Integrand size = 30, antiderivative size = 91 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1126, 14} \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )} \]
[In]
[Out]
Rule 14
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{(d x)^{7/2}} \, dx}{a b+b^2 x^2} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a b}{(d x)^{7/2}}+\frac {b^2}{d^2 (d x)^{3/2}}\right ) \, dx}{a b+b^2 x^2} \\ & = -\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 \sqrt {d x} \left (a+b x^2\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 x \sqrt {\left (a+b x^2\right )^2} \left (a+5 b x^2\right )}{5 (d x)^{7/2} \left (a+b x^2\right )} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.41
method | result | size |
gosper | \(-\frac {2 x \left (5 b \,x^{2}+a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 \left (b \,x^{2}+a \right ) \left (d x \right )^{\frac {7}{2}}}\) | \(37\) |
default | \(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (5 b \,x^{2}+a \right )}{5 d \left (b \,x^{2}+a \right ) \left (d x \right )^{\frac {5}{2}}}\) | \(39\) |
risch | \(-\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (5 b \,x^{2}+a \right )}{5 d^{3} \left (b \,x^{2}+a \right ) x^{2} \sqrt {d x}}\) | \(42\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (5 \, b x^{2} + a\right )} \sqrt {d x}}{5 \, d^{4} x^{3}} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (5 \, b d^{2} x^{2} + a d^{2}\right )}}{5 \, \left (d x\right )^{\frac {5}{2}} d^{3}} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {2 \, {\left (5 \, b d^{3} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a d^{3} \mathrm {sgn}\left (b x^{2} + a\right )\right )}}{5 \, \sqrt {d x} d^{6} x^{2}} \]
[In]
[Out]
Time = 13.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{7/2}} \, dx=-\frac {\left (\frac {2\,x^2}{d^3}+\frac {2\,a}{5\,b\,d^3}\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^4\,\sqrt {d\,x}+\frac {a\,x^2\,\sqrt {d\,x}}{b}} \]
[In]
[Out]