Optimal. Leaf size=57 \[ \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {1687, 1602, 12,
1121, 627} \begin {gather*} \frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 627
Rule 1121
Rule 1602
Rule 1687
Rubi steps
\begin {align*} \int \frac {a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \frac {e x}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx+\int \frac {a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac {g x}{\sqrt {a+b x^2+c x^4}}+e \int \frac {x}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac {g x}{\sqrt {a+b x^2+c x^4}}+\frac {1}{2} e \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {g x}{\sqrt {a+b x^2+c x^4}}-\frac {e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 10.10, size = 51, normalized size = 0.89 \begin {gather*} \frac {-b e+b^2 g x-4 a c g x-2 c e x^2}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
2.
time = 0.05, size = 974, normalized size = 17.09
method | result | size |
gosper | \(\frac {4 a c g x -b^{2} g x +2 c e \,x^{2}+e b}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(52\) |
trager | \(\frac {4 a c g x -b^{2} g x +2 c e \,x^{2}+e b}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(52\) |
elliptic | \(\frac {e \left (2 c \,x^{2}+b \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}+\frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(55\) |
default | \(-c g \left (-\frac {2 c \left (\frac {b \,x^{3}}{2 c \left (4 a c -b^{2}\right )}+\frac {a x}{c \left (4 a c -b^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+a g \left (-\frac {2 c \left (\frac {b \,x^{3}}{2 a \left (4 a c -b^{2}\right )}-\frac {\left (2 a c -b^{2}\right ) x}{2 a \left (4 a c -b^{2}\right ) c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {1}{a}-\frac {2 a c -b^{2}}{a \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\EllipticF \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\EllipticE \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )+\frac {e \left (2 c \,x^{2}+b \right )}{\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (4 a c -b^{2}\right )}\) | \(974\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.33, size = 53, normalized size = 0.93 \begin {gather*} -\frac {2 \, c x^{2} e - {\left (b^{2} g - 4 \, a c g\right )} x + b e}{\sqrt {c x^{4} + b x^{2} + a} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.40, size = 82, normalized size = 1.44 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c e x^{2} - {\left (b^{2} - 4 \, a c\right )} g x + b e\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {a g}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac {e x}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c g x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs.
\(2 (54) = 108\).
time = 4.65, size = 142, normalized size = 2.49 \begin {gather*} -\frac {{\left (\frac {2 \, {\left (b^{2} c e - 4 \, a c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} - \frac {b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{3} e - 4 \, a b c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.93, size = 51, normalized size = 0.89 \begin {gather*} \frac {-g\,b^2\,x+e\,b+2\,c\,e\,x^2+4\,a\,c\,g\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^4+b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________