Integrand size = 19, antiderivative size = 78 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {8 b^2 \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x^3}-\frac {4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac {x \left (b x^2+c x^4\right )^{3/2}}{7 c} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 2025} \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {8 b^2 \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x^3}-\frac {4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac {x \left (b x^2+c x^4\right )^{3/2}}{7 c} \]
[In]
[Out]
Rule 2025
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (b x^2+c x^4\right )^{3/2}}{7 c}-\frac {(4 b) \int x^2 \sqrt {b x^2+c x^4} \, dx}{7 c} \\ & = -\frac {4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac {x \left (b x^2+c x^4\right )^{3/2}}{7 c}+\frac {\left (8 b^2\right ) \int \sqrt {b x^2+c x^4} \, dx}{35 c^2} \\ & = \frac {8 b^2 \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x^3}-\frac {4 b \left (b x^2+c x^4\right )^{3/2}}{35 c^2 x}+\frac {x \left (b x^2+c x^4\right )^{3/2}}{7 c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.73 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {\sqrt {x^2 \left (b+c x^2\right )} \left (8 b^3-4 b^2 c x^2+3 b c^2 x^4+15 c^3 x^6\right )}{105 c^3 x} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (15 c^{2} x^{4}-12 b c \,x^{2}+8 b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{105 c^{3} x}\) | \(50\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (15 c^{2} x^{4}-12 b c \,x^{2}+8 b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{105 c^{3} x}\) | \(50\) |
trager | \(\frac {\left (15 c^{3} x^{6}+3 b \,c^{2} x^{4}-4 b^{2} c \,x^{2}+8 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{105 c^{3} x}\) | \(54\) |
risch | \(\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (15 c^{3} x^{6}+3 b \,c^{2} x^{4}-4 b^{2} c \,x^{2}+8 b^{3}\right )}{105 x \,c^{3}}\) | \(54\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.68 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{105 \, c^{3} x} \]
[In]
[Out]
\[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\int x^{4} \sqrt {x^{2} \left (b + c x^{2}\right )}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.59 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{2} + b}}{105 \, c^{3}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.77 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=-\frac {8 \, b^{\frac {7}{2}} \mathrm {sgn}\left (x\right )}{105 \, c^{3}} + \frac {15 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 42 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b \mathrm {sgn}\left (x\right ) + 35 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right )}{105 \, c^{3}} \]
[In]
[Out]
Time = 12.85 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.68 \[ \int x^4 \sqrt {b x^2+c x^4} \, dx=\frac {\sqrt {c\,x^4+b\,x^2}\,\left (8\,b^3-4\,b^2\,c\,x^2+3\,b\,c^2\,x^4+15\,c^3\,x^6\right )}{105\,c^3\,x} \]
[In]
[Out]