3.30.4 \(\int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x^4} \, dx\) [2904]

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

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(875\) vs. \(2(323)=646\).
time = 2.27, 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 = {2081, 6857, 918, 52, 65, 338, 304, 209, 212, 21, 920, 95} \begin {gather*} -\frac {2 \sqrt [4]{a x^4+b x^3} \text {ArcTan}\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} \text {ArcTan}\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} \text {ArcTan}\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} \text {ArcTan}\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} \text {ArcTan}\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 52

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 65

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 95

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 209

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

Rule 212

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

Rule 304

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 338

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 918

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 920

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 2081

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 6857

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 ) \text {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 ) \text {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} \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.00, size = 340, normalized size = 1.05 \begin {gather*} -\frac {x^{9/4} (b+a x)^{3/4} \left (32 \left (\text {ArcTan}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )+a^{3/4} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {a^3 \log (x)-b^3 \log (x)-4 a^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 b^3 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-3 a^2 \log (x) \text {$\#$1}^4+12 a^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4+3 a \log (x) \text {$\#$1}^8-12 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^8-\log (x) \text {$\#$1}^{12}+4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^{12}}{a^3 \text {$\#$1}^3-3 a^2 \text {$\#$1}^7+3 a \text {$\#$1}^{11}-\text {$\#$1}^{15}}\&\right ]\right )}{16 a^{3/4} \left (x^3 (b+a x)\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]

-1/16*(x^(9/4)*(b + a*x)^(3/4)*(32*(ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] - ArcTanh[(a^(1/4)*x^(1/4))/(b +
a*x)^(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] - 4*a^3*Log[(b + a*x)^(1/4) - x^(1/4)*#1] + 4*b^3*Log[(b + a*x)^(1/4) - x^(1/4)*#1] - 3*a^2*Log[x]*#1
^4 + 12*a^2*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^4 + 3*a*Log[x]*#1^8 - 12*a*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*
#1^8 - Log[x]*#1^12 + 4*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^12)/(a^3*#1^3 - 3*a^2*#1^7 + 3*a*#1^11 - #1^15) &
]))/(a^(3/4)*(x^3*(b + a*x))^(3/4))

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Maple [F]
time = 0.00, 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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 11.20, size = 15336, normalized size = 47.48 \begin {gather*} \text {Too large to display} \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]

1/16*(8*(2*a^24*sqrt(b^3/a^15) - a^18 - a^15*b^3)*sqrt((2*a^3*b^3 - 2*b^6 + 2*(a^15 - a^12*b^3)*sqrt((a^6*sqrt
(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) + 2*(a^12 - a^9*b^3)*sqrt(b^3/a^15) + sqrt(2)*(a^9*b^3 - 2*(a^21*sqrt(b^3/
a^15) - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6) - (2*a^18 - a^15*b^3)*sqrt(b^3/a^15))*(a^6*sqrt(b^3/a^15)
+ 1)*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/(a^3
*b^3 - b^6))*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)*((
a^6*sqrt(b^3/a^15) + 1)/a^6)^(13/8)*(b^3/a^15)^(1/4)*arctan(-(sqrt(2)*sqrt(1/2)*(sqrt(2)*((a^23*x*sqrt(b^3/a^1
5) - a^14*b^3*x)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*(b^3/a^15)^(1/4) + (a^20*x*sqrt(b^3/a^15) - a^11*b^3*x)*(b
^3/a^15)^(1/4))*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)
*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6) + ((a^23*x*sqrt(b^3/a^15) - a^14*b^3*x)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)
*(b^3/a^15)^(1/4) + (a^20*x*sqrt(b^3/a^15) - a^11*b^3*x)*(b^3/a^15)^(1/4))*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6))
*sqrt((2*a^3*b^3 - 2*b^6 + 2*(a^15 - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) + 2*(a^12 - a
^9*b^3)*sqrt(b^3/a^15) + sqrt(2)*(a^9*b^3 - 2*(a^21*sqrt(b^3/a^15) - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a
^6) - (2*a^18 - a^15*b^3)*sqrt(b^3/a^15))*(a^6*sqrt(b^3/a^15) + 1)*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a
^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/(a^3*b^3 - b^6))*sqrt((2*(a^8 - 2*a^5*b^3 + a^2*b^6)*
x^2*((a^6*sqrt(b^3/a^15) + 1)/a^6)^(1/4) + sqrt(2)*(2*(a^7 - 2*a^4*b^3 + a*b^6)*(a*x^4 + b*x^3)^(1/4)*x - sqrt
(2)*(a^6*sqrt(b^3/a^15) + 1)*((2*a^22*x*sqrt(b^3/a^15) - (a^16 + a^13*b^3)*x)*(a*x^4 + b*x^3)^(1/4)*sqrt((a^6*
sqrt(b^3/a^15) + 1)/a^6) + (a*x^4 + b*x^3)^(1/4)*((a^19 - a^16*b^3)*x*sqrt(b^3/a^15) - (a^13 - a^10*b^3)*x))*s
qrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)*sqrt((2*a^3
*b^3 - 2*b^6 + 2*(a^15 - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) + 2*(a^12 - a^9*b^3)*sqrt
(b^3/a^15) + sqrt(2)*(a^9*b^3 - 2*(a^21*sqrt(b^3/a^15) - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6) - (2*a^1
8 - a^15*b^3)*sqrt(b^3/a^15))*(a^6*sqrt(b^3/a^15) + 1)*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3
/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/(a^3*b^3 - b^6))*((a^6*sqrt(b^3/a^15) + 1)/a^6)^(1/8) + 2*(a^6 -
2*a^3*b^3 + b^6)*sqrt(a*x^4 + b*x^3))/((a^6 - 2*a^3*b^3 + b^6)*x^2))*((a^6*sqrt(b^3/a^15) + 1)/a^6)^(3/8) - (a
^18 - a^15*b^3)*x*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*(b^3/a^15)^(3/4) - (a^15 - a^12*b^3)*x*(b^3/a^15)^(3/4) -
sqrt(2)*(sqrt(2)*((a^23*sqrt(b^3/a^15) - a^14*b^3)*(a*x^4 + b*x^3)^(1/4)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*(
b^3/a^15)^(1/4) + (a^20*sqrt(b^3/a^15) - a^11*b^3)*(a*x^4 + b*x^3)^(1/4)*(b^3/a^15)^(1/4))*sqrt(-(a^12*sqrt((a
^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)
+ ((a^23*sqrt(b^3/a^15) - a^14*b^3)*(a*x^4 + b*x^3)^(1/4)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*(b^3/a^15)^(1/4)
+ (a^20*sqrt(b^3/a^15) - a^11*b^3)*(a*x^4 + b*x^3)^(1/4)*(b^3/a^15)^(1/4))*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)
)*sqrt((2*a^3*b^3 - 2*b^6 + 2*(a^15 - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) + 2*(a^12 -
a^9*b^3)*sqrt(b^3/a^15) + sqrt(2)*(a^9*b^3 - 2*(a^21*sqrt(b^3/a^15) - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/
a^6) - (2*a^18 - a^15*b^3)*sqrt(b^3/a^15))*(a^6*sqrt(b^3/a^15) + 1)*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15) + 1)/
a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/(a^3*b^3 - b^6))*((a^6*sqrt(b^3/a^15) + 1)/a^6)^(3/8
) - sqrt(2)*(a^6*sqrt(b^3/a^15) + 1)*((a^24*x*sqrt(b^3/a^15) - a^15*b^3*x)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*
(b^3/a^15)^(1/4) + (a^21*x*sqrt(b^3/a^15) - a^12*b^3*x)*(b^3/a^15)^(1/4))*sqrt(-(a^12*sqrt((a^6*sqrt(b^3/a^15)
+ 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/((a^3*b^3 - b^6)*x)) + 8*(2*a^24*sqrt(b^3/a^15
) - a^18 - a^15*b^3)*sqrt((2*a^3*b^3 - 2*b^6 + 2*(a^15 - a^12*b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3
/a^15) + 2*(a^12 - a^9*b^3)*sqrt(b^3/a^15) + sqrt(2)*(a^9*b^3 - 2*(a^21*sqrt(b^3/a^15) - a^12*b^3)*sqrt((a^6*s
qrt(b^3/a^15) + 1)/a^6) - (2*a^18 - a^15*b^3)*sqrt(b^3/a^15))*(a^6*sqrt(b^3/a^15) + 1)*sqrt(-(a^12*sqrt((a^6*s
qrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)/a^6)/(a^3*b^3 - b^6))*sqrt(-(a^12*sqrt
((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)*((a^6*sqrt(b^3/a^15) + 1)/a^6)^
(13/8)*(b^3/a^15)^(1/4)*arctan(-(sqrt(2)*sqrt(1/2)*(sqrt(2)*((a^23*x*sqrt(b^3/a^15) - a^14*b^3*x)*sqrt((a^6*sq
rt(b^3/a^15) + 1)/a^6)*(b^3/a^15)^(1/4) + (a^20*x*sqrt(b^3/a^15) - a^11*b^3*x)*(b^3/a^15)^(1/4))*sqrt(-(a^12*s
qrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*sqrt(b^3/a^15) - a^9*sqrt(b^3/a^15) - b^3)/b^3)*sqrt((a^6*sqrt(b^3/a^15) + 1
)/a^6) + ((a^23*x*sqrt(b^3/a^15) - a^14*b^3*x)*sqrt((a^6*sqrt(b^3/a^15) + 1)/a^6)*(b^3/a^15)^(1/4) + (a^20*x*s
qrt(b^3/a^15) - a^11*b^3*x)*(b^3/a^15)^(1/4))*s...

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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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Giac [F(-1)] Timed out
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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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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