Integrand size = 26, antiderivative size = 164 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1126, 272, 45} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {3 a^2 b \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}-\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )} \]
[In]
[Out]
Rule 45
Rule 272
Rule 1126
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {\left (a b+b^2 x^2\right )^3}{x^3} \, dx}{b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^3}{x^2} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \text {Subst}\left (\int \left (3 a b^5+\frac {a^3 b^3}{x^2}+\frac {3 a^2 b^4}{x}+b^6 x\right ) \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {3 a b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.38 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (-2 a^3+6 a b^2 x^4+b^3 x^6+12 a^2 b x^2 \log (x)\right )}{4 x^2 \left (a+b x^2\right )} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.29
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (-\frac {b^{3} x^{6}}{2}-3 b^{2} x^{4} a -3 a^{2} b \ln \left (x^{2}\right ) x^{2}+a^{3}\right )}{2 x^{2}}\) | \(48\) |
default | \(\frac {{\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}} \left (b^{3} x^{6}+6 b^{2} x^{4} a +12 \ln \left (x \right ) x^{2} a^{2} b -2 a^{3}\right )}{4 \left (b \,x^{2}+a \right )^{3} x^{2}}\) | \(59\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, b \left (b \,x^{2}+3 a \right )^{2}}{4 b \,x^{2}+4 a}-\frac {a^{3} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{2 x^{2} \left (b \,x^{2}+a \right )}+\frac {3 a^{2} b \ln \left (x \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\) | \(92\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\frac {b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \left (x\right ) - 2 \, a^{3}}{4 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b \log \left (x\right ) - \frac {a^{3}}{2 \, x^{2}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.53 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\frac {1}{4} \, b^{3} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a b^{2} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {3 \, a^{2} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, x^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}}{x^3} \,d x \]
[In]
[Out]