Integrand size = 32, antiderivative size = 118 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {c \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 b c-5 a d) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {\left (8 b^2 c-10 a b d+15 a^2 e\right ) \sqrt {a+b x^2}}{15 a^3 x}+\frac {f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1821, 1599, 1279, 462, 223, 212} \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} (4 b c-5 a d)}{15 a^2 x^3}-\frac {\sqrt {a+b x^2} \left (15 a^2 e-10 a b d+8 b^2 c\right )}{15 a^3 x}+\frac {f \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {c \sqrt {a+b x^2}}{5 a x^5} \]
[In]
[Out]
Rule 212
Rule 223
Rule 462
Rule 1279
Rule 1599
Rule 1821
Rubi steps \begin{align*} \text {integral}& = -\frac {c \sqrt {a+b x^2}}{5 a x^5}-\frac {\int \frac {(4 b c-5 a d) x-5 a e x^3-5 a f x^5}{x^5 \sqrt {a+b x^2}} \, dx}{5 a} \\ & = -\frac {c \sqrt {a+b x^2}}{5 a x^5}-\frac {\int \frac {4 b c-5 a d-5 a e x^2-5 a f x^4}{x^4 \sqrt {a+b x^2}} \, dx}{5 a} \\ & = -\frac {c \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 b c-5 a d) \sqrt {a+b x^2}}{15 a^2 x^3}+\frac {\int \frac {8 b^2 c-10 a b d+15 a^2 e+15 a^2 f x^2}{x^2 \sqrt {a+b x^2}} \, dx}{15 a^2} \\ & = -\frac {c \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 b c-5 a d) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {\left (8 b^2 c-10 a b d+15 a^2 e\right ) \sqrt {a+b x^2}}{15 a^3 x}+f \int \frac {1}{\sqrt {a+b x^2}} \, dx \\ & = -\frac {c \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 b c-5 a d) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {\left (8 b^2 c-10 a b d+15 a^2 e\right ) \sqrt {a+b x^2}}{15 a^3 x}+f \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right ) \\ & = -\frac {c \sqrt {a+b x^2}}{5 a x^5}+\frac {(4 b c-5 a d) \sqrt {a+b x^2}}{15 a^2 x^3}-\frac {\left (8 b^2 c-10 a b d+15 a^2 e\right ) \sqrt {a+b x^2}}{15 a^3 x}+\frac {f \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a+b x^2} \left (8 b^2 c x^4-2 a b x^2 \left (2 c+5 d x^2\right )+a^2 \left (3 c+5 d x^2+15 e x^4\right )\right )}{15 a^3 x^5}-\frac {f \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}} \]
[In]
[Out]
Time = 3.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (15 a^{2} e \,x^{4}-10 a b d \,x^{4}+8 b^{2} c \,x^{4}+5 a^{2} d \,x^{2}-4 a b c \,x^{2}+3 a^{2} c \right )}{15 a^{3} x^{5}}+\frac {f \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}\) | \(90\) |
pseudoelliptic | \(\frac {f \,a^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right ) x^{5}-\frac {\sqrt {b \,x^{2}+a}\, \left (-\frac {4 x^{2} \left (\frac {5 d \,x^{2}}{2}+c \right ) a \,b^{\frac {3}{2}}}{3}+\frac {8 b^{\frac {5}{2}} c \,x^{4}}{3}+a^{2} \sqrt {b}\, \left (5 e \,x^{4}+\frac {5}{3} d \,x^{2}+c \right )\right )}{5}}{\sqrt {b}\, x^{5} a^{3}}\) | \(96\) |
default | \(\frac {f \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{\sqrt {b}}+d \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )-\frac {e \sqrt {b \,x^{2}+a}}{a x}+c \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )\) | \(141\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.87 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=\left [\frac {15 \, a^{3} \sqrt {b} f x^{5} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left ({\left (8 \, b^{3} c - 10 \, a b^{2} d + 15 \, a^{2} b e\right )} x^{4} + 3 \, a^{2} b c - {\left (4 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{30 \, a^{3} b x^{5}}, -\frac {15 \, a^{3} \sqrt {-b} f x^{5} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left ({\left (8 \, b^{3} c - 10 \, a b^{2} d + 15 \, a^{2} b e\right )} x^{4} + 3 \, a^{2} b c - {\left (4 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{15 \, a^{3} b x^{5}}\right ] \]
[In]
[Out]
Time = 1.49 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.62 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=- \frac {3 a^{4} b^{\frac {9}{2}} c \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {2 a^{3} b^{\frac {11}{2}} c x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {3 a^{2} b^{\frac {13}{2}} c x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {12 a b^{\frac {15}{2}} c x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} - \frac {8 b^{\frac {17}{2}} c x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{15 a^{5} b^{4} x^{4} + 30 a^{4} b^{5} x^{6} + 15 a^{3} b^{6} x^{8}} + f \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \wedge b \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {a}} & \text {otherwise} \end {cases}\right ) - \frac {\sqrt {b} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 a x^{2}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{2}} + 1}}{a} + \frac {2 b^{\frac {3}{2}} d \sqrt {\frac {a}{b x^{2}} + 1}}{3 a^{2}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=\frac {f \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} c}{15 \, a^{3} x} + \frac {2 \, \sqrt {b x^{2} + a} b d}{3 \, a^{2} x} - \frac {\sqrt {b x^{2} + a} e}{a x} + \frac {4 \, \sqrt {b x^{2} + a} b c}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} d}{3 \, a x^{3}} - \frac {\sqrt {b x^{2} + a} c}{5 \, a x^{5}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (100) = 200\).
Time = 0.33 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.70 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=-\frac {f \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{2 \, \sqrt {b}} + \frac {2 \, {\left (15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} \sqrt {b} e + 30 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {3}{2}} d - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a \sqrt {b} e + 80 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {5}{2}} c - 70 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {3}{2}} d + 90 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} \sqrt {b} e - 40 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {5}{2}} c + 50 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {3}{2}} d - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} \sqrt {b} e + 8 \, a^{2} b^{\frac {5}{2}} c - 10 \, a^{3} b^{\frac {3}{2}} d + 15 \, a^{4} \sqrt {b} e\right )}}{15 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
[In]
[Out]
Time = 6.65 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \sqrt {a+b x^2}} \, dx=\frac {f\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{\sqrt {b}}-\frac {e\,\sqrt {b\,x^2+a}}{a\,x}-\frac {d\,\sqrt {b\,x^2+a}\,\left (a-2\,b\,x^2\right )}{3\,a^2\,x^3}-\frac {c\,\sqrt {b\,x^2+a}\,\left (3\,a^2-4\,a\,b\,x^2+8\,b^2\,x^4\right )}{15\,a^3\,x^5} \]
[In]
[Out]