Integrand size = 19, antiderivative size = 58 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{12} \sqrt {-x+x^4} \left (-a x+4 b x+2 a x^4\right )+\frac {1}{12} (-a-4 b) \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.29 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {x^2 \left (-1+x^3\right ) \left (4 b+a \left (-1+2 x^3\right )\right )-(a+4 b) \sqrt {x} \sqrt {-1+x^3} \log \left (x^{3/2}+\sqrt {-1+x^3}\right )}{12 \sqrt {x \left (-1+x^3\right )}} \]
(x^2*(-1 + x^3)*(4*b + a*(-1 + 2*x^3)) - (a + 4*b)*Sqrt[x]*Sqrt[-1 + x^3]* Log[x^(3/2) + Sqrt[-1 + x^3]])/(12*Sqrt[x*(-1 + x^3)])
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.66, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2450, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {x^4-x} \left (a x^3+b\right ) \, dx\) |
\(\Big \downarrow \) 2450 |
\(\displaystyle \int \left (a \sqrt {x^4-x} x^3+b \sqrt {x^4-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{12} a \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {1}{6} a \sqrt {x^4-x} x^4-\frac {1}{12} a \sqrt {x^4-x} x-\frac {1}{3} b \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {1}{3} b \sqrt {x^4-x} x\) |
-1/12*(a*x*Sqrt[-x + x^4]) + (b*x*Sqrt[-x + x^4])/3 + (a*x^4*Sqrt[-x + x^4 ])/6 - (a*ArcTanh[x^2/Sqrt[-x + x^4]])/12 - (b*ArcTanh[x^2/Sqrt[-x + x^4]] )/3
3.8.49.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po lyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
Time = 3.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93
method | result | size |
trager | \(\frac {x \left (2 a \,x^{3}-a +4 b \right ) \sqrt {x^{4}-x}}{12}+\frac {\left (a +4 b \right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{24}\) | \(54\) |
risch | \(\frac {x^{2} \left (2 a \,x^{3}-a +4 b \right ) \left (x^{3}-1\right )}{12 \sqrt {x \left (x^{3}-1\right )}}-\frac {\left (-\frac {b}{2}-\frac {a}{8}\right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{3}\) | \(63\) |
pseudoelliptic | \(\frac {\left (4 \sqrt {x^{4}-x}\, x \left (a \,x^{3}-\frac {1}{2} a +2 b \right )+\left (a +4 b \right ) \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )\right ) x^{2}}{24 \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} \left (x^{2}-\sqrt {x^{4}-x}\right )^{2}}\) | \(110\) |
meijerg | \(-\frac {i a \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \left (-6 x^{3}+3\right ) \sqrt {-x^{3}+1}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )}{2}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}+\frac {i b \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\) | \(117\) |
default | \(b \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )+\frac {a \,x^{2} \left (\left (4 x^{4}-2 x \right ) \sqrt {x^{4}-x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )}{24 \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} \left (x^{2}-\sqrt {x^{4}-x}\right )^{2}}\) | \(157\) |
elliptic | \(\frac {a \,x^{4} \sqrt {x^{4}-x}}{6}+\left (-\frac {a}{12}+\frac {b}{3}\right ) x \sqrt {x^{4}-x}+\frac {2 \left (-\frac {b}{2}-\frac {a}{8}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(329\) |
Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{24} \, {\left (a + 4 \, b\right )} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + \frac {1}{12} \, {\left (2 \, a x^{4} - {\left (a - 4 \, b\right )} x\right )} \sqrt {x^{4} - x} \]
1/24*(a + 4*b)*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1) + 1/12*(2*a*x^4 - (a - 4 *b)*x)*sqrt(x^4 - x)
\[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{3} + b\right )\, dx \]
\[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int { {\left (a x^{3} + b\right )} \sqrt {x^{4} - x} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{12} \, {\left (2 \, a x^{3} - a + 4 \, b\right )} \sqrt {x^{4} - x} x - \frac {1}{24} \, {\left (a + 4 \, b\right )} {\left (\log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) - \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )\right )} \]
1/12*(2*a*x^3 - a + 4*b)*sqrt(x^4 - x)*x - 1/24*(a + 4*b)*(log(sqrt(-1/x^3 + 1) + 1) - log(abs(sqrt(-1/x^3 + 1) - 1)))
Timed out. \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int \sqrt {x^4-x}\,\left (a\,x^3+b\right ) \,d x \]