Integrand size = 26, antiderivative size = 165 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 276} \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}-\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )} \]
[In]
[Out]
Rule 276
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \frac {\left (a b+b^2 x^3\right )^3}{x^{11}} \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (\frac {a^3 b^3}{x^{11}}+\frac {3 a^2 b^4}{x^8}+\frac {3 a b^5}{x^5}+\frac {b^6}{x^2}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )} \\ & = -\frac {a^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{10 x^{10} \left (a+b x^3\right )}-\frac {3 a^2 b \sqrt {a^2+2 a b x^3+b^2 x^6}}{7 x^7 \left (a+b x^3\right )}-\frac {3 a b^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac {b^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.37 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {\sqrt {\left (a+b x^3\right )^2} \left (14 a^3+60 a^2 b x^3+105 a b^2 x^6+140 b^3 x^9\right )}{140 x^{10} \left (a+b x^3\right )} \]
[In]
[Out]
Time = 12.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.35
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-b^{3} x^{9}-\frac {3}{4} b^{2} x^{6} a -\frac {3}{7} a^{2} b \,x^{3}-\frac {1}{10} a^{3}\right )}{\left (b \,x^{3}+a \right ) x^{10}}\) | \(57\) |
gosper | \(-\frac {\left (140 b^{3} x^{9}+105 b^{2} x^{6} a +60 a^{2} b \,x^{3}+14 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{140 x^{10} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
default | \(-\frac {\left (140 b^{3} x^{9}+105 b^{2} x^{6} a +60 a^{2} b \,x^{3}+14 a^{3}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}{140 x^{10} \left (b \,x^{3}+a \right )^{3}}\) | \(58\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {140 \, b^{3} x^{9} + 105 \, a b^{2} x^{6} + 60 \, a^{2} b x^{3} + 14 \, a^{3}}{140 \, x^{10}} \]
[In]
[Out]
\[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=\int \frac {\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}{x^{11}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.22 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {140 \, b^{3} x^{9} + 105 \, a b^{2} x^{6} + 60 \, a^{2} b x^{3} + 14 \, a^{3}}{140 \, x^{10}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {140 \, b^{3} x^{9} \mathrm {sgn}\left (b x^{3} + a\right ) + 105 \, a b^{2} x^{6} \mathrm {sgn}\left (b x^{3} + a\right ) + 60 \, a^{2} b x^{3} \mathrm {sgn}\left (b x^{3} + a\right ) + 14 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )}{140 \, x^{10}} \]
[In]
[Out]
Time = 8.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}}{x^{11}} \, dx=-\frac {a^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{10\,x^{10}\,\left (b\,x^3+a\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{x\,\left (b\,x^3+a\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{4\,x^4\,\left (b\,x^3+a\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6}}{7\,x^7\,\left (b\,x^3+a\right )} \]
[In]
[Out]