Integrand size = 17, antiderivative size = 53 \[ \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx=\frac {1}{12} \sqrt {x+x^4} \left (a x+4 b x+2 a x^4\right )+\frac {1}{12} (-a+4 b) \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \]
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.28 \[ \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx=\frac {\sqrt {x+x^4} \left (x^{3/2} \left (a+4 b+2 a x^3\right )-\frac {(a-4 b) \log \left (x^{3/2}+\sqrt {1+x^3}\right )}{\sqrt {1+x^3}}\right )}{12 \sqrt {x}} \]
(Sqrt[x + x^4]*(x^(3/2)*(a + 4*b + 2*a*x^3) - ((a - 4*b)*Log[x^(3/2) + Sqr t[1 + x^3]])/Sqrt[1 + x^3]))/(12*Sqrt[x])
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.62, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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 |
\(\displaystyle -\frac {1}{12} a \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {1}{6} a \sqrt {x^4+x} x^4+\frac {1}{12} a \sqrt {x^4+x} x+\frac {1}{3} b \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {1}{3} b \sqrt {x^4+x} x\) |
(a*x*Sqrt[x + x^4])/12 + (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
3.7.69.3.1 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 = 2.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91
method | result | size |
trager | \(\frac {x \left (2 a \,x^{3}+a +4 b \right ) \sqrt {x^{4}+x}}{12}+\frac {\left (a -4 b \right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{24}\) | \(48\) |
risch | \(\frac {x^{2} \left (2 a \,x^{3}+a +4 b \right ) \left (x^{3}+1\right )}{12 \sqrt {x \left (x^{3}+1\right )}}-\frac {\left (\frac {b}{2}-\frac {a}{8}\right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}\) | \(59\) |
meijerg | \(-\frac {a \left (-\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \left (6 x^{3}+3\right ) \sqrt {x^{3}+1}}{6}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )}{2}\right )}{6 \sqrt {\pi }}-\frac {b \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}\) | \(71\) |
pseudoelliptic | \(\frac {\left (4 x \sqrt {x^{4}+x}\, \left (a \,x^{3}+\frac {1}{2} a +2 b \right )+\left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) \left (a -4 b \right )\right ) x^{2}}{24 \left (x^{2}+\sqrt {x^{4}+x}\right )^{2} \left (x^{2}-\sqrt {x^{4}+x}\right )^{2}}\) | \(100\) |
default | \(b \left (\frac {x \sqrt {x^{4}+x}}{3}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}\right )+\frac {a \,x^{2} \left (\left (4 x^{4}+2 x \right ) \sqrt {x^{4}+x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right )}{24 \left (x^{2}+\sqrt {x^{4}+x}\right )^{2} \left (x^{2}-\sqrt {x^{4}+x}\right )^{2}}\) | \(141\) |
elliptic | \(\frac {a \,x^{4} \sqrt {x^{4}+x}}{6}+\left (\frac {a}{12}+\frac {b}{3}\right ) x \sqrt {x^{4}+x}-\frac {2 \left (\frac {b}{2}-\frac {a}{8}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(327\) |
Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx=-\frac {1}{24} \, {\left (a - 4 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \frac {1}{12} \, {\left (2 \, a x^{4} + {\left (a + 4 \, b\right )} x\right )} \sqrt {x^{4} + x} \]
-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 = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00 \[ \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx=\frac {1}{12} \, {\left (2 \, a x^{3} + a + 4 \, b\right )} \sqrt {x^{4} + x} x - \frac {1}{24} \, {\left (a - 4 \, b\right )} {\left (\log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )\right )} \]
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. \[ \int \left (b+a x^3\right ) \sqrt {x+x^4} \, dx=\int \left (a\,x^3+b\right )\,\sqrt {x^4+x} \,d x \]