Integrand size = 20, antiderivative size = 163 \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1917, 654, 626, 635, 212} \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}}-\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x} \]
[In]
[Out]
Rule 212
Rule 626
Rule 635
Rule 654
Rule 1917
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a x^2+b x^3+c x^4} \int x \sqrt {a+b x+c x^2} \, dx}{x \sqrt {a+b x+c x^2}} \\ & = \frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}-\frac {\left (b \sqrt {a x^2+b x^3+c x^4}\right ) \int \sqrt {a+b x+c x^2} \, dx}{2 c x \sqrt {a+b x+c x^2}} \\ & = -\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {\left (b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 x \sqrt {a+b x+c x^2}} \\ & = -\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {\left (b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c^2 x \sqrt {a+b x+c x^2}} \\ & = -\frac {b (b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{8 c^2 x}+\frac {\left (a+b x+c x^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c x}+\frac {b \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} x \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.74 \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {2 \sqrt {c} x (a+x (b+c x)) \left (-3 b^2+2 b c x+8 c \left (a+c x^2\right )\right )-3 \left (b^3-4 a b c\right ) x \sqrt {a+x (b+c x)} \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{48 c^{5/2} \sqrt {x^2 (a+x (b+c x))}} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {\left (8 c^{2} x^{2}+2 b c x +8 a c -3 b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{24 c^{2} x}-\frac {b \left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{16 c^{\frac {5}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(121\) |
pseudoelliptic | \(\frac {16 x^{2} \sqrt {c \,x^{2}+b x +a}\, c^{\frac {5}{2}}+4 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b x +16 a \,c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}-6 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}\, b^{2}-12 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) a b c +3 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b^{3}}{48 c^{\frac {5}{2}}}\) | \(142\) |
default | \(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (16 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {5}{2}}-12 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b x -6 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{2}-12 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a b \,c^{2}+3 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{3} c \right )}{48 x \sqrt {c \,x^{2}+b x +a}\, c^{\frac {7}{2}}}\) | \(167\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.60 \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\left [-\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{96 \, c^{3} x}, -\frac {3 \, {\left (b^{3} - 4 \, a b c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (8 \, c^{3} x^{2} + 2 \, b c^{2} x - 3 \, b^{2} c + 8 \, a c^{2}\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{48 \, c^{3} x}\right ] \]
[In]
[Out]
\[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\int \sqrt {a x^{2} + b x^{3} + c x^{4}}\, dx \]
[In]
[Out]
\[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{3} + a x^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, x \mathrm {sgn}\left (x\right ) + \frac {b \mathrm {sgn}\left (x\right )}{c}\right )} x - \frac {3 \, b^{2} \mathrm {sgn}\left (x\right ) - 8 \, a c \mathrm {sgn}\left (x\right )}{c^{2}}\right )} - \frac {{\left (b^{3} \mathrm {sgn}\left (x\right ) - 4 \, a b c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {5}{2}}} + \frac {{\left (3 \, b^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 12 \, a b c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6 \, \sqrt {a} b^{2} \sqrt {c} - 16 \, a^{\frac {3}{2}} c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{48 \, c^{\frac {5}{2}}} \]
[In]
[Out]
Timed out. \[ \int \sqrt {a x^2+b x^3+c x^4} \, dx=\int \sqrt {c\,x^4+b\,x^3+a\,x^2} \,d x \]
[In]
[Out]