Integrand size = 22, antiderivative size = 52 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {x+x^4} \left (2 b+2 b x^3+3 a x^6\right )}{9 x^5}+\frac {1}{3} a \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \] Output:
1/9*(x^4+x)^(1/2)*(3*a*x^6+2*b*x^3+2*b)/x^5+1/3*a*arctanh(x^2/(x^4+x)^(1/2 ))
Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {\sqrt {x+x^4} \left (2 b+2 b x^3+3 a x^6\right )}{9 x^5}+\frac {a \sqrt {x+x^4} \log \left (x^{3/2}+\sqrt {1+x^3}\right )}{3 \sqrt {x} \sqrt {1+x^3}} \] Input:
Integrate[(Sqrt[x + x^4]*(-b + a*x^6))/x^6,x]
Output:
(Sqrt[x + x^4]*(2*b + 2*b*x^3 + 3*a*x^6))/(9*x^5) + (a*Sqrt[x + x^4]*Log[x ^(3/2) + Sqrt[1 + x^3]])/(3*Sqrt[x]*Sqrt[1 + x^3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2445, 2449, 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.
\(\displaystyle \int \frac {\sqrt {x^4+x} \left (a x^6-b\right )}{x^6} \, dx\) |
\(\Big \downarrow \) 2445 |
\(\displaystyle \int \frac {\left (a x^2-b\right ) \sqrt {x^4+x}}{x^6}dx+\frac {a \left (x^4+x\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 2449 |
\(\displaystyle \int \left (\frac {a \sqrt {x^4+x}}{x^4}-\frac {b \sqrt {x^4+x}}{x^6}\right )dx+\frac {a \left (x^4+x\right )^{3/2}}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3^{3/4} a x (x+1) \sqrt {\frac {x^2-x+1}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) x+1}{\left (1+\sqrt {3}\right ) x+1}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt {\frac {x (x+1)}{\left (\left (1+\sqrt {3}\right ) x+1\right )^2}} \sqrt {x^4+x}}+\frac {a \left (x^4+x\right )^{3/2}}{3 x^3}-\frac {2 a \sqrt {x^4+x}}{5 x^3}+\frac {2 b \left (x^4+x\right )^{3/2}}{9 x^6}\) |
Input:
Int[(Sqrt[x + x^4]*(-b + a*x^6))/x^6,x]
Output:
(-2*a*Sqrt[x + x^4])/(5*x^3) + (2*b*(x + x^4)^(3/2))/(9*x^6) + (a*(x + x^4 )^(3/2))/(3*x^3) + (3^(3/4)*a*x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[ 3])*x)^2]*EllipticF[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3])*x)], ( 2 + Sqrt[3])/4])/(5*Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[x + x^4 ])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> With[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[(c*x)^ m*ExpandToSum[Pq - Pqq*x^q - a*Pqq*(m + q - n + 1)*(x^(q - n)/(b*(m + q + n *p + 1))), x]*(a*x^j + b*x^n)^p, x] + Simp[Pqq*(c*x)^(m + q - n + 1)*((a*x^ j + b*x^n)^(p + 1)/(b*c^(q - n + 1)*(m + q + n*p + 1))), x]] /; GtQ[q, n - 1] && NeQ[m + q + n*p + 1, 0] && (IntegerQ[2*p] || IntegerQ[p + (q + 1)/(2* n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && !IntegerQ[p] && IGtQ [j, 0] && IGtQ[n, 0] && LtQ[j, n]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Time = 0.72 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88
method | result | size |
meijerg | \(-\frac {a \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}+\frac {2 b \left (x^{3}+1\right )^{\frac {3}{2}}}{9 x^{\frac {9}{2}}}\) | \(46\) |
trager | \(\frac {\sqrt {x^{4}+x}\, \left (3 a \,x^{6}+2 b \,x^{3}+2 b \right )}{9 x^{5}}-\frac {a \ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{6}\) | \(51\) |
risch | \(\frac {\left (x^{3}+1\right ) \left (3 a \,x^{6}+2 b \,x^{3}+2 b \right )}{9 x^{4} \sqrt {x \left (x^{3}+1\right )}}-\frac {a \ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{6}\) | \(58\) |
pseudoelliptic | \(\frac {\left (6 a \,x^{6}+4 b \,x^{3}+4 b \right ) \sqrt {x^{4}+x}+3 \left (\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) a \,x^{5}}{18 x^{5}}\) | \(74\) |
default | \(a \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}\right )-b \left (-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}-\frac {2 \sqrt {x^{4}+x}}{9 x^{2}}\right )\) | \(81\) |
elliptic | \(\frac {2 b \sqrt {x^{4}+x}}{9 x^{5}}+\frac {2 b \sqrt {x^{4}+x}}{9 x^{2}}+\frac {a x \sqrt {x^{4}+x}}{3}-\frac {a \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(329\) |
Input:
int((x^4+x)^(1/2)*(a*x^6-b)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/6*a/Pi^(1/2)*(-2*Pi^(1/2)*x^(3/2)*(x^3+1)^(1/2)-2*Pi^(1/2)*arcsinh(x^(3 /2)))+2/9*b/x^(9/2)*(x^3+1)^(3/2)
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {3 \, a x^{5} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + 2 \, {\left (3 \, a x^{6} + 2 \, b x^{3} + 2 \, b\right )} \sqrt {x^{4} + x}}{18 \, x^{5}} \] Input:
integrate((x^4+x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="fricas")
Output:
1/18*(3*a*x^5*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + 2*(3*a*x^6 + 2*b*x^3 + 2*b)*sqrt(x^4 + x))/x^5
\[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{6} - b\right )}{x^{6}}\, dx \] Input:
integrate((x**4+x)**(1/2)*(a*x**6-b)/x**6,x)
Output:
Integral(sqrt(x*(x + 1)*(x**2 - x + 1))*(a*x**6 - b)/x**6, x)
\[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\int { \frac {{\left (a x^{6} - b\right )} \sqrt {x^{4} + x}}{x^{6}} \,d x } \] Input:
integrate((x^4+x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="maxima")
Output:
integrate((a*x^6 - b)*sqrt(x^4 + x)/x^6, x)
Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {1}{3} \, \sqrt {x^{4} + x} a x + \frac {2}{9} \, b {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} + \frac {1}{6} \, a \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{6} \, a \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \] Input:
integrate((x^4+x)^(1/2)*(a*x^6-b)/x^6,x, algorithm="giac")
Output:
1/3*sqrt(x^4 + x)*a*x + 2/9*b*(1/x^3 + 1)^(3/2) + 1/6*a*log(sqrt(1/x^3 + 1 ) + 1) - 1/6*a*log(abs(sqrt(1/x^3 + 1) - 1))
Time = 8.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {2\,b\,\left (x^3+1\right )\,\sqrt {x^4+x}}{9\,x^5}+\frac {2\,a\,x\,\sqrt {x^4+x}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {1}{2};\ \frac {3}{2};\ -x^3\right )}{3\,\sqrt {x^3+1}} \] Input:
int(-((b - a*x^6)*(x + x^4)^(1/2))/x^6,x)
Output:
(2*b*(x^3 + 1)*(x + x^4)^(1/2))/(9*x^5) + (2*a*x*(x + x^4)^(1/2)*hypergeom ([-1/2, 1/2], 3/2, -x^3))/(3*(x^3 + 1)^(1/2))
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {x+x^4} \left (-b+a x^6\right )}{x^6} \, dx=\frac {6 \sqrt {x^{3}+1}\, a \,x^{6}+4 \sqrt {x^{3}+1}\, b \,x^{3}+4 \sqrt {x^{3}+1}\, b -3 \sqrt {x}\, \mathrm {log}\left (\sqrt {x^{3}+1}-\sqrt {x}\, x \right ) a \,x^{4}+3 \sqrt {x}\, \mathrm {log}\left (\sqrt {x^{3}+1}+\sqrt {x}\, x \right ) a \,x^{4}}{18 \sqrt {x}\, x^{4}} \] Input:
int((x^4+x)^(1/2)*(a*x^6-b)/x^6,x)
Output:
(6*sqrt(x**3 + 1)*a*x**6 + 4*sqrt(x**3 + 1)*b*x**3 + 4*sqrt(x**3 + 1)*b - 3*sqrt(x)*log(sqrt(x**3 + 1) - sqrt(x)*x)*a*x**4 + 3*sqrt(x)*log(sqrt(x**3 + 1) + sqrt(x)*x)*a*x**4)/(18*sqrt(x)*x**4)