\(\int \frac {(b+a x^2) \sqrt [4]{b x^3+a x^4}}{x^2 (-b+a x^2)} \, dx\) [2552]

Optimal result

Integrand size = 37, antiderivative size = 215 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-2 \sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+2 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+a \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+b \log (x)+a \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}^3-\text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(491\) vs. \(2(215)=430\).

Time = 1.45 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.28, number of steps used = 23, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2081, 6857, 49, 65, 338, 304, 209, 212, 922, 37, 6820, 12, 95} \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {2 \sqrt [4]{a} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{a x^4+b x^3} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{x} \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x+b}}\right )}{x^{3/4} \sqrt [4]{a x+b}}+\frac {4 \sqrt [4]{a x^4+b x^3}}{x} \]

[In]

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

[Out]

(4*(b*x^3 + a*x^4)^(1/4))/x - (2*a^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(
3/4)*(b + a*x)^(1/4)) + (2*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/8)*(Sqrt[a] -
Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) + (2*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*(b*
x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/
4)) + (2*a^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) -
(2*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x^(1/4)
)/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4)) - (2*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*Arc
Tanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(x^(3/4)*(b + a*x)^(1/4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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 922

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(-g)*((e*f
- d*g)/(c*f^2 + a*g^2)), Int[(d + e*x)^(m - 1)*(f + g*x)^n, x], x] + Dist[1/(c*f^2 + a*g^2), Int[Simp[c*d*f +
a*e*g + c*(e*f - 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] && LtQ[n, -1]

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 6820

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

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*} \text {integral}& = \frac {\sqrt [4]{b x^3+a x^4} \int \frac {\sqrt [4]{b+a x} \left (b+a x^2\right )}{x^{5/4} \left (-b+a x^2\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}}{x^{5/4}}+\frac {2 b \sqrt [4]{b+a x}}{x^{5/4} \left (-b+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 {\sqrt [4]{b+a x}}{x^{5/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\sqrt [4]{b+a x}}{x^{5/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = -\frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {-a b-a b x}{\sqrt [4]{x} (b+a x)^{3/4} \left (-b+a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{x^{5/4} (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {\left (2 \sqrt [4]{b x^3+a x^4}\right ) \int \frac {a b (1+x)}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (4 a \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 {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (4 a \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 (2 a b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1+x}{\sqrt [4]{x} (b+a x)^{3/4} \left (b-a x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}+\frac {\left (2 \sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \sqrt {a} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a b \sqrt [4]{b x^3+a x^4}\right ) \int \left (\frac {\sqrt {b}+\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}}+\frac {\sqrt {b}-\frac {b}{\sqrt {a}}}{2 b \sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}+\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt {b}-\sqrt {a} x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 a \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (4 a \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) b \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (-\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}-\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 a^{3/4} \left (\frac {1}{\sqrt {a}}+\frac {1}{\sqrt {b}}\right ) \sqrt {b} \sqrt [4]{b x^3+a x^4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {\sqrt {a}+\sqrt {b}} x^{3/4} \sqrt [4]{b+a x}} \\ & = \frac {4 \sqrt [4]{b x^3+a x^4}}{x}-\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \arctan \left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {2 \sqrt [4]{a} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {2 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{b x^3+a x^4} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.12 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {x^2 (b+a x)^{3/4} \left (4 \sqrt [4]{b+a x}-2 \sqrt [4]{a} \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+2 \sqrt [4]{a} \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-\frac {1}{4} a \sqrt [4]{x} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+b \log (x)+4 a \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-4 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}^3+\text {$\#$1}^7}\&\right ]\right )}{\left (x^3 (b+a x)\right )^{3/4}} \]

[In]

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

[Out]

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

Maple [N/A]

Time = 0.41 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}-a b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a +b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (\textit {\_R}^{4}-a \right )}\right ) x +a^{\frac {1}{4}} x \ln \left (\frac {x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-x \,a^{\frac {1}{4}}+\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right )+2 a^{\frac {1}{4}} x \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\)	\(154\)

[In]

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

[Out]

(a*sum((_R^4-a+b)*ln((-_R*x+(x^3*(a*x+b))^(1/4))/x)/_R^3/(_R^4-a),_R=RootOf(_Z^8-2*_Z^4*a+a^2-a*b))*x+a^(1/4)*
x*ln((x*a^(1/4)+(x^3*(a*x+b))^(1/4))/(-x*a^(1/4)+(x^3*(a*x+b))^(1/4)))+2*a^(1/4)*x*arctan(1/a^(1/4)/x*(x^3*(a*
x+b))^(1/4))+4*(x^3*(a*x+b))^(1/4))/x

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.31 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.52 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\frac {x \sqrt {-\sqrt {a + \sqrt {a b}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {a b}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a + \sqrt {a b}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a + \sqrt {a b}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + x \sqrt {-\sqrt {a - \sqrt {a b}}} \log \left (\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {a b}}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - x \sqrt {-\sqrt {a - \sqrt {a b}}} \log \left (-\frac {2 \, {\left (x \sqrt {-\sqrt {a - \sqrt {a b}}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a + \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) + {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - {\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x \log \left (-\frac {2 \, {\left ({\left (a - \sqrt {a b}\right )}^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}\right )}}{x}\right ) - a^{\frac {1}{4}} x \log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + a^{\frac {1}{4}} x \log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - i \, a^{\frac {1}{4}} x \log \left (\frac {i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + i \, a^{\frac {1}{4}} x \log \left (\frac {-i \, a^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x} \]

[In]

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

[Out]

-(x*sqrt(-sqrt(a + sqrt(a*b)))*log(2*(x*sqrt(-sqrt(a + sqrt(a*b))) + (a*x^4 + b*x^3)^(1/4))/x) - x*sqrt(-sqrt(
a + sqrt(a*b)))*log(-2*(x*sqrt(-sqrt(a + sqrt(a*b))) - (a*x^4 + b*x^3)^(1/4))/x) + x*sqrt(-sqrt(a - sqrt(a*b))
)*log(2*(x*sqrt(-sqrt(a - sqrt(a*b))) + (a*x^4 + b*x^3)^(1/4))/x) - x*sqrt(-sqrt(a - sqrt(a*b)))*log(-2*(x*sqr
t(-sqrt(a - sqrt(a*b))) - (a*x^4 + b*x^3)^(1/4))/x) + (a + sqrt(a*b))^(1/4)*x*log(2*((a + sqrt(a*b))^(1/4)*x +
(a*x^4 + b*x^3)^(1/4))/x) - (a + sqrt(a*b))^(1/4)*x*log(-2*((a + sqrt(a*b))^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/
x) + (a - sqrt(a*b))^(1/4)*x*log(2*((a - sqrt(a*b))^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - (a - sqrt(a*b))^(1/4
)*x*log(-2*((a - sqrt(a*b))^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) - a^(1/4)*x*log((a^(1/4)*x + (a*x^4 + b*x^3)^(
1/4))/x) + a^(1/4)*x*log(-(a^(1/4)*x - (a*x^4 + b*x^3)^(1/4))/x) - I*a^(1/4)*x*log((I*a^(1/4)*x + (a*x^4 + b*x
^3)^(1/4))/x) + I*a^(1/4)*x*log((-I*a^(1/4)*x + (a*x^4 + b*x^3)^(1/4))/x) - 4*(a*x^4 + b*x^3)^(1/4))/x

Sympy [N/A]

Not integrable

Time = 4.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.13 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (a x^{2} + b\right )}{x^{2} \left (a x^{2} - b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.17 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 8.60 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.18 \[ \int \frac {\left (b+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\int \frac {\left (a\,x^2+b\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2\,\left (b-a\,x^2\right )} \,d x \]

[In]

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

[Out]

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