Integrand size = 24, antiderivative size = 252 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a b^4 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {b^5 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 252, 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 = {1369, 276} \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {b^5 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac {5 a b^4 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )} \]
[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 x \left (a b+b^2 x^3\right )^5 \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a^5 b^5 x+5 a^4 b^6 x^4+10 a^3 b^7 x^7+10 a^2 b^8 x^{10}+5 a b^9 x^{13}+b^{10} x^{16}\right ) \, dx}{b^4 \left (a b+b^2 x^3\right )} \\ & = \frac {a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}{2 \left (a+b x^3\right )}+\frac {a^4 b x^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}{a+b x^3}+\frac {5 a^3 b^2 x^8 \sqrt {a^2+2 a b x^3+b^2 x^6}}{4 \left (a+b x^3\right )}+\frac {10 a^2 b^3 x^{11} \sqrt {a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )}+\frac {5 a b^4 x^{14} \sqrt {a^2+2 a b x^3+b^2 x^6}}{14 \left (a+b x^3\right )}+\frac {b^5 x^{17} \sqrt {a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.33 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {x^2 \sqrt {\left (a+b x^3\right )^2} \left (2618 a^5+5236 a^4 b x^3+6545 a^3 b^2 x^6+4760 a^2 b^3 x^9+1870 a b^4 x^{12}+308 b^5 x^{15}\right )}{5236 \left (a+b x^3\right )} \]
[In]
[Out]
Time = 1.91 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.32
method | result | size |
gosper | \(\frac {x^{2} \left (308 b^{5} x^{15}+1870 a \,b^{4} x^{12}+4760 a^{2} b^{3} x^{9}+6545 a^{3} b^{2} x^{6}+5236 a^{4} b \,x^{3}+2618 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{5236 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
default | \(\frac {x^{2} \left (308 b^{5} x^{15}+1870 a \,b^{4} x^{12}+4760 a^{2} b^{3} x^{9}+6545 a^{3} b^{2} x^{6}+5236 a^{4} b \,x^{3}+2618 a^{5}\right ) {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}{5236 \left (b \,x^{3}+a \right )^{5}}\) | \(80\) |
risch | \(\frac {a^{5} x^{2} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{2 b \,x^{3}+2 a}+\frac {a^{4} b \,x^{5} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{b \,x^{3}+a}+\frac {5 a^{3} b^{2} x^{8} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{4 \left (b \,x^{3}+a \right )}+\frac {10 a^{2} b^{3} x^{11} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{11 \left (b \,x^{3}+a \right )}+\frac {5 a \,b^{4} x^{14} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{14 \left (b \,x^{3}+a \right )}+\frac {b^{5} x^{17} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{17 b \,x^{3}+17 a}\) | \(177\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.22 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} + \frac {5}{14} \, a b^{4} x^{14} + \frac {10}{11} \, a^{2} b^{3} x^{11} + \frac {5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac {1}{2} \, a^{5} x^{2} \]
[In]
[Out]
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.22 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} + \frac {5}{14} \, a b^{4} x^{14} + \frac {10}{11} \, a^{2} b^{3} x^{11} + \frac {5}{4} \, a^{3} b^{2} x^{8} + a^{4} b x^{5} + \frac {1}{2} \, a^{5} x^{2} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.41 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\frac {1}{17} \, b^{5} x^{17} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{14} \, a b^{4} x^{14} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {10}{11} \, a^{2} b^{3} x^{11} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {5}{4} \, a^{3} b^{2} x^{8} \mathrm {sgn}\left (b x^{3} + a\right ) + a^{4} b x^{5} \mathrm {sgn}\left (b x^{3} + a\right ) + \frac {1}{2} \, a^{5} x^{2} \mathrm {sgn}\left (b x^{3} + a\right ) \]
[In]
[Out]
Timed out. \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx=\int x\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2} \,d x \]
[In]
[Out]