\(\int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx\) [2693]

Optimal result

Integrand size = 24, antiderivative size = 244 \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt [4]{-1} \sqrt {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}-i b} \sqrt {-x+x^4}}{\sqrt {2} \left (\sqrt {a}-i \sqrt {b}\right ) x^2}\right )}{3 a^{3/2}}-\frac {(-1)^{3/4} \sqrt {\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}+i b} x \sqrt {-x+x^4}}{\sqrt {2} \sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a^{3/2}}-\frac {\text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a} \]

[Out]

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)*arctan((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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.11, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {2081, 1507, 1505, 1306, 201, 223, 212, 1189, 399, 385, 214} \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=-\frac {\sqrt [4]{b} \sqrt {x^4-x} \sqrt {\sqrt {-a}+\sqrt {b}} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x^3-1}}\right )}{3 (-a)^{3/2} \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt [4]{b} \sqrt {x^4-x} \sqrt {\sqrt {-a} \sqrt {b}+a} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x^3-1}}\right )}{3 (-a)^{7/4} \sqrt {x^3-1} \sqrt {x}}-\frac {\sqrt {x^4-x} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 a \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt {x^4-x} x}{3 a} \]

[In]

Int[(x^6*Sqrt[-x + x^4])/(b + a*x^6),x]

[Out]

(x*Sqrt[-x + x^4])/(3*a) - (Sqrt[-x + x^4]*ArcTanh[x^(3/2)/Sqrt[-1 + x^3]])/(3*a*Sqrt[x]*Sqrt[-1 + x^3]) - (Sq
rt[Sqrt[-a] + Sqrt[b]]*b^(1/4)*Sqrt[-x + x^4]*ArcTanh[(Sqrt[Sqrt[-a] + Sqrt[b]]*x^(3/2))/(b^(1/4)*Sqrt[-1 + x^
3])])/(3*(-a)^(3/2)*Sqrt[x]*Sqrt[-1 + x^3]) + (Sqrt[a + Sqrt[-a]*Sqrt[b]]*b^(1/4)*Sqrt[-x + x^4]*ArcTanh[(Sqrt
[a + Sqrt[-a]*Sqrt[b]]*x^(3/2))/((-a)^(1/4)*b^(1/4)*Sqrt[-1 + x^3])])/(3*(-a)^(7/4)*Sqrt[x]*Sqrt[-1 + x^3])

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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]

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 1189

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Dist[-c/(2*r), In
t[(d + e*x^2)^q/(r - c*x^2), x], x] - Dist[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]

Rule 1306

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (c_.)*(x_)^4), x_Symbol] :> Dist[f^4/c, Int[(f*x)
^(m - 4)*(d + e*x^2)^q, x], x] - Dist[a*(f^4/c), Int[(f*x)^(m - 4)*((d + e*x^2)^q/(a + c*x^4)), x], x] /; Free
Q[{a, c, d, e, f, q}, x] &&  !IntegerQ[q] && GtQ[m, 3]

Rule 1505

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m +
1, n]}, Dist[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]

Rule 1507

Int[((f_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = D
enominator[m]}, Dist[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
]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^4} \int \frac {x^{13/2} \sqrt {-1+x^3}}{b+a x^6} \, dx}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {x^{14} \sqrt {-1+x^6}}{b+a x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {-1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 b \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt {-x+x^4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {-a} \sqrt {b}-a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {-a} \sqrt {b}+a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt {-x+x^4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {-a} \sqrt {b}-a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {-a} \sqrt {b}+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (-a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {x \sqrt {-x+x^4}}{3 a}-\frac {\sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\sqrt {\sqrt {-a}+\sqrt {b}} \sqrt [4]{b} \sqrt {-x+x^4} \text {arctanh}\left (\frac {\sqrt {\sqrt {-a}+\sqrt {b}} x^{3/2}}{\sqrt [4]{b} \sqrt {-1+x^3}}\right )}{3 (-a)^{3/2} \sqrt {x} \sqrt {-1+x^3}}+\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} \sqrt [4]{b} \sqrt {-x+x^4} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} x^{3/2}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {-1+x^3}}\right )}{3 (-a)^{7/4} \sqrt {x} \sqrt {-1+x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.74 \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x} \sqrt {-1+x^3} \left (x^{3/2} \sqrt {-1+x^3}-\log \left (x^{3/2}+\sqrt {-1+x^3}\right )-b \text {RootSum}\left [16 a+16 b+32 a \text {$\#$1}+32 b \text {$\#$1}+24 a \text {$\#$1}^2+16 b \text {$\#$1}^2+8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (-2+2 x^3+2 x^{3/2} \sqrt {-1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{8 a+8 b+12 a \text {$\#$1}+8 b \text {$\#$1}+6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]\right )}{3 a \sqrt {x \left (-1+x^3\right )}} \]

[In]

Integrate[(x^6*Sqrt[-x + x^4])/(b + a*x^6),x]

[Out]

(Sqrt[x]*Sqrt[-1 + x^3]*(x^(3/2)*Sqrt[-1 + x^3] - Log[x^(3/2) + Sqrt[-1 + x^3]] - b*RootSum[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*(-1 + x^3)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(392\) vs. \(2(177)=354\).

Time = 3.07 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.61

method	result	size
pseudoelliptic	\(\frac {\left (\frac {\left (\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )\right ) \left (-\sqrt {b \left (a +b \right )}+b \right ) \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}}{2}+\left (2 x \sqrt {x^{4}-x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right ) a \sqrt {b}\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}-2 a b \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}-x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}-x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )\right )}{6 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, \sqrt {b}\, a^{2}}\)	\(393\)
default	\(\frac {\frac {x \sqrt {x^{4}-x}}{3}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}}{a}+\frac {\left (-\sqrt {b \left (a +b \right )}+b \right ) \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \left (\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}{4}+\left (\sqrt {b \left (a +b \right )}+b \right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}-x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}-x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )\right )\right )}{3 \sqrt {b}\, a^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\)	\(399\)
risch	\(\frac {x^{2} \left (x^{3}-1\right )}{3 a \sqrt {x \left (x^{3}-1\right )}}+\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{6 a}+\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}-\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}-x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}-x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )}{12 a^{2}}+\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}-x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x -\sqrt {b}}{x^{3}}\right )}{12 a^{2}}\)	\(563\)
elliptic	\(\text {Expression too large to display}\)	\(668\)

[In]

int(x^6*(x^4-x)^(1/2)/(a*x^6+b),x,method=_RETURNVERBOSE)

[Out]

1/6/(4*(a+b)^(1/2)*b^(1/2)-2*(b*(a+b))^(1/2)-2*b)^(1/2)/b^(1/2)*((1/2*(ln((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))/x^3)-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))/x^2/(4*(a+b)^(1/2)*b^(1/2)-
2*(b*(a+b))^(1/2)-2*b)^(1/2))-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)))/a^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1800 vs. \(2 (168) = 336\).

Time = 53.18 (sec) , antiderivative size = 1800, normalized size of antiderivative = 7.38 \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\text {Too large to display} \]

[In]

integrate(x^6*(x^4-x)^(1/2)/(a*x^6+b),x, algorithm="fricas")

[Out]

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 - 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 + 24
3*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 - 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)) + 4*sqrt(x^4 - x)*x + 2*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1))/a

Sympy [F]

\[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\int \frac {x^{6} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{6} + b}\, dx \]

[In]

integrate(x**6*(x**4-x)**(1/2)/(a*x**6+b),x)

[Out]

Integral(x**6*sqrt(x*(x - 1)*(x**2 + x + 1))/(a*x**6 + b), x)

Maxima [F]

\[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\int { \frac {\sqrt {x^{4} - x} x^{6}}{a x^{6} + b} \,d x } \]

[In]

integrate(x^6*(x^4-x)^(1/2)/(a*x^6+b),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x)*x^6/(a*x^6 + b), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (168) = 336\).

Time = 1.04 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.85 \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x^{4} - x} x}{3 \, a} - \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b + \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} + \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b - \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} - \frac {\log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a} + \frac {\log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a} \]

[In]

integrate(x^6*(x^4-x)^(1/2)/(a*x^6+b),x, algorithm="giac")

[Out]

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)*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)) - 1/6*log(sqrt(-1/x^3 + 1) + 1)/a + 1/6*log(abs(sqrt(-1/x^3 + 1) - 1))/a

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \sqrt {-x+x^4}}{b+a x^6} \, dx=\int \frac {x^6\,\sqrt {x^4-x}}{a\,x^6+b} \,d x \]

[In]

int((x^6*(x^4 - x)^(1/2))/(b + a*x^6),x)

[Out]

int((x^6*(x^4 - x)^(1/2))/(b + a*x^6), x)