Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \log (x) \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}-\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)} \]
[In]
[Out]
Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5}{x^5} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (b^{10}+\frac {a^5 b^5}{x^5}+\frac {5 a^4 b^6}{x^4}+\frac {10 a^3 b^7}{x^3}+\frac {10 a^2 b^8}{x^2}+\frac {5 a b^9}{x}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {a^5 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {5 a^4 b \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {5 a^3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{x^2 (a+b x)}-\frac {10 a^2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac {b^5 x \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {5 a b^4 \sqrt {a^2+2 a b x+b^2 x^2} \log (x)}{a+b x} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(634\) vs. \(2(219)=438\).
Time = 1.35 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {48 a^6 \sqrt {a^2}+368 a^5 \sqrt {a^2} b x+1280 a^4 \sqrt {a^2} b^2 x^2+2880 a^3 \sqrt {a^2} b^3 x^3+2677 \left (a^2\right )^{3/2} b^4 x^4+565 a \sqrt {a^2} b^5 x^5-192 \sqrt {a^2} b^6 x^6-48 a^6 \sqrt {(a+b x)^2}-320 a^5 b x \sqrt {(a+b x)^2}-960 a^4 b^2 x^2 \sqrt {(a+b x)^2}-1920 a^3 b^3 x^3 \sqrt {(a+b x)^2}-757 a^2 b^4 x^4 \sqrt {(a+b x)^2}+192 a b^5 x^5 \sqrt {(a+b x)^2}-960 a b^4 x^4 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right ) \text {arctanh}\left (\frac {b x}{\sqrt {a^2}-\sqrt {(a+b x)^2}}\right )-960 b^4 x^4 \left (\left (a^2\right )^{3/2}+a \sqrt {a^2} b x-a^2 \sqrt {(a+b x)^2}\right ) \log (x)+480 \left (a^2\right )^{3/2} b^4 x^4 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+480 a \sqrt {a^2} b^5 x^5 \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )-480 a^2 b^4 x^4 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )+480 \left (a^2\right )^{3/2} b^4 x^4 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )+480 a \sqrt {a^2} b^5 x^5 \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )-480 a^2 b^4 x^4 \sqrt {(a+b x)^2} \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{192 x^4 \left (a^2+a b x-\sqrt {a^2} \sqrt {(a+b x)^2}\right )} \]
[In]
[Out]
Time = 2.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (60 a \,b^{4} \ln \left (x \right ) x^{4}+12 b^{5} x^{5}-120 a^{2} b^{3} x^{3}-60 a^{3} b^{2} x^{2}-20 a^{4} b x -3 a^{5}\right )}{12 x^{4} \left (b x +a \right )^{5}}\) | \(76\) |
risch | \(\frac {b^{5} x \sqrt {\left (b x +a \right )^{2}}}{b x +a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-10 a^{2} b^{3} x^{3}-5 a^{3} b^{2} x^{2}-\frac {5}{3} a^{4} b x -\frac {1}{4} a^{5}\right )}{\left (b x +a \right ) x^{4}}+\frac {5 a \,b^{4} \ln \left (x \right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(102\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\frac {12 \, b^{5} x^{5} + 60 \, a b^{4} x^{4} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{3} - 60 \, a^{3} b^{2} x^{2} - 20 \, a^{4} b x - 3 \, a^{5}}{12 \, x^{4}} \]
[In]
[Out]
\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{5}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (149) = 298\).
Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} a b^{4} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} a b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5} x}{2 \, a} + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5} x}{4 \, a^{3}} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{4}}{12 \, a^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{3 \, a^{4}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{3 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{3 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{4 \, a^{2} x^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=b^{5} x \mathrm {sgn}\left (b x + a\right ) + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (b x + a\right ) - \frac {120 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 60 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} \mathrm {sgn}\left (b x + a\right )}{12 \, x^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^5} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{x^5} \,d x \]
[In]
[Out]