Integrand size = 26, antiderivative size = 250 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {5 a^3 b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {b^5 x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {5 a^4 b \sqrt {a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 250, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\frac {b^5 x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {5 a^4 b \log (x) \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {5 a^3 b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
[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 )^5}{x^3} \, dx}{b^4 \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 )^5}{x^2} \, dx,x,x^2\right )}{2 b^4 \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 (10 a^3 b^7+\frac {a^5 b^5}{x^2}+\frac {5 a^4 b^6}{x}+10 a^2 b^8 x+5 a b^9 x^2+b^{10} x^3\right ) \, dx,x,x^2\right )}{2 b^4 \left (a b+b^2 x^2\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac {5 a^3 b^2 x^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac {5 a^2 b^3 x^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac {5 a b^4 x^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}{6 \left (a+b x^2\right )}+\frac {b^5 x^8 \sqrt {a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )}+\frac {5 a^4 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 = 85, normalized size of antiderivative = 0.34 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\frac {\sqrt {\left (a+b x^2\right )^2} \left (-12 a^5+120 a^3 b^2 x^4+60 a^2 b^3 x^6+20 a b^4 x^8+3 b^5 x^{10}+120 a^4 b x^2 \log (x)\right )}{24 x^2 \left (a+b x^2\right )} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.28
method | result | size |
pseudoelliptic | \(-\frac {\operatorname {csgn}\left (b \,x^{2}+a \right ) \left (-\frac {x^{10} b^{5}}{4}-\frac {5 a \,x^{8} b^{4}}{3}-5 a^{2} x^{6} b^{3}-10 a^{3} x^{4} b^{2}-5 b \,a^{4} \ln \left (x^{2}\right ) x^{2}+a^{5}\right )}{2 x^{2}}\) | \(70\) |
default | \(\frac {{\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}} \left (3 x^{10} b^{5}+20 a \,x^{8} b^{4}+60 a^{2} x^{6} b^{3}+120 a^{3} x^{4} b^{2}+120 \ln \left (x \right ) x^{2} a^{4} b -12 a^{5}\right )}{24 \left (b \,x^{2}+a \right )^{5} x^{2}}\) | \(82\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, b^{2} \left (\frac {1}{8} b^{3} x^{8}+\frac {5}{6} b^{2} x^{6} a +\frac {5}{2} a^{2} b \,x^{4}+5 a^{3} x^{2}\right )}{b \,x^{2}+a}-\frac {a^{5} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{2 x^{2} \left (b \,x^{2}+a \right )}+\frac {5 a^{4} b \ln \left (x \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b \,x^{2}+a}\) | \(117\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.24 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\frac {3 \, b^{5} x^{10} + 20 \, a b^{4} x^{8} + 60 \, a^{2} b^{3} x^{6} + 120 \, a^{3} b^{2} x^{4} + 120 \, a^{4} b x^{2} \log \left (x\right ) - 12 \, a^{5}}{24 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\frac {1}{8} \, b^{5} x^{8} + \frac {5}{6} \, a b^{4} x^{6} + \frac {5}{2} \, a^{2} b^{3} x^{4} + 5 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b \log \left (x\right ) - \frac {a^{5}}{2 \, x^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.50 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\frac {1}{8} \, b^{5} x^{8} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{6} \, a b^{4} x^{6} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{2} \, a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 5 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {5}{2} \, a^{4} b \log \left (x^{2}\right ) \mathrm {sgn}\left (b x^{2} + a\right ) - \frac {5 \, a^{4} b x^{2} \mathrm {sgn}\left (b x^{2} + a\right ) + a^{5} \mathrm {sgn}\left (b x^{2} + a\right )}{2 \, x^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}}{x^3} \,d x \]
[In]
[Out]