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

Optimal. Leaf size=82 \[ \frac {1}{3} \text {RootSum}\left [-\text {$\#$1}^{12}+3 \text {$\#$1}^8 a-3 \text {$\#$1}^4 a^2+a^3+a b^2\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4-b x^3}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-a}\& \right ] \]

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Rubi [B]  time = 1.26, antiderivative size = 1101, normalized size of antiderivative = 13.43, number of steps used = 33, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2056, 6725, 105, 63, 331, 298, 203, 206, 93} \begin {gather*} -\frac {2 (-1)^{2/3} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 (-1)^{2/3} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{a x^4-b x^3} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{a x-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/12)*b^(2/3)*x^(3/4)*(-b + a*x
)^(1/4)) + (2*(-1)^(1/3)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/12)*b^(2
/3)*x^(3/4)*(-b + a*x)^(1/4)) - (2*(-1)^(2/3)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/
4)])/(3*a^(1/12)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) + (2*(a^(2/3) + b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*Arc
Tan[(a^(1/12)*(a^(2/3) + b^(2/3))^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4
)) + (2*(-1)^(2/3)*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/12)*(a^(2/3) - (
-1)^(1/3)*b^(2/3))^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) - (2*(-1)^(1
/3)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTan[(a^(1/12)*(a^(2/3) + (-1)^(2/3)*b^(2/
3))^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) + (2*(-(b*x^3) + a*x^4)^(1/
4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/12)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) - (2*(-1)^(1/3)*
(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/12)*b^(2/3)*x^(3/4)*(-b + a*x)^(
1/4)) + (2*(-1)^(2/3)*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/12)*b^(2/3
)*x^(3/4)*(-b + a*x)^(1/4)) - (2*(a^(2/3) + b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[(a^(1/12)*(a^(2/3)
+ b^(2/3))^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) - (2*(-1)^(2/3)*(a^
(2/3) - (-1)^(1/3)*b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[(a^(1/12)*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1
/4)*x^(1/4))/(-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4)) + (2*(-1)^(1/3)*(a^(2/3) + (-1)^(
2/3)*b^(2/3))^(1/4)*(-(b*x^3) + a*x^4)^(1/4)*ArcTanh[(a^(1/12)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/4)*x^(1/4))/(
-b + a*x)^(1/4)])/(3*a^(1/4)*b^(2/3)*x^(3/4)*(-b + a*x)^(1/4))

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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt [4]{-b x^3+a x^4}}{x \left (b+a x^3\right )} \, dx &=\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (b+a x^3\right )} \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\sqrt [4]{-b x^3+a x^4} \int \left (-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}-\frac {\sqrt [4]{-b+a x}}{3 b^{2/3} \sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\sqrt [4]{-b x^3+a x^4} \int \frac {\sqrt [4]{-b+a x}}{\sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\sqrt [3]{-1} a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left ((-1)^{2/3} a^{2/3} \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (-b+a x)^{3/4}} \, dx}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (\left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left (a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (\left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) (-b+a x)^{3/4}} \, dx}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \sqrt [3]{-1} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 (-1)^{2/3} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 \left (a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}-\sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-\sqrt [3]{b}-\left (-a \sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [3]{b} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {\left (4 a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (4 \sqrt [3]{-1} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (4 (-1)^{2/3} a^{2/3} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt {a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}+b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt {a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}+b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}-\sqrt [3]{-1} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \left (\sqrt [3]{-1} a^{2/3}-b^{2/3}\right ) \sqrt [4]{-b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [6]{a} \sqrt {a^{2/3}+(-1)^{2/3} b^{2/3}} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 \sqrt [3]{-1} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 \sqrt [3]{-1} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {\left (2 (-1)^{2/3} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {\left (2 (-1)^{2/3} \sqrt [6]{a} \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 )}{3 b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ &=-\frac {2 \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [3]{-1} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 (-1)^{2/3} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [12]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}-\frac {2 (-1)^{2/3} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}+\frac {2 \sqrt [3]{-1} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{-b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [12]{a} \sqrt [4]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )}{3 \sqrt [4]{a} b^{2/3} x^{3/4} \sqrt [4]{-b+a x}}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 194, normalized size = 2.37 \begin {gather*} \frac {4 \sqrt [4]{x^3 (a x-b)} \left (\left (a^{2/3}+b^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [3]{a} b^{2/3} x}{a x-b}\right )+\left ((-1)^{2/3} a^{2/3}+b^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\sqrt [3]{-1} \sqrt [3]{a} b^{2/3} x-a x}{b-a x}\right )+\left (b^{2/3}-\sqrt [3]{-1} a^{2/3}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+(-1)^{2/3} \sqrt [3]{a} b^{2/3} x}{a x-b}\right )\right )}{9 b^{2/3} (b-a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 82, normalized size = 1.00 \begin {gather*} \frac {1}{3} \text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^4+3 a \text {$\#$1}^8-\text {$\#$1}^{12}\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[a^3 + a*b^2 - 3*a^2*#1^4 + 3*a*#1^8 - #1^12 & , (-(Log[x]*#1) + Log[(-(b*x^3) + a*x^4)^(1/4) - x*#1]*#
1)/(-a + #1^4) & ]/3

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

[Out]

Timed out

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giac [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 (a x^{3} + b\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 (a x^{3} + b\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x\,\left (a\,x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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