3.27.63 \(\int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx\)

Optimal. Leaf size=238 \[ \frac {\sqrt [4]{-1} \sqrt {\sqrt {b} \left (\sqrt {a}-i \sqrt {b}\right )} \tan ^{-1}\left (\frac {(1+i) \sqrt {x^4+x} \sqrt {\sqrt {a} \sqrt {b}-i b}}{\sqrt {2} x^2 \left (\sqrt {a}-i \sqrt {b}\right )}\right )}{3 a^{3/2}}+\frac {(-1)^{3/4} \sqrt {\sqrt {b} \left (\sqrt {a}+i \sqrt {b}\right )} \tan ^{-1}\left (\frac {(1+i) x \sqrt {x^4+x} \sqrt {\sqrt {a} \sqrt {b}+i b}}{\sqrt {2} \sqrt {b} (x+1) \left (x^2-x+1\right )}\right )}{3 a^{3/2}}+\frac {\sqrt {x^4+x} x}{3 a}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 a} \]

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Rubi [A]  time = 0.63, antiderivative size = 253, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2056, 1493, 1491, 1292, 195, 215, 1175, 402, 377, 208} \begin {gather*} \frac {\sqrt [4]{b} \sqrt {x^4+x} \sqrt {\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\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} \tanh ^{-1}\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} x}{3 a}+\frac {\sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x^3+1} \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[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]*ArcSinh[x^(3/2)])/(3*a*Sqrt[x]*Sqrt[1 + x^3]) + (Sqrt[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 195

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 208

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

Rule 215

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]

Rule 377

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 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(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])

Rule 1175

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 1292

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 1491

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 1493

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 2056

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*} \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \operatorname {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 ) \operatorname {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 ) \operatorname {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} \sinh ^{-1}\left (x^{3/2}\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 ) \operatorname {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 ) \operatorname {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} \sinh ^{-1}\left (x^{3/2}\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 ) \operatorname {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 ) \operatorname {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} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\sqrt {\sqrt {-a}+\sqrt {b}} \sqrt [4]{b} \sqrt {x+x^4} \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [A]  time = 2.79, size = 176, normalized size = 0.74 \begin {gather*} \frac {x \sqrt {x^4+x} \left (\frac {\sqrt {\frac {1}{x^3}+1} \left (\sqrt [4]{b} \left (\sqrt {\sqrt {-a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {\frac {1}{x^3}+1}}{\sqrt {\sqrt {-a}-\sqrt {b}}}\right )-\sqrt {\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {\frac {1}{x^3}+1}}{\sqrt {\sqrt {-a}+\sqrt {b}}}\right )\right )+\sqrt {-a} \tanh ^{-1}\left (\sqrt {\frac {1}{x^3}+1}\right )\right )}{\sqrt {-a} \left (x^3+1\right )}+1\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[x + x^4]*(1 + (Sqrt[1 + x^(-3)]*(Sqrt[-a]*ArcTanh[Sqrt[1 + x^(-3)]] + b^(1/4)*(Sqrt[Sqrt[-a] - Sqrt[b]
]*ArcTan[(b^(1/4)*Sqrt[1 + x^(-3)])/Sqrt[Sqrt[-a] - Sqrt[b]]] - Sqrt[Sqrt[-a] + Sqrt[b]]*ArcTanh[(b^(1/4)*Sqrt
[1 + x^(-3)])/Sqrt[Sqrt[-a] + Sqrt[b]]])))/(Sqrt[-a]*(1 + x^3))))/(3*a)

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IntegrateAlgebraic [A]  time = 1.33, size = 233, normalized size = 0.98 \begin {gather*} \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt [4]{-1} \sqrt {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {b}} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {\sqrt {a} \sqrt {b}-i b} x \sqrt {x+x^4}}{\sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^{3/2}}+\frac {(-1)^{3/4} \sqrt {\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {\sqrt {a} \sqrt {b}+i b} x \sqrt {x+x^4}}{\sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^{3/2}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[x + x^4])/(3*a) + ((-1)^(1/4)*Sqrt[(Sqrt[a] - I*Sqrt[b])*Sqrt[b]]*ArcTan[((-1)^(3/4)*Sqrt[Sqrt[a]*Sqrt
[b] - I*b]*x*Sqrt[x + x^4])/(Sqrt[b]*(1 + x)*(1 - x + x^2))])/(3*a^(3/2)) + ((-1)^(3/4)*Sqrt[(Sqrt[a] + I*Sqrt
[b])*Sqrt[b]]*ArcTan[((-1)^(1/4)*Sqrt[Sqrt[a]*Sqrt[b] + I*b]*x*Sqrt[x + x^4])/(Sqrt[b]*(1 + x)*(1 - x + x^2))]
)/(3*a^(3/2)) + ArcTanh[x^2/Sqrt[x + x^4]]/(3*a)

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fricas [B]  time = 101.29, size = 1784, normalized size = 7.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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 - 1
2*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 + 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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-34,84]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [a,b]=[-24,-10]Warning, need to choose a branch for the root of a polynomial with parameters. This m
ight be wrong.The choice was done assuming [a,b]=[80,82]Warning, need to choose a branch for the root of a pol
ynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[57,-56]Warning, need to choos
e a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b
]=[45,-8]-1/6/a*ln(abs(sqrt((1/x)^3+1)-1))+1/6/a*ln(sqrt((1/x)^3+1)+1)+((-2*a*b^2+4*a*sqrt(-a*b)*sqrt(-b^2-b*s
qrt(-a*b))+2*b*a*b-5*b*sqrt(-a*b)*sqrt(-b^2-b*sqrt(-a*b)))*a^2*abs(b)+(-2*a^2*b^3+4*a^2*b*sqrt(-a*b)*sqrt(-b^2
-b*sqrt(-a*b))+2*a*b^2*a*b-5*a*b^2*sqrt(-a*b)*sqrt(-b^2-b*sqrt(-a*b)))*abs(b))/(12*a^4*b^2-3*a^3*b^3-15*a^2*b^
4)/abs(a)*atan(sqrt((1/x)^3+1)/sqrt(-(6*a*b+sqrt(6*a*b*6*a*b-12*a*b*(3*a^2+3*a*b)))/2/3/a/b))-((-2*a*b^2+4*a*s
qrt(-a*b)*sqrt(-b^2+b*sqrt(-a*b))+2*b*a*b-5*b*sqrt(-a*b)*sqrt(-b^2+b*sqrt(-a*b)))*a^2*abs(b)+(-2*a^2*b^3+4*a^2
*b*sqrt(-a*b)*sqrt(-b^2+b*sqrt(-a*b))+2*a*b^2*a*b-5*a*b^2*sqrt(-a*b)*sqrt(-b^2+b*sqrt(-a*b)))*abs(b))/(12*a^4*
b^2-3*a^3*b^3-15*a^2*b^4)/abs(a)*atan(sqrt((1/x)^3+1)/sqrt(-(6*a*b-sqrt(6*a*b*6*a*b-12*a*b*(3*a^2+3*a*b)))/2/3
/a/b))+8*a*1/24/a^2*x*sqrt(x^4+x)

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maple [C]  time = 0.56, size = 675, normalized size = 2.84

method	result	size
elliptic	\(\frac {x \sqrt {x^{4}+x}}{3 a}-\frac {\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 (-\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 )+\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 )}{a \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 )}}+\frac {b \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{6}+b \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b -3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a +b \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3 a}\)	\(675\)
default	\(\frac {\frac {x \sqrt {x^{4}+x}}{3}-\frac {\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 (-\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 )+\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 )}}}{a}-\frac {b \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{6}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b -3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a +b \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3 a}\)	\(676\)
risch	\(\frac {x^{2} \left (x^{3}+1\right )}{3 a \sqrt {x \left (x^{3}+1\right )}}+\frac {-\frac {2 \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 (-\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 )+\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 )}}-\frac {2 b \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{6}+b \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\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 )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} a \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 {i \underline {\hspace {1.25 ex}}\alpha ^{5} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{5} a -i \sqrt {3}\, b -3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (a +b \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3}}{2 a}\)	\(686\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x*(x^4+x)^(1/2)/a-1/a*(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^2
*(-(x-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^(
1/2)/(3/2+1/2*I*3^(1/2))/(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*I*3^(1/2)))^(1/2)*(-EllipticF(((3/2+1/2*I*3
^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3
/2-1/2*I*3^(1/2)))^(1/2))+EllipticPi(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),(1/2+1/2*I*3^(1/2
))/(3/2+1/2*I*3^(1/2)),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(
1/2)))+1/3*b/a*4^(1/2)*sum((_alpha^3+1)/_alpha^4*(1+x)^2*(_alpha^5-_alpha^4+_alpha^3-_alpha^2+_alpha-1)/(a+b)*
(-1-I*3^(1/2))*(x/(1+x)*(3+I*3^(1/2))/(1+I*3^(1/2)))^(1/2)*(-1/(1+x)*(I*3^(1/2)+2*x-1)/(1-I*3^(1/2)))^(1/2)*(-
1/(1+x)*(-1+2*x-I*3^(1/2))/(1+I*3^(1/2)))^(1/2)/(3+I*3^(1/2))/(x*(1+x)*(I*3^(1/2)+2*x-1)*(-1+2*x-I*3^(1/2)))^(
1/2)*(EllipticF(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1
/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))-_alpha^5*a/b*EllipticPi(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2
*I*3^(1/2))/(1+x))^(1/2),-1/6*(I*_alpha^5*3^(1/2)*a+3*_alpha^5*a-I*3^(1/2)*b-3*b)/b,((-3/2+1/2*I*3^(1/2))*(-1/
2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))),_alpha=RootOf(_Z^6*a+b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + x} x^{6}}{a x^{6} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\sqrt {x^4+x}}{a\,x^6+b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{a x^{6} + b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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