Integrand size = 28, antiderivative size = 97 \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac {b (d x)^{4+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1369, 14} \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]
[In]
[Out]
Rule 14
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int (d x)^m \left (a b+b^2 x^3\right ) \, dx}{a b+b^2 x^3} \\ & = \frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a b (d x)^m+\frac {b^2 (d x)^{3+m}}{d^3}\right ) \, dx}{a b+b^2 x^3} \\ & = \frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac {b (d x)^{4+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.55 \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {x (d x)^m \sqrt {\left (a+b x^3\right )^2} \left (a (4+m)+b (1+m) x^3\right )}{(1+m) (4+m) \left (a+b x^3\right )} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(\frac {x \left (b m \,x^{3}+b \,x^{3}+a m +4 a \right ) \left (d x \right )^{m} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{\left (4+m \right ) \left (1+m \right ) \left (b \,x^{3}+a \right )}\) | \(56\) |
risch | \(\frac {x \left (b m \,x^{3}+b \,x^{3}+a m +4 a \right ) \left (d x \right )^{m} \sqrt {\left (b \,x^{3}+a \right )^{2}}}{\left (4+m \right ) \left (1+m \right ) \left (b \,x^{3}+a \right )}\) | \(56\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {{\left ({\left (b m + b\right )} x^{4} + {\left (a m + 4 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 4} \]
[In]
[Out]
\[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\int \left (d x\right )^{m} \sqrt {\left (a + b x^{3}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.36 \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {{\left (b d^{m} {\left (m + 1\right )} x^{4} + a d^{m} {\left (m + 4\right )} x\right )} x^{m}}{m^{2} + 5 \, m + 4} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\frac {\left (d x\right )^{m} b m x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + \left (d x\right )^{m} b x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + \left (d x\right )^{m} a m x \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, \left (d x\right )^{m} a x \mathrm {sgn}\left (b x^{3} + a\right )}{m^{2} + 5 \, m + 4} \]
[In]
[Out]
Timed out. \[ \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx=\int {\left (d\,x\right )}^m\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6} \,d x \]
[In]
[Out]