Integrand size = 28, antiderivative size = 227 \[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\sqrt {-\left (\left (\sqrt {c}+\sqrt {d}\right ) \sqrt {d}\right )} (b c+a d) \arctan \left (\frac {\sqrt {-\sqrt {c} \sqrt {d}-d} x \sqrt {x+x^4}}{\sqrt {d} (1+x) \left (1-x+x^2\right )}\right )}{3 c^{3/2} d}-\frac {\sqrt {\left (\sqrt {c}-\sqrt {d}\right ) \sqrt {d}} (b c+a d) \arctan \left (\frac {\sqrt {\sqrt {c} \sqrt {d}-d} x \sqrt {x+x^4}}{\sqrt {d} (1+x) \left (1-x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {a \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 c} \]
1/3*a*x*(x^4+x)^(1/2)/c+1/3*(-(c^(1/2)+d^(1/2))*d^(1/2))^(1/2)*(a*d+b*c)*a rctan((-c^(1/2)*d^(1/2)-d)^(1/2)*x*(x^4+x)^(1/2)/d^(1/2)/(1+x)/(x^2-x+1))/ c^(3/2)/d-1/3*((c^(1/2)-d^(1/2))*d^(1/2))^(1/2)*(a*d+b*c)*arctan((c^(1/2)* d^(1/2)-d)^(1/2)*x*(x^4+x)^(1/2)/d^(1/2)/(1+x)/(x^2-x+1))/c^(3/2)/d+1/3*a* arctanh(x^2/(x^4+x)^(1/2))/c
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.44 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x+x^4} \left (a \left (x^{3/2} \sqrt {1+x^3}+\log \left (x^{3/2}+\sqrt {1+x^3}\right )\right )+(b c+a d) \text {RootSum}\left [16 c-16 d-32 c \text {$\#$1}+32 d \text {$\#$1}+24 c \text {$\#$1}^2-16 d \text {$\#$1}^2-8 c \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (2+2 x^3+2 x^{3/2} \sqrt {1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 c+8 d+12 c \text {$\#$1}-8 d \text {$\#$1}-6 c \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]\right )}{3 c \sqrt {x} \sqrt {1+x^3}} \]
(Sqrt[x + x^4]*(a*(x^(3/2)*Sqrt[1 + x^3] + Log[x^(3/2) + Sqrt[1 + x^3]]) + (b*c + a*d)*RootSum[16*c - 16*d - 32*c*#1 + 32*d*#1 + 24*c*#1^2 - 16*d*#1 ^2 - 8*c*#1^3 + c*#1^4 & , (Log[2 + 2*x^3 + 2*x^(3/2)*Sqrt[1 + x^3] - #1]* #1^2)/(-8*c + 8*d + 12*c*#1 - 8*d*#1 - 6*c*#1^2 + c*#1^3) & ]))/(3*c*Sqrt[ x]*Sqrt[1 + x^3])
Time = 0.93 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2467, 25, 2035, 7266, 2257, 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.
| \(\displaystyle \int \frac {\sqrt {x^4+x} \left (a x^6+b\right )}{c x^6-d} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x^4+x} \int -\frac {\sqrt {x} \sqrt {x^3+1} \left (a x^6+b\right )}{d-c x^6}dx}{\sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x^4+x} \int \frac {\sqrt {x} \sqrt {x^3+1} \left (a x^6+b\right )}{d-c x^6}dx}{\sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x^4+x} \int \frac {x \sqrt {x^3+1} \left (a x^6+b\right )}{d-c x^6}d\sqrt {x}}{\sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle -\frac {2 \sqrt {x^4+x} \int \frac {\sqrt {x+1} \left (a x^2+b\right )}{d-c x^2}dx^{3/2}}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle -\frac {2 \sqrt {x^4+x} \int \left (-\frac {\sqrt {x+1} a}{c}-\frac {(-b c-a d) \sqrt {x+1}}{c \left (d-c x^2\right )}\right )dx^{3/2}}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x^4+x} \left (-\frac {a \text {arcsinh}\left (x^{3/2}\right )}{2 c}+\frac {\sqrt {\sqrt {c}-\sqrt {d}} (a d+b c) \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {c}-\sqrt {d}}}{\sqrt [4]{d} \sqrt {x+1}}\right )}{2 c^{3/2} d^{3/4}}+\frac {\sqrt {\sqrt {c}+\sqrt {d}} (a d+b c) \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {c}+\sqrt {d}}}{\sqrt [4]{d} \sqrt {x+1}}\right )}{2 c^{3/2} d^{3/4}}-\frac {a \sqrt {x+1} x^{3/2}}{2 c}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
(-2*Sqrt[x + x^4]*(-1/2*(a*x^(3/2)*Sqrt[1 + x])/c - (a*ArcSinh[x^(3/2)])/( 2*c) + (Sqrt[Sqrt[c] - Sqrt[d]]*(b*c + a*d)*ArcTan[(Sqrt[Sqrt[c] - Sqrt[d] ]*x^(3/2))/(d^(1/4)*Sqrt[1 + x])])/(2*c^(3/2)*d^(3/4)) + (Sqrt[Sqrt[c] + S qrt[d]]*(b*c + a*d)*ArcTanh[(Sqrt[Sqrt[c] + Sqrt[d]]*x^(3/2))/(d^(1/4)*Sqr t[1 + x])])/(2*c^(3/2)*d^(3/4))))/(3*Sqrt[x]*Sqrt[1 + x^3])
3.27.11.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Time = 1.99 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {a \,x^{2} \left (x^{3}+1\right )}{3 c \sqrt {x \left (x^{3}+1\right )}}+\frac {\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{3}+\frac {2 \left (a d +b c \right ) \left (-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}+1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}+\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}+1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{3 \sqrt {c d}}}{2 c}\) | \(164\) |
| default | \(\frac {a \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}\right )}{c}+\frac {\left (a d +b c \right ) \left (-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}+1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}+\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}+1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{3 c \sqrt {c d}}\) | \(171\) |
| pseudoelliptic | \(\frac {x \left (2 \left (a d +b c \right ) \sqrt {\left (\sqrt {c d}+d \right ) d}\, \left (-\sqrt {c d}+d \right ) \arctan \left (\frac {d \sqrt {x^{4}+x}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )+\sqrt {\left (\sqrt {c d}-d \right ) d}\, \left (2 \left (\sqrt {c d}+d \right ) \left (a d +b c \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x^{4}+x}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )+a \left (-2 x \sqrt {x^{4}+x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) \sqrt {\left (\sqrt {c d}+d \right ) d}\, \sqrt {c d}\right )\right )}{6 \sqrt {\left (\sqrt {c d}+d \right ) d}\, \sqrt {c d}\, \sqrt {\left (\sqrt {c d}-d \right ) d}\, c \left (x^{2}+\sqrt {x^{4}+x}\right ) \left (x^{2}-\sqrt {x^{4}+x}\right )}\) | \(240\) |
| elliptic | \(\text {Expression too large to display}\) | \(686\) |
1/3*a*x^2/c*(x^3+1)/(x*(x^3+1))^(1/2)+1/2/c*(1/3*a*ln(-2*x^3-2*x*(x^4+x)^( 1/2)-1)+2/3*(a*d+b*c)/(c*d)^(1/2)*(-((c*d)^(1/2)+d)/(((c*d)^(1/2)+d)*d)^(1 /2)*arctanh(d*(x*(x^3+1))^(1/2)/x^2/(((c*d)^(1/2)+d)*d)^(1/2))+((c*d)^(1/2 )-d)/(((c*d)^(1/2)-d)*d)^(1/2)*arctan(d*(x*(x^3+1))^(1/2)/x^2/(((c*d)^(1/2 )-d)*d)^(1/2))))
Timed out. \[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{6} + b\right )}{c x^{6} - d}\, dx \]
\[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int { \frac {{\left (a x^{6} + b\right )} \sqrt {x^{4} + x}}{c x^{6} - d} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 680 vs. \(2 (175) = 350\).
Time = 1.42 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.00 \[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x^{4} + x} a x}{3 \, c} + \frac {a \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, c} - \frac {a \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, c} - \frac {{\left ({\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} d^{2}\right )} a c^{2} {\left | d \right |} + {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d\right )} b c^{2} {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} d^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c d^{3}\right )} a {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{3} d + 5 \, \sqrt {c d} \sqrt {-d^{2} - \sqrt {c d} d} c^{2} d^{2}\right )} b {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {c d + \sqrt {c^{2} d^{2} + {\left (c^{2} - c d\right )} c d}}{c d}}}\right )}{3 \, {\left (4 \, c^{4} d^{3} + c^{3} d^{4} - 5 \, c^{2} d^{5}\right )} {\left | c \right |}} + \frac {{\left ({\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} d^{2}\right )} a c^{2} {\left | d \right |} + {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d\right )} b c^{2} {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} d^{2} + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c d^{3}\right )} a {\left | d \right |} - {\left (4 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{3} d + 5 \, \sqrt {c d} \sqrt {-d^{2} + \sqrt {c d} d} c^{2} d^{2}\right )} b {\left | d \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {c d - \sqrt {c^{2} d^{2} + {\left (c^{2} - c d\right )} c d}}{c d}}}\right )}{3 \, {\left (4 \, c^{4} d^{3} + c^{3} d^{4} - 5 \, c^{2} d^{5}\right )} {\left | c \right |}} \]
1/3*sqrt(x^4 + x)*a*x/c + 1/6*a*log(sqrt(1/x^3 + 1) + 1)/c - 1/6*a*log(abs (sqrt(1/x^3 + 1) - 1))/c - 1/3*((4*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d)*d)*c*d + 5*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d)*d)*d^2)*a*c^2*abs(d) + (4*sqrt(c*d)*sq rt(-d^2 - sqrt(c*d)*d)*c^2 + 5*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d)*d)*c*d)*b*c ^2*abs(d) - (4*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d)*d)*c^2*d^2 + 5*sqrt(c*d)*sq rt(-d^2 - sqrt(c*d)*d)*c*d^3)*a*abs(d) - (4*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d )*d)*c^3*d + 5*sqrt(c*d)*sqrt(-d^2 - sqrt(c*d)*d)*c^2*d^2)*b*abs(d))*arcta n(sqrt(1/x^3 + 1)/sqrt(-(c*d + sqrt(c^2*d^2 + (c^2 - c*d)*c*d))/(c*d)))/(( 4*c^4*d^3 + c^3*d^4 - 5*c^2*d^5)*abs(c)) + 1/3*((4*sqrt(c*d)*sqrt(-d^2 + s qrt(c*d)*d)*c*d + 5*sqrt(c*d)*sqrt(-d^2 + sqrt(c*d)*d)*d^2)*a*c^2*abs(d) + (4*sqrt(c*d)*sqrt(-d^2 + sqrt(c*d)*d)*c^2 + 5*sqrt(c*d)*sqrt(-d^2 + sqrt( c*d)*d)*c*d)*b*c^2*abs(d) - (4*sqrt(c*d)*sqrt(-d^2 + sqrt(c*d)*d)*c^2*d^2 + 5*sqrt(c*d)*sqrt(-d^2 + sqrt(c*d)*d)*c*d^3)*a*abs(d) - (4*sqrt(c*d)*sqrt (-d^2 + sqrt(c*d)*d)*c^3*d + 5*sqrt(c*d)*sqrt(-d^2 + sqrt(c*d)*d)*c^2*d^2) *b*abs(d))*arctan(sqrt(1/x^3 + 1)/sqrt(-(c*d - sqrt(c^2*d^2 + (c^2 - c*d)* c*d))/(c*d)))/((4*c^4*d^3 + c^3*d^4 - 5*c^2*d^5)*abs(c))
Timed out. \[ \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\int -\frac {\left (a\,x^6+b\right )\,\sqrt {x^4+x}}{d-c\,x^6} \,d x \]