Optimal. Leaf size=739 \[ \frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4} \]
[Out]
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Rubi [A]
time = 0.67, antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps
used = 23, 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 {3 b^{11/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-5 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-5 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 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)^4}+\frac {3 b^{11/4} (b c-5 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)^4}-\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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} (b c-a d)^4}+\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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} (b c-a d)^4}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}-\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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} (b c-a d)^4}+\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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} (b c-a d)^4}+\frac {b \sqrt {x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {d \sqrt {x} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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 )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, 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 )^2}-\frac {\text {Subst}\left (\int \frac {-3 b c+4 a d-11 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 \left (6 b^2 c^2-16 a b c d+7 a^2 d^2\right )-28 b d (2 b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 a c (b c-a d)^2}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-12 \left (8 b^3 c^3-32 a b^2 c^2 d+23 a^2 b c d^2-7 a^3 d^3\right )-12 b d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 a c^2 (b c-a d)^3}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^3 (b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^3 (b c-5 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)^4}+\frac {\left (3 b^3 (b c-5 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)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^{5/2} (b c-5 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)^4}+\frac {\left (3 b^{5/2} (b c-5 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)^4}-\frac {\left (3 b^{11/4} (b c-5 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)^4}-\frac {\left (3 b^{11/4} (b c-5 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)^4}+\frac {\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}+\frac {\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 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)^4}+\frac {3 b^{11/4} (b c-5 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)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}+\frac {\left (3 b^{11/4} (b c-5 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)^4}-\frac {\left (3 b^{11/4} (b c-5 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)^4}+\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\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} (b c-a d)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 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)^4}+\frac {3 b^{11/4} (b c-5 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)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 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)^4}+\frac {3 b^{11/4} (b c-5 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)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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} (b c-a d)^4}\\ \end {align*}
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Mathematica [A]
time = 3.20, size = 450, normalized size = 0.61 \begin {gather*} \frac {1}{64} \left (-\frac {4 \sqrt {x} \left (8 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (11 c+7 d x^2\right )+a b^2 c d^2 x^2 \left (27 c+23 d x^2\right )+a^2 b d^2 \left (27 c^2+12 c d x^2-7 d^2 x^4\right )\right )}{a c^2 (-b c+a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {24 \sqrt {2} b^{11/4} (-b c+5 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)^4}-\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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 )}{c^{11/4} (b c-a d)^4}+\frac {24 \sqrt {2} b^{11/4} (b c-5 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)^4}+\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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 )}{c^{11/4} (b c-a d)^4}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.19, size = 375, normalized size = 0.51
method | result | size |
derivativedivides | \(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}\) | \(375\) |
default | \(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}\) | \(375\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 951, normalized size = 1.29 \begin {gather*} \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b c - 5 \, 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 (b c - 5 \, 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 (b c - 5 \, 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 (b c - 5 \, 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^{3}}{16 \, {\left (a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + 6 \, a^{3} b^{2} c^{2} d^{2} - 4 \, a^{4} b c d^{3} + a^{5} d^{4}\right )}} + \frac {{\left (8 \, b^{3} c^{2} d^{2} + 23 \, a b^{2} c d^{3} - 7 \, a^{2} b d^{4}\right )} x^{\frac {9}{2}} + {\left (16 \, b^{3} c^{3} d + 27 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{\frac {5}{2}} + {\left (8 \, b^{3} c^{4} + 27 \, a^{2} b c^{2} d^{2} - 11 \, a^{3} c d^{3}\right )} \sqrt {x}}{16 \, {\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} + {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{6} + {\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{4} + {\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x^{2}\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 )}}{128 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\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. 1253 vs.
\(2 (583) = 1166\).
time = 2.59, size = 1253, normalized size = 1.70 \begin {gather*} \frac {b^{3} \sqrt {x}}{2 \, {\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 )} {\left (b x^{2} + a\right )}} + \frac {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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 )}{32 \, {\left (\sqrt {2} b^{4} c^{7} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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 )}{32 \, {\left (\sqrt {2} b^{4} c^{7} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} - \frac {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{4} c^{7} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} - \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{4} c^{7} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {23 \, b c d^{3} x^{\frac {5}{2}} - 7 \, a d^{4} x^{\frac {5}{2}} + 27 \, b c^{2} d^{2} \sqrt {x} - 11 \, a c d^{3} \sqrt {x}}{16 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.10, size = 2500, normalized size = 3.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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