\(\int \frac {\sqrt {-b+a^2 x^2} (d+c x^4) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx\) [2994]

Optimal result

Integrand size = 49, antiderivative size = 397 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {2 \sqrt {-b+a^2 x^2} \left (-1368 a b^4 c x+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+18480 a^9 d x^5-8100 a^7 b c x^7+5040 a^9 c x^9\right )+2 \left (304 b^5 c-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+18480 a^{10} d x^6-10620 a^8 b c x^8+5040 a^{10} c x^{10}\right )}{3465 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )+\sqrt {2} b^{3/4} d \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right ) \]

[Out]

1/3465*(2*(a^2*x^2-b)^(1/2)*(5040*a^9*c*x^9-8100*a^7*b*c*x^7+18480*a^9*d*x^5+1335*a^5*b^2*c*x^5-32340*a^7*b*d*
x^3+3705*a^3*b^3*c*x^3+10395*a^5*b^2*d*x-1368*a*b^4*c*x)+10080*a^10*c*x^10-21240*a^8*b*c*x^8+36960*a^10*d*x^6+
9510*a^6*b^2*c*x^6-83160*a^8*b*d*x^4+7470*a^4*b^3*c*x^4+48510*a^6*b^2*d*x^2-6156*a^2*b^4*c*x^2-4620*a^4*b^3*d+
608*b^5*c)/a^4/(a*x+(a^2*x^2-b)^(1/2))^(9/2)+2^(1/2)*b^(3/4)*d*arctan(2^(1/2)*b^(1/4)*(a*x+(a^2*x^2-b)^(1/2))^
(1/2)/(-b^(1/2)+a*x+(a^2*x^2-b)^(1/2)))+2^(1/2)*b^(3/4)*d*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*a*x*2^(1/2)/b^(1/4)
+1/2*(a^2*x^2-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x+(a^2*x^2-b)^(1/2))^(1/2))

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.26, number of steps used = 18, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.245, Rules used = {6874, 2145, 473, 470, 335, 303, 1176, 631, 210, 1179, 642, 459} \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\sqrt {2} b^{3/4} d \arctan \left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )-\frac {b^{3/4} d \log \left (\sqrt {a^2 x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {a^2 x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}+a x+\sqrt {b}\right )}{\sqrt {2}}+\frac {1}{3} d \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}-\frac {b d}{\sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {b^5 c}{144 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {\sqrt {a^2 x^2-b}+a x}}-\frac {b^2 c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/2}}{24 a^4}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{11/2}}{176 a^4}+\frac {b c \left (\sqrt {a^2 x^2-b}+a x\right )^{7/2}}{112 a^4} \]

[In]

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

[Out]

-1/144*(b^5*c)/(a^4*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) - (b^4*c)/(80*a^4*(a*x + Sqrt[-b + a^2*x^2])^(5/2)) + (b
^3*c)/(8*a^4*Sqrt[a*x + Sqrt[-b + a^2*x^2]]) - (b*d)/Sqrt[a*x + Sqrt[-b + a^2*x^2]] - (b^2*c*(a*x + Sqrt[-b +
a^2*x^2])^(3/2))/(24*a^4) + (d*(a*x + Sqrt[-b + a^2*x^2])^(3/2))/3 + (b*c*(a*x + Sqrt[-b + a^2*x^2])^(7/2))/(1
12*a^4) + (c*(a*x + Sqrt[-b + a^2*x^2])^(11/2))/(176*a^4) + Sqrt[2]*b^(3/4)*d*ArcTan[1 - (Sqrt[2]*Sqrt[a*x + S
qrt[-b + a^2*x^2]])/b^(1/4)] - Sqrt[2]*b^(3/4)*d*ArcTan[1 + (Sqrt[2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/b^(1/4)]
- (b^(3/4)*d*Log[Sqrt[b] + a*x + Sqrt[-b + a^2*x^2] - Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/Sqrt[2]
+ (b^(3/4)*d*Log[Sqrt[b] + a*x + Sqrt[-b + a^2*x^2] + Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/Sqrt[2
]

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 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 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 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 470

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

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[c^2*(e*x)^(m
+ 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

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 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x}+c x^3 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \, dx \\ & = c \int x^3 \sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \, dx+d \int \frac {\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx \\ & = \frac {c \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2 \left (b+x^2\right )^3}{x^{11/2}} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{32 a^4}+\frac {1}{2} d \text {Subst}\left (\int \frac {\left (-b+x^2\right )^2}{x^{3/2} \left (b+x^2\right )} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = -\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\frac {c \text {Subst}\left (\int \left (\frac {b^5}{x^{11/2}}+\frac {b^4}{x^{7/2}}-\frac {2 b^3}{x^{3/2}}-2 b^2 \sqrt {x}+b x^{5/2}+x^{9/2}\right ) \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{32 a^4}+\frac {d \text {Subst}\left (\int \frac {\sqrt {x} \left (-\frac {3 b^2}{2}+\frac {b x^2}{2}\right )}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right )}{b} \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-(2 b d) \text {Subst}\left (\int \frac {\sqrt {x}}{b+x^2} \, dx,x,a x+\sqrt {-b+a^2 x^2}\right ) \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-(4 b d) \text {Subst}\left (\int \frac {x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}+(2 b d) \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-(2 b d) \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{b+x^4} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-\frac {\left (b^{3/4} d\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\frac {\left (b^{3/4} d\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-(b d) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )-(b d) \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt {a x+\sqrt {-b+a^2 x^2}}\right ) \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}-\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}-\left (\sqrt {2} b^{3/4} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )+\left (\sqrt {2} b^{3/4} d\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right ) \\ & = -\frac {b^5 c}{144 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}-\frac {b^4 c}{80 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {b^3 c}{8 a^4 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b d}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}}{24 a^4}+\frac {1}{3} d \left (a x+\sqrt {-b+a^2 x^2}\right )^{3/2}+\frac {b c \left (a x+\sqrt {-b+a^2 x^2}\right )^{7/2}}{112 a^4}+\frac {c \left (a x+\sqrt {-b+a^2 x^2}\right )^{11/2}}{176 a^4}+\sqrt {2} b^{3/4} d \arctan \left (1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \arctan \left (1+\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )-\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}-\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}}+\frac {b^{3/4} d \log \left (\sqrt {b}+a x+\sqrt {-b+a^2 x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {608 b^5 c+3360 a^9 x^5 \left (11 d+3 c x^4\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )-684 a b^4 c x \left (9 a x+4 \sqrt {-b+a^2 x^2}\right )+30 a^3 b^3 \left (-154 a d+249 a c x^4+247 c x^3 \sqrt {-b+a^2 x^2}\right )-120 a^7 b x^3 \left (\sqrt {-b+a^2 x^2} \left (539 d+135 c x^4\right )+3 a \left (231 d x+59 c x^5\right )\right )+30 a^5 b^2 x \left (\sqrt {-b+a^2 x^2} \left (693 d+89 c x^4\right )+a \left (1617 d x+317 c x^5\right )\right )}{3465 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )+\sqrt {2} b^{3/4} d \text {arctanh}\left (\frac {\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}\right ) \]

[In]

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

[Out]

(608*b^5*c + 3360*a^9*x^5*(11*d + 3*c*x^4)*(a*x + Sqrt[-b + a^2*x^2]) - 684*a*b^4*c*x*(9*a*x + 4*Sqrt[-b + a^2
*x^2]) + 30*a^3*b^3*(-154*a*d + 249*a*c*x^4 + 247*c*x^3*Sqrt[-b + a^2*x^2]) - 120*a^7*b*x^3*(Sqrt[-b + a^2*x^2
]*(539*d + 135*c*x^4) + 3*a*(231*d*x + 59*c*x^5)) + 30*a^5*b^2*x*(Sqrt[-b + a^2*x^2]*(693*d + 89*c*x^4) + a*(1
617*d*x + 317*c*x^5)))/(3465*a^4*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + Sqrt[2]*b^(3/4)*d*ArcTan[(Sqrt[2]*b^(1/4)
*Sqrt[a*x + Sqrt[-b + a^2*x^2]])/(-Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])] + Sqrt[2]*b^(3/4)*d*ArcTanh[(Sqrt[b] +
a*x + Sqrt[-b + a^2*x^2])/(Sqrt[2]*b^(1/4)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])]

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=-\frac {3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} + \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) - 3465 i \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} + i \, \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) + 3465 i \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - i \, \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) - 3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (35 \, a^{5} c x^{5} - 19 \, a^{3} b c x^{3} + {\left (1155 \, a^{5} d - 152 \, a b^{2} c\right )} x - 2 \, {\left (175 \, a^{4} c x^{4} - 57 \, a^{2} b c x^{2} + 1155 \, a^{4} d - 152 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{3465 \, a^{4}} \]

[In]

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

[Out]

-1/3465*(3465*(-b^3*d^4)^(1/4)*a^4*log(sqrt(a*x + sqrt(a^2*x^2 - b))*b^2*d^3 + (-b^3*d^4)^(3/4)) - 3465*I*(-b^
3*d^4)^(1/4)*a^4*log(sqrt(a*x + sqrt(a^2*x^2 - b))*b^2*d^3 + I*(-b^3*d^4)^(3/4)) + 3465*I*(-b^3*d^4)^(1/4)*a^4
*log(sqrt(a*x + sqrt(a^2*x^2 - b))*b^2*d^3 - I*(-b^3*d^4)^(3/4)) - 3465*(-b^3*d^4)^(1/4)*a^4*log(sqrt(a*x + sq
rt(a^2*x^2 - b))*b^2*d^3 - (-b^3*d^4)^(3/4)) + 2*(35*a^5*c*x^5 - 19*a^3*b*c*x^3 + (1155*a^5*d - 152*a*b^2*c)*x
- 2*(175*a^4*c*x^4 - 57*a^2*b*c*x^2 + 1155*a^4*d - 152*b^2*c)*sqrt(a^2*x^2 - b))*sqrt(a*x + sqrt(a^2*x^2 - b)
))/a^4

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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