Integrand size = 22, antiderivative size = 238 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt [4]{-1} \sqrt {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}-i b} \sqrt {x+x^4}}{\sqrt {2} \left (\sqrt {a}-i \sqrt {b}\right ) x^2}\right )}{3 a^{3/2}}+\frac {(-1)^{3/4} \sqrt {\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}+i b} x \sqrt {x+x^4}}{\sqrt {2} \sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^{3/2}}+\frac {\text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 a} \]
1/3*x*(x^4+x)^(1/2)/a+1/3*(-1)^(1/4)*((a^(1/2)-I*b^(1/2))*b^(1/2))^(1/2)*a rctan((1/2+1/2*I)*(a^(1/2)*b^(1/2)-I*b)^(1/2)*(x^4+x)^(1/2)*2^(1/2)/(a^(1/ 2)-I*b^(1/2))/x^2)/a^(3/2)+1/3*(-1)^(3/4)*((a^(1/2)+I*b^(1/2))*b^(1/2))^(1 /2)*arctan((1/2+1/2*I)*(a^(1/2)*b^(1/2)+I*b)^(1/2)*x*(x^4+x)^(1/2)*2^(1/2) /b^(1/2)/(1+x)/(x^2-x+1))/a^(3/2)+1/3*arctanh(x^2/(x^4+x)^(1/2))/a
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.74 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x+x^4} \left (x^{3/2} \sqrt {1+x^3}+\log \left (x^{3/2}+\sqrt {1+x^3}\right )-b \text {RootSum}\left [16 a+16 b-32 a \text {$\#$1}-32 b \text {$\#$1}+24 a \text {$\#$1}^2+16 b \text {$\#$1}^2-8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (2+2 x^3+2 x^{3/2} \sqrt {1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 a-8 b+12 a \text {$\#$1}+8 b \text {$\#$1}-6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]\right )}{3 a \sqrt {x} \sqrt {1+x^3}} \]
(Sqrt[x + x^4]*(x^(3/2)*Sqrt[1 + x^3] + Log[x^(3/2) + Sqrt[1 + x^3]] - b*R ootSum[16*a + 16*b - 32*a*#1 - 32*b*#1 + 24*a*#1^2 + 16*b*#1^2 - 8*a*#1^3 + a*#1^4 & , (Log[2 + 2*x^3 + 2*x^(3/2)*Sqrt[1 + x^3] - #1]*#1^2)/(-8*a - 8*b + 12*a*#1 + 8*b*#1 - 6*a*#1^2 + a*#1^3) & ]))/(3*a*Sqrt[x]*Sqrt[1 + x^ 3])
Time = 0.66 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2467, 1817, 1815, 1615, 211, 222, 1489, 301, 222, 291, 221}
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.
| \(\displaystyle \int \frac {x^6 \sqrt {x^4+x}}{a x^6+b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x^4+x} \int \frac {x^{13/2} \sqrt {x^3+1}}{a x^6+b}dx}{\sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 1817 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \int \frac {x^7 \sqrt {x^3+1}}{a x^6+b}d\sqrt {x}}{\sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 1815 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \int \frac {x^2 \sqrt {x+1}}{a x^2+b}dx^{3/2}}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 1615 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\int \sqrt {x+1}dx^{3/2}}{a}-\frac {b \int \frac {\sqrt {x+1}}{a x^2+b}dx^{3/2}}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \int \frac {1}{\sqrt {x+1}}dx^{3/2}+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \int \frac {\sqrt {x+1}}{a x^2+b}dx^{3/2}}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \int \frac {\sqrt {x+1}}{a x^2+b}dx^{3/2}}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 1489 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \left (\frac {\sqrt {-a} \int \frac {\sqrt {x+1}}{\sqrt {-a} \sqrt {b}-a x}dx^{3/2}}{2 \sqrt {b}}+\frac {\sqrt {-a} \int \frac {\sqrt {x+1}}{a x+\sqrt {-a} \sqrt {b}}dx^{3/2}}{2 \sqrt {b}}\right )}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 301 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \left (\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a} \sqrt {b}+a\right ) \int \frac {1}{\sqrt {x+1} \left (\sqrt {-a} \sqrt {b}-a x\right )}dx^{3/2}}{a}-\frac {\int \frac {1}{\sqrt {x+1}}dx^{3/2}}{a}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\sqrt {x+1} \left (a x+\sqrt {-a} \sqrt {b}\right )}dx^{3/2}}{\sqrt {-a}}+\frac {\int \frac {1}{\sqrt {x+1}}dx^{3/2}}{a}\right )}{2 \sqrt {b}}\right )}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \left (\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a} \sqrt {b}+a\right ) \int \frac {1}{\sqrt {x+1} \left (\sqrt {-a} \sqrt {b}-a x\right )}dx^{3/2}}{a}-\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\sqrt {x+1} \left (a x+\sqrt {-a} \sqrt {b}\right )}dx^{3/2}}{\sqrt {-a}}+\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}\right )}{2 \sqrt {b}}\right )}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \left (\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} \sqrt {b}-a\right ) x}d\frac {x^{3/2}}{\sqrt {x+1}}}{\sqrt {-a}}+\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {\left (\sqrt {-a} \sqrt {b}+a\right ) \int \frac {1}{\sqrt {-a} \sqrt {b}-\left (a+\sqrt {-a} \sqrt {b}\right ) x}d\frac {x^{3/2}}{\sqrt {x+1}}}{a}-\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}\right )}{2 \sqrt {b}}\right )}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {x^4+x} \left (\frac {\frac {1}{2} \text {arcsinh}\left (x^{3/2}\right )+\frac {1}{2} \sqrt {x+1} x^{3/2}}{a}-\frac {b \left (\frac {\sqrt {-a} \left (\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}-\frac {\sqrt {\sqrt {-a}+\sqrt {b}} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x+1}}\right )}{a \sqrt [4]{b}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {\sqrt {\sqrt {-a} \sqrt {b}+a} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x+1}}\right )}{\sqrt [4]{-a} a \sqrt [4]{b}}-\frac {\text {arcsinh}\left (x^{3/2}\right )}{a}\right )}{2 \sqrt {b}}\right )}{a}\right )}{3 \sqrt {x} \sqrt {x^3+1}}\) |
(2*Sqrt[x + x^4]*(((x^(3/2)*Sqrt[1 + x])/2 + ArcSinh[x^(3/2)]/2)/a - (b*(( Sqrt[-a]*(ArcSinh[x^(3/2)]/a - (Sqrt[Sqrt[-a] + Sqrt[b]]*ArcTanh[(Sqrt[Sqr t[-a] + Sqrt[b]]*x^(3/2))/(b^(1/4)*Sqrt[1 + x])])/(a*b^(1/4))))/(2*Sqrt[b] ) + (Sqrt[-a]*(-(ArcSinh[x^(3/2)]/a) + (Sqrt[a + Sqrt[-a]*Sqrt[b]]*ArcTanh [(Sqrt[a + Sqrt[-a]*Sqrt[b]]*x^(3/2))/((-a)^(1/4)*b^(1/4)*Sqrt[1 + x])])/( (-a)^(1/4)*a*b^(1/4))))/(2*Sqrt[b])))/a))/(3*Sqrt[x]*Sqrt[1 + x^3])
3.27.63.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S imp[c/(2*r) Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (c_.)*(x_)^4), x_Symbol] :> Simp[f^4/c Int[(f*x)^(m - 4)*(d + e*x^2)^q, x], x] - Simp[a* (f^4/c) Int[(f*x)^(m - 4)*((d + e*x^2)^q/(a + c*x^4)), x], x] /; FreeQ[{a , c, d, e, f, q}, x] && !IntegerQ[q] && GtQ[m, 3]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_ .), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/ k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m ]
Int[((f_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n _))^(q_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[Int[x^( k*(m + 1) - 1)*(d + e*(x^(k*n)/f))^q*(a + c*(x^(2*k*n)/f))^p, x], x, (f*x)^ (1/k)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0 ] && FractionQ[m] && 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]
Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(171)=342\).
Time = 3.02 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.58
| method | result | size |
| pseudoelliptic | \(-\frac {\left (-\frac {\left (-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )+\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )\right ) \left (-\sqrt {b \left (a +b \right )}+b \right ) \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}}{2}+\left (-2 x \sqrt {x^{4}+x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) a \sqrt {b}\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}+2 a b \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )\right )}{6 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, \sqrt {b}\, a^{2}}\) | \(375\) |
| default | \(\frac {\frac {x \sqrt {x^{4}+x}}{3}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}}{a}+\frac {\left (-\sqrt {b \left (a +b \right )}+b \right ) \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \left (-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )+\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}{4}+\left (\sqrt {b \left (a +b \right )}+b \right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )\right )\right )}{3 \sqrt {b}\, a^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\) | \(381\) |
| risch | \(\frac {x^{2} \left (x^{3}+1\right )}{3 a \sqrt {x \left (x^{3}+1\right )}}+\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{6 a}+\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}-\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2}}+\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2}}\) | \(541\) |
| elliptic | \(\text {Expression too large to display}\) | \(675\) |
-1/6*((-1/2*(-ln(1/x^3*(b^(1/2)*x^3+(a+b)^(1/2)*x^3-(x^4+x)^(1/2)*(2*(b*(a +b))^(1/2)+2*b)^(1/2)*x+b^(1/2)))+ln(1/x^3*(b^(1/2)*x^3+(a+b)^(1/2)*x^3+(x ^4+x)^(1/2)*(2*(b*(a+b))^(1/2)+2*b)^(1/2)*x+b^(1/2))))*(-(b*(a+b))^(1/2)+b )*(2*(b*(a+b))^(1/2)+2*b)^(1/2)+(-2*x*(x^4+x)^(1/2)+ln((-x^2+(x^4+x)^(1/2) )/x^2)-ln((x^2+(x^4+x)^(1/2))/x^2))*a*b^(1/2))*(4*(a+b)^(1/2)*b^(1/2)-2*(b *(a+b))^(1/2)-2*b)^(1/2)+2*a*b*(arctan(((2*(b*(a+b))^(1/2)+2*b)^(1/2)*x^2- 2*b^(1/2)*(x^4+x)^(1/2))/(4*(a+b)^(1/2)*b^(1/2)-2*(b*(a+b))^(1/2)-2*b)^(1/ 2)/x^2)-arctan(((2*(b*(a+b))^(1/2)+2*b)^(1/2)*x^2+2*b^(1/2)*(x^4+x)^(1/2)) /x^2/(4*(a+b)^(1/2)*b^(1/2)-2*(b*(a+b))^(1/2)-2*b)^(1/2))))/(4*(a+b)^(1/2) *b^(1/2)-2*(b*(a+b))^(1/2)-2*b)^(1/2)/b^(1/2)/a^2
Leaf count of result is larger than twice the leaf count of optimal. 1784 vs. \(2 (162) = 324\).
Time = 54.75 (sec) , antiderivative size = 1784, normalized size of antiderivative = 7.50 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\text {Too large to display} \]
-1/12*(a*sqrt((a^3*sqrt(-b/a^5) - b)/a^3)*log((2*((9*a^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4)*x^4 - (a^4*b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b ^4 + 243*b^5)*x + ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*a^3*b ^4)*x^4 + (9*a^6*b + 73*a^5*b^2 + 279*a^4*b^3 + 567*a^3*b^4)*x)*sqrt(-b/a^ 5))*sqrt(x^4 + x) + ((a^6 - 2*a^5*b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b ^4)*x^6 - a^5*b + 22*a^4*b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5*b + 73 *a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 + (10*a^7*b + 51*a^6*b^2 + 108*a ^5*b^3 + 243*a^4*b^4 - (8*a^8 + 95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^8 - 12*a^7*b - 120*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^3)*sqrt(- b/a^5))*sqrt((a^3*sqrt(-b/a^5) - b)/a^3))/(a*x^6 + b)) - a*sqrt((a^3*sqrt( -b/a^5) - b)/a^3)*log((2*((9*a^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4) *x^4 - (a^4*b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b^4 + 243*b^5)*x + ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*a^3*b^4)*x^4 + (9*a^6*b + 73* a^5*b^2 + 279*a^4*b^3 + 567*a^3*b^4)*x)*sqrt(-b/a^5))*sqrt(x^4 + x) - ((a^ 6 - 2*a^5*b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b^4)*x^6 - a^5*b + 22*a^4 *b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5*b + 73*a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 + (10*a^7*b + 51*a^6*b^2 + 108*a^5*b^3 + 243*a^4*b^4 - ( 8*a^8 + 95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^8 - 12*a^7*b - 12 0*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^3)*sqrt(-b/a^5))*sqrt((a^3*sqrt(- b/a^5) - b)/a^3))/(a*x^6 + b)) + a*sqrt(-(a^3*sqrt(-b/a^5) + b)/a^3)*lo...
\[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int \frac {x^{6} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{a x^{6} + b}\, dx \]
\[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int { \frac {\sqrt {x^{4} + x} x^{6}}{a x^{6} + b} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (162) = 324\).
Time = 1.00 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x^{4} + x} x}{3 \, a} + \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b + \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} - \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b - \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} + \frac {\log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a} - \frac {\log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a} \]
1/3*sqrt(x^4 + x)*x/a + 1/3*((4*sqrt(-a*b)*sqrt(-b^2 - sqrt(-a*b)*b)*a - 5 *sqrt(-a*b)*sqrt(-b^2 - sqrt(-a*b)*b)*b)*a^2*abs(b) + (4*sqrt(-a*b)*sqrt(- b^2 - sqrt(-a*b)*b)*a^2*b - 5*sqrt(-a*b)*sqrt(-b^2 - sqrt(-a*b)*b)*a*b^2)* abs(b))*arctan(sqrt(1/x^3 + 1)/sqrt(-(a*b + sqrt(a^2*b^2 - (a^2 + a*b)*a*b ))/(a*b)))/((4*a^4*b^2 - a^3*b^3 - 5*a^2*b^4)*abs(a)) - 1/3*((4*sqrt(-a*b) *sqrt(-b^2 + sqrt(-a*b)*b)*a - 5*sqrt(-a*b)*sqrt(-b^2 + sqrt(-a*b)*b)*b)*a ^2*abs(b) + (4*sqrt(-a*b)*sqrt(-b^2 + sqrt(-a*b)*b)*a^2*b - 5*sqrt(-a*b)*s qrt(-b^2 + sqrt(-a*b)*b)*a*b^2)*abs(b))*arctan(sqrt(1/x^3 + 1)/sqrt(-(a*b - sqrt(a^2*b^2 - (a^2 + a*b)*a*b))/(a*b)))/((4*a^4*b^2 - a^3*b^3 - 5*a^2*b ^4)*abs(a)) + 1/6*log(sqrt(1/x^3 + 1) + 1)/a - 1/6*log(abs(sqrt(1/x^3 + 1) - 1))/a
Timed out. \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int \frac {x^6\,\sqrt {x^4+x}}{a\,x^6+b} \,d x \]