3.29.11 \(\int \frac {(-b-a x+x^4) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx\)

Optimal. Leaf size=276 \[ \frac {\left (19712 a^4 b^2-9843 b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{23/4}}+\frac {\left (9843 b^5-19712 a^4 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{23/4}}+\frac {\sqrt [4]{a x^4+b x^3} \left (-15360 a^5 x-65280 a^4 b+6144 a^4 x^4+8064 a^3 b x^3+10400 a^2 b^2 x^2+16420 a b^3 x+32705 b^4\right )}{30720 a^5} \]

________________________________________________________________________________________

Rubi [C]  time = 1.76, antiderivative size = 707, normalized size of antiderivative = 2.56, number of steps used = 41, number of rules used = 11, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2056, 6733, 6725, 279, 331, 298, 203, 206, 321, 511, 510} \begin {gather*} -\frac {371 b^5 \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {371 b^5 \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {371 b^4 \sqrt [4]{a x^4+b x^3}}{6144 a^5}+\frac {53 b^3 x \sqrt [4]{a x^4+b x^3}}{1536 a^4}-\frac {1}{2} x \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{a x^4+b x^3}+\frac {65 b^2 x^2 \sqrt [4]{a x^4+b x^3}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{a x^4+b x^3}}{80 a^2}-\frac {3 b^2 \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {3 b^2 \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{a x+b}}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{\frac {a x}{b}+1}}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{a x^4+b x^3}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{a x^4+b x^3}}{a^5}+\frac {x^4 \sqrt [4]{a x^4+b x^3}}{5 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]

[Out]

(-371*b^4*(b*x^3 + a*x^4)^(1/4))/(6144*a^5) - (b*(a^4 - b^3)*(b*x^3 + a*x^4)^(1/4))/(8*a^5) - (b*(2*a^4 - b^3)
*(b*x^3 + a*x^4)^(1/4))/a^5 + (53*b^3*x*(b*x^3 + a*x^4)^(1/4))/(1536*a^4) - ((1 - b^3/a^4)*x*(b*x^3 + a*x^4)^(
1/4))/2 + (65*b^2*x^2*(b*x^3 + a*x^4)^(1/4))/(192*a^3) + (21*b*x^3*(b*x^3 + a*x^4)^(1/4))/(80*a^2) + (x^4*(b*x
^3 + a*x^4)^(1/4))/(5*a) + (4*b*(2*a^4 - b^3)*(b*x^3 + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (a*x)/b, -((a*
x)/b)])/(3*a^5*(1 + (a*x)/b)^(1/4)) - (371*b^5*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]
)/(4096*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) - (3*b^2*(a^4 - b^3)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/
(b + a*x)^(1/4)])/(16*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) + (b^2*(2*a^4 - b^3)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(
1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) + (371*b^5*(b*x^3 + a*x^4)^(1/4)*ArcTanh[
(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(4096*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) + (3*b^2*(a^4 - b^3)*(b*x^3 + a*x^
4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16*a^(23/4)*x^(3/4)*(b + a*x)^(1/4)) - (b^2*(2*a^4 - b^3
)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(23/4)*x^(3/4)*(b + a*x)^(1/4))

Rule 203

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

Rule 206

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

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 298

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

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]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps

\begin {align*} \int \frac {\left (-b-a x+x^4\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} \left (-b-a x+x^4\right )}{-b+a x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{b+a x^4} \left (-b-a x^4+x^{16}\right )}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {b \left (2 a^4-b^3\right ) x^2 \sqrt [4]{b+a x^4}}{a^5}-\left (1-\frac {b^3}{a^4}\right ) x^6 \sqrt [4]{b+a x^4}+\frac {b^2 x^{10} \sqrt [4]{b+a x^4}}{a^3}+\frac {b x^{14} \sqrt [4]{b+a x^4}}{a^2}+\frac {x^{18} \sqrt [4]{b+a x^4}}{a}-\frac {\left (2 a^4 b^2-b^5\right ) x^2 \sqrt [4]{b+a x^4}}{a^5 \left (-b+a x^4\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{18} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{14} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^{10} \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^2 \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int x^6 \sqrt [4]{b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {b^2 x^2 \sqrt [4]{b x^3+a x^4}}{3 a^3}+\frac {b x^3 \sqrt [4]{b x^3+a x^4}}{4 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {\left (b \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{18}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{5 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{4 a^2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{3 a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{-b+a x^4} \, dx,x,\sqrt [4]{x}\right )}{a^5 x^{3/4} \sqrt [4]{1+\frac {a x}{b}}}\\ &=-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {b^3 x \sqrt [4]{b x^3+a x^4}}{24 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {17 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{48 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {\left (3 b^2 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{16 a^2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (11 b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{48 a^3 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (7 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{24 a^4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 a x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {7 b^4 \sqrt [4]{b x^3+a x^4}}{96 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {5 b^3 x \sqrt [4]{b x^3+a x^4}}{384 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {\left (11 b^3 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{64 a^3 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (77 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{384 a^4 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{8 a x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {35 b^4 \sqrt [4]{b x^3+a x^4}}{1536 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^4 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^5 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (3 b^2 \left (-1+\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/2} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (7 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{512 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {7 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {7 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (77 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2048 a^5 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {35 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {35 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (231 b^5 \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{11/2} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {371 b^4 \sqrt [4]{b x^3+a x^4}}{6144 a^5}-\frac {b \left (a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{8 a^5}-\frac {b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4}}{a^5}+\frac {53 b^3 x \sqrt [4]{b x^3+a x^4}}{1536 a^4}-\frac {1}{2} \left (1-\frac {b^3}{a^4}\right ) x \sqrt [4]{b x^3+a x^4}+\frac {65 b^2 x^2 \sqrt [4]{b x^3+a x^4}}{192 a^3}+\frac {21 b x^3 \sqrt [4]{b x^3+a x^4}}{80 a^2}+\frac {x^4 \sqrt [4]{b x^3+a x^4}}{5 a}+\frac {4 b \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {a x}{b},-\frac {a x}{b}\right )}{3 a^5 \sqrt [4]{1+\frac {a x}{b}}}-\frac {371 b^5 \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {371 b^5 \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{4096 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {b^2 \left (2 a^4-b^3\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{23/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b^2 \left (1-\frac {b^3}{a^4}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{7/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.75, size = 232, normalized size = 0.84 \begin {gather*} \frac {x^3 \left (35 b^2 \left (13056 a^4-6541 b^3\right ) \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {2 a x}{b-a x}\right )+15 a b x \left (19712 a^4-9843 b^3\right ) \left (1-\frac {a^2 x^2}{b^2}\right )^{3/4} F_1\left (\frac {7}{4};\frac {3}{4},1;\frac {11}{4};-\frac {a x}{b},\frac {a x}{b}\right )+7 \left (1-\frac {a x}{b}\right )^{3/4} \left (-15360 a^6 x^2+768 a^5 \left (8 x^5-105 b x\right )-384 a^4 b \left (170 b-37 x^4\right )+18464 a^3 b^2 x^3+26820 a^2 b^3 x^2+49125 a b^4 x+32705 b^5\right )\right )}{215040 a^5 \left (x^3 (a x+b)\right )^{3/4} \left (1-\frac {a x}{b}\right )^{3/4}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]

[Out]

(x^3*(7*(1 - (a*x)/b)^(3/4)*(32705*b^5 + 49125*a*b^4*x - 15360*a^6*x^2 + 26820*a^2*b^3*x^2 + 18464*a^3*b^2*x^3
- 384*a^4*b*(170*b - 37*x^4) + 768*a^5*(-105*b*x + 8*x^5)) + 15*a*b*(19712*a^4 - 9843*b^3)*x*(1 - (a^2*x^2)/b
^2)^(3/4)*AppellF1[7/4, 3/4, 1, 11/4, -((a*x)/b), (a*x)/b] + 35*b^2*(13056*a^4 - 6541*b^3)*(1 + (a*x)/b)^(3/4)
*Hypergeometric2F1[3/4, 3/4, 7/4, (-2*a*x)/(b - a*x)]))/(215040*a^5*(x^3*(b + a*x))^(3/4)*(1 - (a*x)/b)^(3/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.38, size = 276, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b x^3+a x^4} \left (-65280 a^4 b+32705 b^4-15360 a^5 x+16420 a b^3 x+10400 a^2 b^2 x^2+8064 a^3 b x^3+6144 a^4 x^4\right )}{30720 a^5}+\frac {\left (19712 a^4 b^2-9843 b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}-\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}}+\frac {\left (-19712 a^4 b^2+9843 b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{4096 a^{23/4}}+\frac {2 \sqrt [4]{2} \left (2 a^4 b^2-b^5\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{23/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-b - a*x + x^4)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x),x]

[Out]

((b*x^3 + a*x^4)^(1/4)*(-65280*a^4*b + 32705*b^4 - 15360*a^5*x + 16420*a*b^3*x + 10400*a^2*b^2*x^2 + 8064*a^3*
b*x^3 + 6144*a^4*x^4))/(30720*a^5) + ((19712*a^4*b^2 - 9843*b^5)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4
096*a^(23/4)) - (2*2^(1/4)*(2*a^4*b^2 - b^5)*ArcTan[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(23/4) + ((-
19712*a^4*b^2 + 9843*b^5)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(4096*a^(23/4)) + (2*2^(1/4)*(2*a^4*b^2
- b^5)*ArcTanh[(2^(1/4)*a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(23/4)

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 1239, normalized size = 4.49

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="fricas")

[Out]

1/122880*(491520*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4)*arcta
n(1/2*(2^(3/4)*a^17*x*sqrt((sqrt(2)*a^12*x^2*sqrt((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^2
0)/a^23) + (4*a^8*b^4 - 4*a^4*b^7 + b^10)*sqrt(a*x^4 + b*x^3))/x^2)*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14
- 8*a^4*b^17 + b^20)/a^23)^(3/4) + 2^(3/4)*(2*a^21*b^2 - a^17*b^5)*(a*x^4 + b*x^3)^(1/4)*((16*a^16*b^8 - 32*a
^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(3/4))/((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^
17 + b^20)*x)) + 122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4
)*log(-(2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^(1/4) + (2*a^4*b^2
- b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 122880*2^(1/4)*a^5*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17
+ b^20)/a^23)^(1/4)*log((2^(1/4)*a^6*x*((16*a^16*b^8 - 32*a^12*b^11 + 24*a^8*b^14 - 8*a^4*b^17 + b^20)/a^23)^
(1/4) - (2*a^4*b^2 - b^5)*(a*x^4 + b*x^3)^(1/4))/x) - 60*a^5*((150981161449947136*a^16*b^8 - 30156403655678361
6*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4)*ar
ctan((a^17*x*sqrt((a^12*x^2*sqrt((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079
936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23) + (388562944*a^8*b^4 - 388050432*a^4*
b^7 + 96884649*b^10)*sqrt(a*x^4 + b*x^3))/x^2)*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 +
225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(3/4) + (19712*a^21*b^
2 - 9843*a^17*b^5)*(a*x^4 + b*x^3)^(1/4)*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874
706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(3/4))/((150981161449947136*
a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 938663521
1853201*b^20)*x)) - 15*a^5*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a
^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4)*log(-(a^6*x*((150981161449947136*a^1
6*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 938663521185
3201*b^20)/a^23)^(1/4) + (19712*a^4*b^2 - 9843*b^5)*(a*x^4 + b*x^3)^(1/4))/x) + 15*a^5*((150981161449947136*a^
16*b^8 - 301564036556783616*a^12*b^11 + 225874706663079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 93866352118
53201*b^20)/a^23)^(1/4)*log((a^6*x*((150981161449947136*a^16*b^8 - 301564036556783616*a^12*b^11 + 225874706663
079936*a^8*b^14 - 75192259797236736*a^4*b^17 + 9386635211853201*b^20)/a^23)^(1/4) - (19712*a^4*b^2 - 9843*b^5)
*(a*x^4 + b*x^3)^(1/4))/x) + 4*(6144*a^4*x^4 + 8064*a^3*b*x^3 + 10400*a^2*b^2*x^2 - 65280*a^4*b + 32705*b^4 -
20*(768*a^5 - 821*a*b^3)*x)*(a*x^4 + b*x^3)^(1/4))/a^5

________________________________________________________________________________________

giac [B]  time = 0.33, size = 704, normalized size = 2.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="giac")

[Out]

1/8192*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/
4))/((-a)^(3/4)*a^5) + 1/8192*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(
a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^5) + 1/16384*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*log(sqrt(2)*(-a)^(1/
4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) - 1/16384*sqrt(2)*(19712*a^4*b^2 - 9843*b^5)*l
og(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) + 1/2*sqrt(2)*(2*2^(1/4)*(
-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^5)*log(2^(3/4)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a
+ b/x))/a^6 - 1/2*sqrt(2)*(2*2^(1/4)*(-a)^(1/4)*a^4*b^2 - 2^(1/4)*(-a)^(1/4)*b^5)*log(-2^(3/4)*(-a)^(1/4)*(a
+ b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x))/a^6 - 1/30720*(65280*(a + b/x)^(17/4)*a^4*b^2 - 245760*(a + b
/x)^(13/4)*a^5*b^2 + 345600*(a + b/x)^(9/4)*a^6*b^2 - 215040*(a + b/x)^(5/4)*a^7*b^2 + 49920*(a + b/x)^(1/4)*a
^8*b^2 - 32705*(a + b/x)^(17/4)*b^5 + 114400*(a + b/x)^(13/4)*a*b^5 - 157370*(a + b/x)^(9/4)*a^2*b^5 + 94296*(
a + b/x)^(5/4)*a^3*b^5 - 24765*(a + b/x)^(1/4)*a^4*b^5)*x^5/(a^5*b^5) + (2*2^(3/4)*(-a)^(1/4)*a^4*b^2 - 2^(3/4
)*(-a)^(1/4)*b^5)*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^6 + (2*2^(3/4)*(-a
)^(1/4)*a^4*b^2 - 2^(3/4)*(-a)^(1/4)*b^5)*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1
/4))/a^6

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-a x -b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x -b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x)

[Out]

int((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-a*x-b)*(a*x^4+b*x^3)^(1/4)/(a*x-b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b*x^3)^(1/4)*(x^4 - a*x - b)/(a*x - b), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x^4 + b*x^3)^(1/4)*(b + a*x - x^4))/(b - a*x),x)

[Out]

int(((a*x^4 + b*x^3)^(1/4)*(b + a*x - x^4))/(b - a*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- a x - b + x^{4}\right )}{a x - b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-a*x-b)*(a*x**4+b*x**3)**(1/4)/(a*x-b),x)

[Out]

Integral((x**3*(a*x + b))**(1/4)*(-a*x - b + x**4)/(a*x - b), x)

________________________________________________________________________________________