{7/4} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 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 c^{3/2} (b c-a d)^3}+\frac {\left (d^{3/2} (11 b c-3 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 c^{3/2} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 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} (b c-a d)^3}+\frac {d^{7/4} (11 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} (b c-a d)^3}+\frac {\left (b^{7/4} (3 b c-11 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} a^{7/4} (b c-a d)^3}-\frac {\left (b^{7/4} (3 b c-11 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} a^{7/4} (b c-a d)^3}+\frac {\left (d^{7/4} (11 b c-3 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^{7/4} (b c-a d)^3}-\frac {\left (d^{7/4} (11 b c-3 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^{7/4} (b c-a d)^3}\\ &=\frac {d (b c+a d) \sqrt {x}}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 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} (b c-a d)^3}+\frac {d^{7/4} (11 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} (b c-a d)^3}-\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}+\frac {b^{7/4} (3 b c-11 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^3}-\frac {d^{7/4} (11 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} (b c-a d)^3}+\frac {d^{7/4} (11 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} (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 1.67, size = 361, normalized size = 0.57 \begin {gather*} \frac {1}{8} \left (\frac {4 \sqrt {x} \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )}{a c (b c-a d)^2 \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\sqrt {2} b^{7/4} (-3 b c+11 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} (b c-a d)^3}+\frac {\sqrt {2} d^{7/4} (-11 b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{7/4} (b c-a d)^3}+\frac {\sqrt {2} b^{7/4} (-3 b c+11 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{7/4} (11 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{7/4} (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 312, normalized size = 0.50
method | result | size |
derivativedivides | \(\frac {2 b^{2} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (11 a d -3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{2} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a d -11 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{\left (a d -b c \right )^{3}}\) | \(312\) |
default | \(\frac {2 b^{2} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (11 a d -3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{3}}+\frac {2 d^{2} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (d \,x^{2}+c \right )}+\frac {\left (3 a d -11 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{\left (a d -b c \right )^{3}}\) | \(312\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 678, normalized size = 1.08 \begin {gather*} \frac {{\left (\frac {2 \, \sqrt {2} {\left (3 \, b c - 11 \, 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 (3 \, b c - 11 \, 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 (3 \, b c - 11 \, 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 (3 \, b c - 11 \, 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 )} b^{2}}{16 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{\frac {5}{2}} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (11 \, b c d^{2} - 3 \, a d^{3}\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 (11 \, b c d^{2} - 3 \, a d^{3}\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 (11 \, b c d^{2} - 3 \, a d^{3}\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 (11 \, b c d^{2} - 3 \, a d^{3}\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 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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 [B] Leaf count of result is larger than twice the leaf count of optimal. 977 vs.
\(2 (476) = 952\).
time = 1.10, size = 977, normalized size = 1.56 \begin {gather*} \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b 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} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b 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} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a 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 )}{4 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a 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 )}{4 \, {\left (\sqrt {2} b^{3} c^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b 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} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} - \frac {{\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 11 \, \left (a b^{3}\right )^{\frac {1}{4}} a b 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} a^{2} b^{3} c^{3} - 3 \, \sqrt {2} a^{3} b^{2} c^{2} d + 3 \, \sqrt {2} a^{4} b c d^{2} - \sqrt {2} a^{5} d^{3}\right )}} + \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\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^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} - \frac {{\left (11 \, \left (c d^{3}\right )^{\frac {1}{4}} b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a d^{2}\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^{5} - 3 \, \sqrt {2} a b^{2} c^{4} d + 3 \, \sqrt {2} a^{2} b c^{3} d^{2} - \sqrt {2} a^{3} c^{2} d^{3}\right )}} + \frac {b^{2} c d x^{\frac {5}{2}} + a b d^{2} x^{\frac {5}{2}} + b^{2} c^{2} \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.33, size = 2500, normalized size = 3.98 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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