\(\int \frac {\sqrt {-x+x^4} (b+a x^6)}{-d+c x^6} \, dx\) [2629]

Optimal result

Integrand size = 30, antiderivative size = 231 \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\sqrt {-\left (\left (\sqrt {c}+\sqrt {d}\right ) \sqrt {d}\right )} (b c+a d) \arctan \left (\frac {\sqrt {-\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {\sqrt {\left (\sqrt {c}-\sqrt {d}\right ) \sqrt {d}} (b c+a d) \arctan \left (\frac {\sqrt {\sqrt {c} \sqrt {d}-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^{3/2} d}-\frac {a \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 c} \]

[Out]

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)*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*((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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.16, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2081, 6847, 1707, 201, 223, 212, 1189, 399, 385, 211, 214} \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x^4-x} \sqrt {\sqrt {c}-\sqrt {d}} (a d+b c) \arctan \left (\frac {x^{3/2} \sqrt {\sqrt {c}-\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}+\frac {\sqrt {x^4-x} \sqrt {\sqrt {c}+\sqrt {d}} (a d+b c) \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {c}+\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3-1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3-1} \sqrt {x}}-\frac {a \sqrt {x^4-x} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 c \sqrt {x^3-1} \sqrt {x}}+\frac {a \sqrt {x^4-x} x}{3 c} \]

[In]

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

[Out]

(a*x*Sqrt[-x + x^4])/(3*c) + (Sqrt[Sqrt[c] - Sqrt[d]]*(b*c + a*d)*Sqrt[-x + x^4]*ArcTan[(Sqrt[Sqrt[c] - Sqrt[d
]]*x^(3/2))/(d^(1/4)*Sqrt[-1 + x^3])])/(3*c^(3/2)*d^(3/4)*Sqrt[x]*Sqrt[-1 + x^3]) - (a*Sqrt[-x + x^4]*ArcTanh[
x^(3/2)/Sqrt[-1 + x^3]])/(3*c*Sqrt[x]*Sqrt[-1 + x^3]) + (Sqrt[Sqrt[c] + Sqrt[d]]*(b*c + a*d)*Sqrt[-x + x^4]*Ar
cTanh[(Sqrt[Sqrt[c] + Sqrt[d]]*x^(3/2))/(d^(1/4)*Sqrt[-1 + x^3])])/(3*c^(3/2)*d^(3/4)*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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 1707

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^2] && NeQ[c*d^2 + a*e^2, 0] && Intege
rQ[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]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3} \left (b+a x^6\right )}{-d+c x^6} \, dx}{\sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2} \left (b+a x^4\right )}{-d+c 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 \left (\frac {a \sqrt {-1+x^2}}{c}+\frac {(b c+a d) \sqrt {-1+x^2}}{c \left (-d+c x^4\right )}\right ) \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {\left (2 a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \sqrt {-1+x^2} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (2 (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}-c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}-\frac {\left ((b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {-1+x^2}}{\sqrt {c} \sqrt {d}+c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {\left (a \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}-c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (\sqrt {c} \sqrt {d}+c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}-\frac {a \sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (-c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}}+\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {-x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {-1+x^3}} \\ & = \frac {a x \sqrt {-x+x^4}}{3 c}+\frac {\sqrt {\sqrt {c}-\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \arctan \left (\frac {\sqrt {\sqrt {c}-\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {-1+x^3}}-\frac {a \sqrt {-x+x^4} \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 c \sqrt {x} \sqrt {-1+x^3}}+\frac {\sqrt {\sqrt {c}+\sqrt {d}} (b c+a d) \sqrt {-x+x^4} \text {arctanh}\left (\frac {\sqrt {\sqrt {c}+\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {-1+x^3}}\right )}{3 c^{3/2} d^{3/4} \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 = 0.45 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\frac {\sqrt {x} \sqrt {-1+x^3} \left (a \left (x^{3/2} \sqrt {-1+x^3}-\log \left (x^{3/2}+\sqrt {-1+x^3}\right )\right )+(b c+a d) \text {RootSum}\left [16 c-16 d+32 c \text {$\#$1}-32 d \text {$\#$1}+24 c \text {$\#$1}^2-16 d \text {$\#$1}^2+8 c \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (-2+2 x^3+2 x^{3/2} \sqrt {-1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{8 c-8 d+12 c \text {$\#$1}-8 d \text {$\#$1}+6 c \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]\right )}{3 c \sqrt {x \left (-1+x^3\right )}} \]

[In]

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

[Out]

(Sqrt[x]*Sqrt[-1 + x^3]*(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*(-1 + x^3)])

Maple [A] (verified)

Time = 1.79 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72

method	result	size
risch	\(\frac {a \,x^{2} \left (x^{3}-1\right )}{3 c \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {a \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{3}+\frac {2 \left (a d +b c \right ) \left (\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{3 \sqrt {c d}}}{2 c}\)	\(166\)
default	\(\frac {a \left (\frac {x \sqrt {x^{4}-x}}{3}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )}{c}-\frac {\left (a d +b c \right ) \left (\frac {\left (\sqrt {c d}-d \right ) \arctan \left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}-d \right ) d}}-\frac {\left (\sqrt {c d}+d \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x \left (x^{3}-1\right )}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{\sqrt {\left (\sqrt {c d}+d \right ) d}}\right )}{3 c \sqrt {c d}}\)	\(177\)
pseudoelliptic	\(\frac {\left (2 \left (a d +b c \right ) \sqrt {\left (\sqrt {c d}+d \right ) d}\, \left (-\sqrt {c d}+d \right ) \arctan \left (\frac {d \sqrt {x^{4}-x}}{x^{2} \sqrt {\left (\sqrt {c d}-d \right ) d}}\right )+\sqrt {\left (\sqrt {c d}-d \right ) d}\, \left (2 \left (\sqrt {c d}+d \right ) \left (a d +b c \right ) \operatorname {arctanh}\left (\frac {d \sqrt {x^{4}-x}}{x^{2} \sqrt {\left (\sqrt {c d}+d \right ) d}}\right )+a \sqrt {\left (\sqrt {c d}+d \right ) d}\, \left (2 x \sqrt {x^{4}-x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right ) \sqrt {c d}\right )\right ) x}{6 \sqrt {\left (\sqrt {c d}+d \right ) d}\, \sqrt {c d}\, \sqrt {\left (\sqrt {c d}-d \right ) d}\, c \left (x^{2}+\sqrt {x^{4}-x}\right ) \left (x^{2}-\sqrt {x^{4}-x}\right )}\)	\(254\)
elliptic	\(\text {Expression too large to display}\)	\(681\)

[In]

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

[Out]

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)*arctan(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)*arctanh(d*(x*(x^3-1))^(1/2)/x^2/(((c*d)^(1/2)+d)*d)^(1/2))))

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {-x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (179) = 358\).

Time = 1.83 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.99 \[ \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 |}} \]

[In]

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

[Out]

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
)*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 - sq
rt(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)) + 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)*sqrt(-d^2 + s
qrt(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))

Mupad [F(-1)]

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

[In]

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

[Out]

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