\(\int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\) [3091]

Optimal result

Integrand size = 49, antiderivative size = 535 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 c \sqrt {-b+a^2 x^2} \left (-416 a b^2 x+455 a^3 b x^3+260 a^5 x^5\right )+4 c \left (128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6\right )}{715 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}+\frac {\sqrt {2+\sqrt {2}} d \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \]

[Out]

1/715*(4*c*(a^2*x^2-b)^(1/2)*(260*a^5*x^5+455*a^3*b*x^3-416*a*b^2*x)+4*c*(260*a^6*x^6+325*a^4*b*x^4-676*a^2*b^
2*x^2+128*b^3))/a^4/(a*x+(a^2*x^2-b)^(1/2))^(13/4)+(2+2^(1/2))^(1/2)*d*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*b^(1/
8)-2*b^(1/8)/(2-2^(1/2))^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)))/b^(5/8
)+(2-2^(1/2))^(1/2)*d*arctan((2+2^(1/2))^(1/2)*b^(1/8)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b
)^(1/2))^(1/2)))/b^(5/8)-(2+2^(1/2))^(1/2)*d*arctanh((b^(1/8)/(2-2^(1/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/
(2-2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4))/b^(5/8)+(2-2^(1/2))^(1/2)*d*arctanh((b^(1/8)/(2+2^(1
/2))^(1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2+2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^2-b)^(1/2))^(1/4))/b^(5/8)

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.92, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6874, 2145, 335, 306, 303, 1176, 631, 210, 1179, 642, 304, 209, 212, 276} \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {2 d \arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {\sqrt {2} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{(-b)^{5/8}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {b^3 c}{26 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}+\frac {b c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{2 a^4}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}}{22 a^4} \]

[In]

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

[Out]

-1/26*(b^3*c)/(a^4*(a*x + Sqrt[-b + a^2*x^2])^(13/4)) - (3*b^2*c)/(10*a^4*(a*x + Sqrt[-b + a^2*x^2])^(5/4)) +
(b*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4))/(2*a^4) + (c*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(22*a^4) + (2*d*ArcTan[
(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)])/(-b)^(5/8) + (Sqrt[2]*d*ArcTan[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2
*x^2])^(1/4))/(-b)^(1/8)])/(-b)^(5/8) - (Sqrt[2]*d*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^
(1/8)])/(-b)^(5/8) - (2*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)])/(-b)^(5/8) - (d*Log[(-b)^(1/4)
- Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(Sqrt[2]*(-b)^(5/8))
+ (d*Log[(-b)^(1/4) + Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/
(Sqrt[2]*(-b)^(5/8))

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 276

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

Rule 303

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

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 306

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b
, 2]]}, Dist[r/(2*a), Int[x^m/(r + s*x^(n/2)), x], x] + Dist[r/(2*a), Int[x^m/(r - s*x^(n/2)), x], x]] /; Free
Q[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] &&  !GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2145

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1/(2^(2*m + p + 1)*e^(p + 1)*f^(2*m)))*(i/c)^m, Subst[Int[x^(n - 2*m - p - 2)*((-a)*f^2 + x^2)^p*(a*f^2
+ x^2)^(2*m + 1), x], x, e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0
] && EqQ[c*g - a*i, 0] && IntegersQ[p, 2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\frac {c x^3}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right ) \, dx \\ & = c \int \frac {x^3}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx+d \int \frac {1}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx \\ & = \frac {c \text {Subst}\left (\int \frac {\left (b+x^2\right )^3}{x^{17/4}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^4}+(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{x} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = \frac {c \text {Subst}\left (\int \left (\frac {b^3}{x^{17/4}}+\frac {3 b^2}{x^{9/4}}+\frac {3 b}{\sqrt [4]{x}}+x^{7/4}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{8 a^4}+(8 d) \text {Subst}\left (\int \frac {x^2}{b+x^8} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}-\frac {(4 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {-b}-x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {(4 d) \text {Subst}\left (\int \frac {x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}} \\ & = -\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}+\frac {(2 d) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}+\frac {(2 d) \text {Subst}\left (\int \frac {\sqrt [4]{-b}-x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {(2 d) \text {Subst}\left (\int \frac {\sqrt [4]{-b}+x^2}{\sqrt {-b}+x^4} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}} \\ & = -\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}+2 x}{-\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {d \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [8]{-b}-2 x}{-\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x-x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}}-\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} x+x^2} \, dx,x,\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {-b}} \\ & = -\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {\left (\sqrt {2} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\left (\sqrt {2} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}} \\ & = -\frac {b^3 c}{26 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/4}}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/4}}{2 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/4}}{22 a^4}+\frac {2 d \arctan \left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {\sqrt {2} d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt [4]{-b}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt [4]{-b}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2} (-b)^{5/8}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.89 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 c \left (128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6+13 a x \sqrt {-b+a^2 x^2} \left (-32 b^2+35 a^2 b x^2+20 a^4 x^4\right )\right )}{715 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}+\frac {\sqrt {2+\sqrt {2}} d \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \]

[In]

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

[Out]

(4*c*(128*b^3 - 676*a^2*b^2*x^2 + 325*a^4*b*x^4 + 260*a^6*x^6 + 13*a*x*Sqrt[-b + a^2*x^2]*(-32*b^2 + 35*a^2*b*
x^2 + 20*a^4*x^4)))/(715*a^4*(a*x + Sqrt[-b + a^2*x^2])^(13/4)) + (Sqrt[2 + Sqrt[2]]*d*ArcTan[(Sqrt[2 - Sqrt[2
]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(b^(1/4) - Sqrt[a*x + Sqrt[-b + a^2*x^2]])])/b^(5/8) + (Sqrt[2 -
Sqrt[2]]*d*ArcTan[(Sqrt[2 + Sqrt[2]]*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b
+ a^2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2
*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))])/b^(5/8) - (Sqrt[2 + Sqrt[2]]*d*ArcTanh[(Sqrt[1 + 1/Sqrt[
2]]*(b^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))])/b^(5/8)

Maple [F]

\[\int \frac {c \,x^{4}+d}{x \sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.05 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {-\left (715 i - 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (i + 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + \left (715 i + 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (i - 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - \left (715 i + 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (i - 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + \left (715 i - 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (i + 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + 1430 i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 1430 i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 8 \, {\left (55 \, a^{4} c x^{4} + 36 \, a^{2} b c x^{2} - 128 \, b^{2} c - {\left (55 \, a^{3} c x^{3} + 96 \, a b c x\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{1430 \, a^{4} b} \]

[In]

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

[Out]

1/1430*(-(715*I - 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 + (I + 1)*sqrt
(2)*(-d^8/b^5)^(3/8)*b^2) + (715*I + 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)
*d^3 - (I - 1)*sqrt(2)*(-d^8/b^5)^(3/8)*b^2) - (715*I + 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(
a^2*x^2 - b))^(1/4)*d^3 + (I - 1)*sqrt(2)*(-d^8/b^5)^(3/8)*b^2) + (715*I - 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b
*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 - (I + 1)*sqrt(2)*(-d^8/b^5)^(3/8)*b^2) - 1430*(-d^8/b^5)^(1/8)*a^4
*b*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 + (-d^8/b^5)^(3/8)*b^2) + 1430*I*(-d^8/b^5)^(1/8)*a^4*b*log((a*x +
sqrt(a^2*x^2 - b))^(1/4)*d^3 + I*(-d^8/b^5)^(3/8)*b^2) - 1430*I*(-d^8/b^5)^(1/8)*a^4*b*log((a*x + sqrt(a^2*x^2
- b))^(1/4)*d^3 - I*(-d^8/b^5)^(3/8)*b^2) + 1430*(-d^8/b^5)^(1/8)*a^4*b*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*d
^3 - (-d^8/b^5)^(3/8)*b^2) - 8*(55*a^4*c*x^4 + 36*a^2*b*c*x^2 - 128*b^2*c - (55*a^3*c*x^3 + 96*a*b*c*x)*sqrt(a
^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4))/(a^4*b)

Sympy [F]

\[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {c x^{4} + d}{x \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \]

[In]

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

[Out]

Integral((c*x**4 + d)/(x*(a*x + sqrt(a**2*x**2 - b))**(1/4)*sqrt(a**2*x**2 - b)), x)

Maxima [F]

\[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int { \frac {c x^{4} + d}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} x} \,d x } \]

[In]

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

[Out]

integrate((c*x^4 + d)/(sqrt(a^2*x^2 - b)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*x), x)

Giac [F(-1)]

Timed out. \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {c\,x^4+d}{x\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \]

[In]

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

[Out]

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