3.30.3 \(\int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx\)

Optimal. Leaf size=323 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-4 \text {$\#$1}^4 a^3+a^4-a b^3\& ,\frac {-\text {$\#$1}^{12} \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^{12} \log (x)+3 \text {$\#$1}^8 a \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-3 \text {$\#$1}^8 a \log (x)-3 \text {$\#$1}^4 a^2 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+3 \text {$\#$1}^4 a^2 \log (x)+a^3 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-b^3 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a^3 \log (x)+b^3 \log (x)}{-\text {$\#$1}^{15}+3 \text {$\#$1}^{11} a-3 \text {$\#$1}^7 a^2+\text {$\#$1}^3 a^3}\& \right ]-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{a^{3/4}} \]

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Rubi [B]  time = 3.50, antiderivative size = 875, normalized size of antiderivative = 2.71, number of steps used = 53, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2056, 6725, 904, 50, 63, 331, 298, 203, 206, 21, 906, 93} \begin {gather*} -\frac {2 \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a^(
3/4) - b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4
)])/(2*a^(15/16)*x^(3/4)*(b + a*x)^(1/4)) + ((a^(3/4) + b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/16)*
(a^(3/4) + b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(15/16)*x^(3/4)*(b + a*x)^(1/4)) + ((a - Sqrt[-Sqrt[
a]]*b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[((a - Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])
/(2*a*x^(3/4)*(b + a*x)^(1/4)) + ((a + Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[((a + Sqrt[-
Sqrt[a]]*b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a*x^(3/4)*(b + a*x)^(1/4)) + (2*(b*x^3 + a*x^4)^(1/4)*Ar
cTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a^(3/4) - b^(3/4))^(1/4)*(b*x^
3 + a*x^4)^(1/4)*ArcTanh[(a^(1/16)*(a^(3/4) - b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(15/16)*x^(3/4)*(
b + a*x)^(1/4)) - ((a^(3/4) + b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/16)*(a^(3/4) + b^(3/4))^(1/4)
*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(15/16)*x^(3/4)*(b + a*x)^(1/4)) - ((a - Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)*(b*x^3
+ a*x^4)^(1/4)*ArcTanh[((a - Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a*x^(3/4)*(b + a*x)^
(1/4)) - ((a + Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((a + Sqrt[-Sqrt[a]]*b^(3/4))^(1/4)
*x^(1/4))/(b + a*x)^(1/4)])/(2*a*x^(3/4)*(b + a*x)^(1/4))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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 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 904

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

Rule 906

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

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{-b+a x^4} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (-\frac {x^{11/4} \sqrt [4]{b+a x}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {x^{11/4} \sqrt [4]{b+a x}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-b^{3/2}-a \sqrt {b} x\right )}{(b+a x)^{3/4} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 \sqrt {a} \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-b^{3/2}-a \sqrt {b} x\right )}{(b+a x)^{3/4} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 \sqrt {a} \sqrt {b} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{\sqrt {b}-\sqrt {a} x^2} \, dx}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x}}{\sqrt {b}+\sqrt {a} x^2} \, dx}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\sqrt [4]{b x^3+a x^4} \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {-a \sqrt {b}-\sqrt {a} b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \frac {-a \sqrt {b}+\sqrt {a} b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a b^{3/4}+\frac {\sqrt {a} b^{3/2}}{\sqrt {-\sqrt {a}}}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}}+\frac {-a b^{3/4}-\frac {\sqrt {a} b^{3/2}}{\sqrt {-\sqrt {a}}}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {-a b^{3/4}-\sqrt [4]{a} b^{3/2}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) (b+a x)^{3/4}}+\frac {-a b^{3/4}+\sqrt [4]{a} b^{3/2}}{2 \sqrt {b} \sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) (b+a x)^{3/4}}\right ) \, dx}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \frac {\left (2 \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 )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}-b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) (b+a x)^{3/4}} \, dx}{4 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}+b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) (b+a x)^{3/4}} \, dx}{4 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}} \, dx}{4 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) (b+a x)^{3/4}} \, dx}{4 a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \left (\frac {\sqrt [4]{b x^3+a x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{b x^3+a x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\left (\left (a^{3/4}-b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}-\sqrt [4]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^{3/4}+b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}+\sqrt [4]{a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}-\sqrt {-\sqrt {a}} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\sqrt {-\sqrt {a}} b^{3/4}\right ) \sqrt [4]{b} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{b}-\left (a \sqrt [4]{b}+\sqrt {-\sqrt {a}} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a x^{3/4} \sqrt [4]{b+a x}}\\ &=2 \left (-\frac {\sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\left (\sqrt {a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [8]{a} \sqrt {a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [8]{a} \sqrt {a^{3/4}-b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [8]{a} \sqrt {a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [8]{a} \sqrt {a^{3/4}+b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{7/8} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-\sqrt {a}} b^{3/4}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}+2 \left (-\frac {\sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}\right )-\frac {\sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}-b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [16]{a} \sqrt [4]{a^{3/4}+b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{15/16} x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}-\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}

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Mathematica [C]  time = 0.54, size = 359, normalized size = 1.11 \begin {gather*} \frac {x^3 \left (\left (b^{3/4}-a^{3/4}\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-\sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-a^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [4]{a} b^{3/4} x}{b+a x}\right )+4 a^{3/4} \left (\frac {a x}{b}+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {a x}{b}\right )+i b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x-i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-i b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i \sqrt [4]{a} b^{3/4} x}{b+a x}\right )-b^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+\sqrt [4]{a} b^{3/4} x}{b+a x}\right )\right )}{3 a^{3/4} \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(4*a^(3/4)*(1 + (a*x)/b)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -((a*x)/b)] + (-a^(3/4) + b^(3/4))*Hyperg
eometric2F1[3/4, 1, 7/4, (a*x - a^(1/4)*b^(3/4)*x)/(b + a*x)] - a^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (a*x -
I*a^(1/4)*b^(3/4)*x)/(b + a*x)] + I*b^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (a*x - I*a^(1/4)*b^(3/4)*x)/(b + a*
x)] - a^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (a*x + I*a^(1/4)*b^(3/4)*x)/(b + a*x)] - I*b^(3/4)*Hypergeometric
2F1[3/4, 1, 7/4, (a*x + I*a^(1/4)*b^(3/4)*x)/(b + a*x)] - a^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (a*x + a^(1/4
)*b^(3/4)*x)/(b + a*x)] - b^(3/4)*Hypergeometric2F1[3/4, 1, 7/4, (a*x + a^(1/4)*b^(3/4)*x)/(b + a*x)]))/(3*a^(
3/4)*(x^3*(b + a*x))^(3/4))

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IntegrateAlgebraic [A]  time = 5.59, size = 322, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{3/4}}+\frac {1}{4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {a^3 \log (x)-b^3 \log (x)-a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+b^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-3 a^2 \log (x) \text {$\#$1}^4+3 a^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+3 a \log (x) \text {$\#$1}^8-3 a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^8-\log (x) \text {$\#$1}^{12}+\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^{12}}{-a^3 \text {$\#$1}^3+3 a^2 \text {$\#$1}^7-3 a \text {$\#$1}^{11}+\text {$\#$1}^{15}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/a^(3/4
) + RootSum[a^4 - a*b^3 - 4*a^3*#1^4 + 6*a^2*#1^8 - 4*a*#1^12 + #1^16 & , (a^3*Log[x] - b^3*Log[x] - a^3*Log[(
b*x^3 + a*x^4)^(1/4) - x*#1] + b^3*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 3*a^2*Log[x]*#1^4 + 3*a^2*Log[(b*x^3 +
a*x^4)^(1/4) - x*#1]*#1^4 + 3*a*Log[x]*#1^8 - 3*a*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^8 - Log[x]*#1^12 + Log[
(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^12)/(-(a^3*#1^3) + 3*a^2*#1^7 - 3*a*#1^11 + #1^15) & ]/4

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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(x^2*(a*x^4+b*x^3)^(1/4)/(a*x^4-b),x, algorithm="fricas")

[Out]

Timed out

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giac [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(x^2*(a*x^4+b*x^3)^(1/4)/(a*x^4-b),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a \,x^{4}-b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a*x^4+b*x^3)^(1/4)/(a*x^4-b),x)

[Out]

int(x^2*(a*x^4+b*x^3)^(1/4)/(a*x^4-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}} x^{2}}{a x^{4} - b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a*x^4+b*x^3)^(1/4)/(a*x^4-b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b*x^3)^(1/4)*x^2/(a*x^4 - b), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(a*x^4 + b*x^3)^(1/4))/(b - a*x^4),x)

[Out]

-int((x^2*(a*x^4 + b*x^3)^(1/4))/(b - a*x^4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x^{4} - b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a*x**4+b*x**3)**(1/4)/(a*x**4-b),x)

[Out]

Integral(x**2*(x**3*(a*x + b))**(1/4)/(a*x**4 - b), x)

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