3.32.22 \(\int \sqrt [4]{\frac {1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}} \, dx\)

Optimal. Leaf size=708 \[ \frac {32 a^3 b^2 x^9-192 a^3 b^2 x^7+384 a^3 b^2 x^5-320 a^3 b^2 x^3+96 a^3 b^2 x+36 a^3 b c x^8-108 a^3 b c x^6+108 a^3 b c x^4-36 a^3 b c x^2+45 a^3 c^2 x^7-135 a^3 c^2 x^5+135 a^3 c^2 x^3-45 a^3 c^2 x+36 a^2 b^2 x^8-204 a^2 b^2 x^6+396 a^2 b^2 x^4-324 a^2 b^2 x^2+96 a^2 b^2+42 a^2 b c x^7-126 a^2 b c x^5+126 a^2 b c x^3-42 a^2 b c x+45 a^2 c^2 x^6-135 a^2 c^2 x^4+135 a^2 c^2 x^2-45 a^2 c^2-3 a b^2 x^7+9 a b^2 x^5-9 a b^2 x^3+3 a b^2 x+6 a b c x^6-18 a b c x^4+18 a b c x^2-6 a b c-7 b^2 x^6+21 b^2 x^4-21 b^2 x^2+7 b^2}{96 a^2 b^3 \left (\frac {a x^9-4 a x^7+6 a x^5-4 a x^3+a x+x^8-4 x^6+6 x^4-4 x^2+1}{b x-c}\right )^{3/4}}+\frac {\left (32 a^3 b^2 c-15 a^3 c^3+32 a^2 b^3-5 a^2 b c^2+3 a b^2 c-7 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} (x-1) (x+1)}{\sqrt [4]{b} \sqrt [4]{\frac {a x^9-4 a x^7+6 a x^5-4 a x^3+a x+x^8-4 x^6+6 x^4-4 x^2+1}{b x-c}}}\right )}{64 a^{11/4} b^{13/4}}+\frac {\left (-32 a^3 b^2 c+15 a^3 c^3-32 a^2 b^3+5 a^2 b c^2-3 a b^2 c+7 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} (x-1) (x+1)}{\sqrt [4]{b} \sqrt [4]{\frac {a x^9-4 a x^7+6 a x^5-4 a x^3+a x+x^8-4 x^6+6 x^4-4 x^2+1}{b x-c}}}\right )}{64 a^{11/4} b^{13/4}} \]

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Rubi [A]  time = 0.41, antiderivative size = 471, normalized size of antiderivative = 0.67, number of steps used = 10, number of rules used = 10, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6688, 6718, 952, 80, 50, 63, 240, 212, 208, 205} \begin {gather*} -\frac {(a x+1) (5 b-3 a c) (c-b x) \sqrt [4]{-\frac {\left (1-x^2\right )^4 (a x+1)}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {\left (\left (7-32 a^2\right ) b^2+15 a^2 c^2-10 a b c\right ) (c-b x) \sqrt [4]{-\frac {\left (1-x^2\right )^4 (a x+1)}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}+\frac {(a x+1)^2 (c-b x) \sqrt [4]{-\frac {\left (1-x^2\right )^4 (a x+1)}{c-b x}}}{3 a^2 b \left (1-x^2\right )}-\frac {(a c+b) \left (\left (7-32 a^2\right ) b^2+15 a^2 c^2-10 a b c\right ) \sqrt [4]{b x-c} \sqrt [4]{-\frac {\left (1-x^2\right )^4 (a x+1)}{c-b x}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{a x+1}}{\sqrt [4]{a} \sqrt [4]{b x-c}}\right )}{64 a^{11/4} b^{13/4} \left (1-x^2\right ) \sqrt [4]{a x+1}}-\frac {(a c+b) \left (\left (7-32 a^2\right ) b^2+15 a^2 c^2-10 a b c\right ) \sqrt [4]{b x-c} \sqrt [4]{-\frac {\left (1-x^2\right )^4 (a x+1)}{c-b x}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{a x+1}}{\sqrt [4]{a} \sqrt [4]{b x-c}}\right )}{64 a^{11/4} b^{13/4} \left (1-x^2\right ) \sqrt [4]{a x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4),x]

[Out]

(((7 - 32*a^2)*b^2 - 10*a*b*c + 15*a^2*c^2)*(c - b*x)*(-(((1 + a*x)*(1 - x^2)^4)/(c - b*x)))^(1/4))/(32*a^2*b^
3*(1 - x^2)) - ((5*b - 3*a*c)*(1 + a*x)*(c - b*x)*(-(((1 + a*x)*(1 - x^2)^4)/(c - b*x)))^(1/4))/(8*a^2*b^2*(1
- x^2)) + ((1 + a*x)^2*(c - b*x)*(-(((1 + a*x)*(1 - x^2)^4)/(c - b*x)))^(1/4))/(3*a^2*b*(1 - x^2)) - ((b + a*c
)*((7 - 32*a^2)*b^2 - 10*a*b*c + 15*a^2*c^2)*(-c + b*x)^(1/4)*(-(((1 + a*x)*(1 - x^2)^4)/(c - b*x)))^(1/4)*Arc
Tan[(b^(1/4)*(1 + a*x)^(1/4))/(a^(1/4)*(-c + b*x)^(1/4))])/(64*a^(11/4)*b^(13/4)*(1 + a*x)^(1/4)*(1 - x^2)) -
((b + a*c)*((7 - 32*a^2)*b^2 - 10*a*b*c + 15*a^2*c^2)*(-c + b*x)^(1/4)*(-(((1 + a*x)*(1 - x^2)^4)/(c - b*x)))^
(1/4)*ArcTanh[(b^(1/4)*(1 + a*x)^(1/4))/(a^(1/4)*(-c + b*x)^(1/4))])/(64*a^(11/4)*b^(13/4)*(1 + a*x)^(1/4)*(1
- x^2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 205

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

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 240

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

Rule 952

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(c^p*(d
+ e*x)^(m + 2*p)*(f + g*x)^(n + 1))/(g*e^(2*p)*(m + n + 2*p + 1)), x] + Dist[1/(g*e^(2*p)*(m + n + 2*p + 1)),
Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^(2*p)*(a + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c
^p*(e*f - d*g)*(m + 2*p)*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0]
&& NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[n] ||  !IntegerQ[m])

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6718

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n*z^q)^FracP
art[p])/(v^(m*FracPart[p])*w^(n*FracPart[p])*z^(q*FracPart[p])), Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; Free
Q[{a, m, n, p, q}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !FreeQ[w, x] &&  !FreeQ[z, x]

Rubi steps

\begin {align*} \int \sqrt [4]{\frac {1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}} \, dx &=\int \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}} \, dx\\ &=\frac {\left (\sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \int \frac {\sqrt [4]{1+a x} \left (-1+x^2\right )}{\sqrt [4]{-c+b x}} \, dx}{\sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left (\sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \int \frac {\sqrt [4]{1+a x} \left (-\frac {3}{4} \left (b+4 a^2 b-3 a c\right )-\frac {3}{4} a (5 b-3 a c) x\right )}{\sqrt [4]{-c+b x}} \, dx}{3 a^2 b \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left (\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \int \frac {\sqrt [4]{1+a x}}{\sqrt [4]{-c+b x}} \, dx}{32 a^2 b^2 \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left ((b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \int \frac {1}{(1+a x)^{3/4} \sqrt [4]{-c+b x}} \, dx}{128 a^2 b^3 \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left ((b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-\frac {b}{a}-c+\frac {b x^4}{a}}} \, dx,x,\sqrt [4]{1+a x}\right )}{32 a^3 b^3 \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left ((b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^4}{a}} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{-c+b x}}\right )}{32 a^3 b^3 \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}+\frac {\left ((b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-\sqrt {b} x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{-c+b x}}\right )}{64 a^{5/2} b^3 \sqrt [4]{1+a x} \left (-1+x^2\right )}+\frac {\left ((b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{\frac {(1+a x) \left (-1+x^2\right )^4}{-c+b x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+\sqrt {b} x^2} \, dx,x,\frac {\sqrt [4]{1+a x}}{\sqrt [4]{-c+b x}}\right )}{64 a^{5/2} b^3 \sqrt [4]{1+a x} \left (-1+x^2\right )}\\ &=\frac {\left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{32 a^2 b^3 \left (1-x^2\right )}-\frac {(5 b-3 a c) (1+a x) (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{8 a^2 b^2 \left (1-x^2\right )}+\frac {(1+a x)^2 (c-b x) \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}}}{3 a^2 b \left (1-x^2\right )}-\frac {(b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}} \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{1+a x}}{\sqrt [4]{a} \sqrt [4]{-c+b x}}\right )}{64 a^{11/4} b^{13/4} \sqrt [4]{1+a x} \left (1-x^2\right )}-\frac {(b+a c) \left (\left (7-32 a^2\right ) b^2-10 a b c+15 a^2 c^2\right ) \sqrt [4]{-c+b x} \sqrt [4]{-\frac {(1+a x) \left (1-x^2\right )^4}{c-b x}} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{1+a x}}{\sqrt [4]{a} \sqrt [4]{-c+b x}}\right )}{64 a^{11/4} b^{13/4} \sqrt [4]{1+a x} \left (1-x^2\right )}\\ \end {align*}

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Mathematica [C]  time = 0.24, size = 171, normalized size = 0.24 \begin {gather*} \frac {4 (a x+1) \sqrt [4]{\frac {a (c-b x)}{a c+b}} \sqrt [4]{\frac {\left (x^2-1\right )^4 (a x+1)}{b x-c}} \left ((a c+b)^2 \, _2F_1\left (-\frac {7}{4},\frac {5}{4};\frac {9}{4};\frac {a x b+b}{b+a c}\right )-a \left (a \left (b^2-c^2\right ) \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {a x b+b}{b+a c}\right )+2 c (a c+b) \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};\frac {a x b+b}{b+a c}\right )\right )\right )}{5 a^3 b^2 \left (x^2-1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4),x]

[Out]

(4*(1 + a*x)*((a*(c - b*x))/(b + a*c))^(1/4)*(((1 + a*x)*(-1 + x^2)^4)/(-c + b*x))^(1/4)*((b + a*c)^2*Hypergeo
metric2F1[-7/4, 5/4, 9/4, (b + a*b*x)/(b + a*c)] - a*(2*c*(b + a*c)*Hypergeometric2F1[-3/4, 5/4, 9/4, (b + a*b
*x)/(b + a*c)] + a*(b^2 - c^2)*Hypergeometric2F1[1/4, 5/4, 9/4, (b + a*b*x)/(b + a*c)])))/(5*a^3*b^2*(-1 + x^2
))

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IntegrateAlgebraic [A]  time = 2.03, size = 708, normalized size = 1.00 \begin {gather*} \frac {7 b^2+96 a^2 b^2-6 a b c-45 a^2 c^2+3 a b^2 x+96 a^3 b^2 x-42 a^2 b c x-45 a^3 c^2 x-21 b^2 x^2-324 a^2 b^2 x^2+18 a b c x^2-36 a^3 b c x^2+135 a^2 c^2 x^2-9 a b^2 x^3-320 a^3 b^2 x^3+126 a^2 b c x^3+135 a^3 c^2 x^3+21 b^2 x^4+396 a^2 b^2 x^4-18 a b c x^4+108 a^3 b c x^4-135 a^2 c^2 x^4+9 a b^2 x^5+384 a^3 b^2 x^5-126 a^2 b c x^5-135 a^3 c^2 x^5-7 b^2 x^6-204 a^2 b^2 x^6+6 a b c x^6-108 a^3 b c x^6+45 a^2 c^2 x^6-3 a b^2 x^7-192 a^3 b^2 x^7+42 a^2 b c x^7+45 a^3 c^2 x^7+36 a^2 b^2 x^8+36 a^3 b c x^8+32 a^3 b^2 x^9}{96 a^2 b^3 \left (\frac {1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}\right )^{3/4}}+\frac {\left (-7 b^3+32 a^2 b^3+3 a b^2 c+32 a^3 b^2 c-5 a^2 b c^2-15 a^3 c^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} (-1+x) (1+x)}{\sqrt [4]{b} \sqrt [4]{\frac {1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right )}{64 a^{11/4} b^{13/4}}+\frac {\left (7 b^3-32 a^2 b^3-3 a b^2 c-32 a^3 b^2 c+5 a^2 b c^2+15 a^3 c^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} (-1+x) (1+x)}{\sqrt [4]{b} \sqrt [4]{\frac {1+a x-4 x^2-4 a x^3+6 x^4+6 a x^5-4 x^6-4 a x^7+x^8+a x^9}{-c+b x}}}\right )}{64 a^{11/4} b^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^
(1/4),x]

[Out]

(7*b^2 + 96*a^2*b^2 - 6*a*b*c - 45*a^2*c^2 + 3*a*b^2*x + 96*a^3*b^2*x - 42*a^2*b*c*x - 45*a^3*c^2*x - 21*b^2*x
^2 - 324*a^2*b^2*x^2 + 18*a*b*c*x^2 - 36*a^3*b*c*x^2 + 135*a^2*c^2*x^2 - 9*a*b^2*x^3 - 320*a^3*b^2*x^3 + 126*a
^2*b*c*x^3 + 135*a^3*c^2*x^3 + 21*b^2*x^4 + 396*a^2*b^2*x^4 - 18*a*b*c*x^4 + 108*a^3*b*c*x^4 - 135*a^2*c^2*x^4
+ 9*a*b^2*x^5 + 384*a^3*b^2*x^5 - 126*a^2*b*c*x^5 - 135*a^3*c^2*x^5 - 7*b^2*x^6 - 204*a^2*b^2*x^6 + 6*a*b*c*x
^6 - 108*a^3*b*c*x^6 + 45*a^2*c^2*x^6 - 3*a*b^2*x^7 - 192*a^3*b^2*x^7 + 42*a^2*b*c*x^7 + 45*a^3*c^2*x^7 + 36*a
^2*b^2*x^8 + 36*a^3*b*c*x^8 + 32*a^3*b^2*x^9)/(96*a^2*b^3*((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^
6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(3/4)) + ((-7*b^3 + 32*a^2*b^3 + 3*a*b^2*c + 32*a^3*b^2*c - 5*a^2*b*c^2
- 15*a^3*c^3)*ArcTan[(a^(1/4)*(-1 + x)*(1 + x))/(b^(1/4)*((1 + a*x - 4*x^2 - 4*a*x^3 + 6*x^4 + 6*a*x^5 - 4*x^
6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4))])/(64*a^(11/4)*b^(13/4)) + ((7*b^3 - 32*a^2*b^3 - 3*a*b^2*c - 32
*a^3*b^2*c + 5*a^2*b*c^2 + 15*a^3*c^3)*ArcTanh[(a^(1/4)*(-1 + x)*(1 + x))/(b^(1/4)*((1 + a*x - 4*x^2 - 4*a*x^3
+ 6*x^4 + 6*a*x^5 - 4*x^6 - 4*a*x^7 + x^8 + a*x^9)/(-c + b*x))^(1/4))])/(64*a^(11/4)*b^(13/4))

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fricas [B]  time = 1.04, size = 4114, normalized size = 5.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^9-4*a*x^7+x^8+6*a*x^5-4*x^6-4*a*x^3+6*x^4+a*x-4*x^2+1)/(b*x-c))^(1/4),x, algorithm="fricas")

[Out]

-1/384*(12*(a^2*b^3*x^2 - a^2*b^3)*((50625*a^12*c^12 + 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*
a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(314572
8*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7
- 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4
- 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*
a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8
- 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10)/(a^11*b^13))^(1/4)*arctan(-((a^8*b^10*x^2
- a^8*b^10)*sqrt(((225*a^6*c^6 + 150*a^5*b*c^5 + (1024*a^4 - 448*a^2 + 49)*b^6 + 2*(1024*a^5 - 128*a^3 - 21*a
)*b^5*c + (1024*a^6 - 128*a^4 + 79*a^2)*b^4*c^2 - 20*(64*a^5 - 9*a^3)*b^3*c^3 - 5*(192*a^6 + 13*a^4)*b^2*c^4)*
sqrt((a*x^9 - 4*a*x^7 + x^8 + 6*a*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c)) + (a^6*b^6*x^4 -
2*a^6*b^6*x^2 + a^6*b^6)*sqrt((50625*a^12*c^12 + 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 -
43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^1
0 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79
104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 20
0*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*
b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 150
0*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10)/(a^11*b^13)))/(x^4 - 2*x^2 + 1))*((50625*a^12*
c^12 + 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 -
589824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 +
4753*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12
- 7471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a
^5)*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a
^7)*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^
12 + a^10)*b^2*c^10)/(a^11*b^13))^(3/4) - (15*a^11*b^10*c^3 + 5*a^10*b^11*c^2 - (32*a^10 - 7*a^8)*b^13 - (32*a
^11 + 3*a^9)*b^12*c)*((a*x^9 - 4*a*x^7 + x^8 + 6*a*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))
^(1/4)*((50625*a^12*c^12 + 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12
+ 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 13516
8*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9
*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^
9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^
11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b
^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10)/(a^11*b^13))^(3/4))/(50625*a^12*c^12 + 67500*a^11*b*c^11 + (1048576*a
^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3 - 1
029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a^11
- 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 40704
*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 141312
*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12672*
a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10 - (50625*a^12*c^1
2 + 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589
824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 475
3*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 -
7471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)
*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)
*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12
+ a^10)*b^2*c^10)*x^2)) - 3*(a^2*b^3*x^2 - a^2*b^3)*((50625*a^12*c^12 + 67500*a^11*b*c^11 + (1048576*a^8 - 917
504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b
^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504
*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 40704*a^6 - 1
5249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 141312*a^10 +
43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3
751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10)/(a^11*b^13))^(1/4)*log((
(15*a^3*c^3 + 5*a^2*b*c^2 - (32*a^2 - 7)*b^3 - (32*a^3 + 3*a)*b^2*c)*((a*x^9 - 4*a*x^7 + x^8 + 6*a*x^5 - 4*x^6
- 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))^(1/4) + (a^3*b^3*x^2 - a^3*b^3)*((50625*a^12*c^12 + 67500*a^1
1*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 860
16*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c
^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10
+ 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20
*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25
*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c
^10)/(a^11*b^13))^(1/4))/(x^2 - 1)) + 3*(a^2*b^3*x^2 - a^2*b^3)*((50625*a^12*c^12 + 67500*a^11*b*c^11 + (10485
76*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 589824*a^7 + 86016*a^5 + 3136*a^3
- 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753*a^2)*b^10*c^2 + 4*(1048576*a
^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7471104*a^10 + 2297856*a^8 - 4
0704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*b^7*c^5 - 20*(98304*a^12 - 14
1312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*b^5*c^7 + 25*(55296*a^12 - 12
672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 + a^10)*b^2*c^10)/(a^11*b^13))
^(1/4)*log(((15*a^3*c^3 + 5*a^2*b*c^2 - (32*a^2 - 7)*b^3 - (32*a^3 + 3*a)*b^2*c)*((a*x^9 - 4*a*x^7 + x^8 + 6*a
*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))^(1/4) - (a^3*b^3*x^2 - a^3*b^3)*((50625*a^12*c^12
+ 67500*a^11*b*c^11 + (1048576*a^8 - 917504*a^6 + 301056*a^4 - 43904*a^2 + 2401)*b^12 + 4*(1048576*a^9 - 5898
24*a^7 + 86016*a^5 + 3136*a^3 - 1029*a)*b^11*c + 2*(3145728*a^10 - 1114112*a^8 + 135168*a^6 - 30912*a^4 + 4753
*a^2)*b^10*c^2 + 4*(1048576*a^11 - 917504*a^9 + 454656*a^7 - 79104*a^5 + 2751*a^3)*b^9*c^3 + (1048576*a^12 - 7
471104*a^10 + 2297856*a^8 - 40704*a^6 - 15249*a^4)*b^8*c^4 - 200*(32768*a^11 - 6144*a^9 + 1344*a^7 - 243*a^5)*
b^7*c^5 - 20*(98304*a^12 - 141312*a^10 + 43712*a^8 - 1579*a^6)*b^6*c^6 + 200*(18432*a^11 - 1664*a^9 - 93*a^7)*
b^5*c^7 + 25*(55296*a^12 - 12672*a^10 + 3751*a^8)*b^4*c^8 - 1500*(576*a^11 - 41*a^9)*b^3*c^9 - 6750*(64*a^12 +
a^10)*b^2*c^10)/(a^11*b^13))^(1/4))/(x^2 - 1)) - 4*(32*a^2*b^3*x^3 - 45*a^2*c^3 + (96*a^2 + 7)*b^2*c - 6*a*b*
c^2 + 4*(a^2*b^2*c + a*b^3)*x^2 + (9*a^2*b*c^2 - (96*a^2 + 7)*b^3 + 2*a*b^2*c)*x)*((a*x^9 - 4*a*x^7 + x^8 + 6*
a*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))^(1/4))/(a^2*b^3*x^2 - a^2*b^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {a x^{9} - 4 \, a x^{7} + x^{8} + 6 \, a x^{5} - 4 \, x^{6} - 4 \, a x^{3} + 6 \, x^{4} + a x - 4 \, x^{2} + 1}{b x - c}\right )^{\frac {1}{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^9-4*a*x^7+x^8+6*a*x^5-4*x^6-4*a*x^3+6*x^4+a*x-4*x^2+1)/(b*x-c))^(1/4),x, algorithm="giac")

[Out]

integrate(((a*x^9 - 4*a*x^7 + x^8 + 6*a*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))^(1/4), x)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \left (\frac {a \,x^{9}-4 a \,x^{7}+x^{8}+6 a \,x^{5}-4 x^{6}-4 a \,x^{3}+6 x^{4}+a x -4 x^{2}+1}{b x -c}\right )^{\frac {1}{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x^9-4*a*x^7+x^8+6*a*x^5-4*x^6-4*a*x^3+6*x^4+a*x-4*x^2+1)/(b*x-c))^(1/4),x)

[Out]

int(((a*x^9-4*a*x^7+x^8+6*a*x^5-4*x^6-4*a*x^3+6*x^4+a*x-4*x^2+1)/(b*x-c))^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\frac {a x^{9} - 4 \, a x^{7} + x^{8} + 6 \, a x^{5} - 4 \, x^{6} - 4 \, a x^{3} + 6 \, x^{4} + a x - 4 \, x^{2} + 1}{b x - c}\right )^{\frac {1}{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x^9-4*a*x^7+x^8+6*a*x^5-4*x^6-4*a*x^3+6*x^4+a*x-4*x^2+1)/(b*x-c))^(1/4),x, algorithm="maxima")

[Out]

integrate(((a*x^9 - 4*a*x^7 + x^8 + 6*a*x^5 - 4*x^6 - 4*a*x^3 + 6*x^4 + a*x - 4*x^2 + 1)/(b*x - c))^(1/4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (-\frac {a\,x^9+x^8-4\,a\,x^7-4\,x^6+6\,a\,x^5+6\,x^4-4\,a\,x^3-4\,x^2+a\,x+1}{c-b\,x}\right )}^{1/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(a*x - 4*a*x^3 + 6*a*x^5 - 4*a*x^7 + a*x^9 - 4*x^2 + 6*x^4 - 4*x^6 + x^8 + 1)/(c - b*x))^(1/4),x)

[Out]

int((-(a*x - 4*a*x^3 + 6*a*x^5 - 4*a*x^7 + a*x^9 - 4*x^2 + 6*x^4 - 4*x^6 + x^8 + 1)/(c - b*x))^(1/4), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((a*x**9-4*a*x**7+x**8+6*a*x**5-4*x**6-4*a*x**3+6*x**4+a*x-4*x**2+1)/(b*x-c))**(1/4),x)

[Out]

Timed out

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