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

Optimal. Leaf size=397 \[ \sqrt {2} b^{3/4} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {a^2 x^2-b}+a x-\sqrt {b}}\right )+\sqrt {2} b^{3/4} d \tanh ^{-1}\left (\frac {\frac {\sqrt {a^2 x^2-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt {\sqrt {a^2 x^2-b}+a x}}\right )+\frac {2 \left (5040 a^{10} c x^{10}+18480 a^{10} d x^6-10620 a^8 b c x^8-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+304 b^5 c\right )+2 \sqrt {a^2 x^2-b} \left (5040 a^9 c x^9+18480 a^9 d x^5-8100 a^7 b c x^7-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-1368 a b^4 c x\right )}{3465 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{9/2}} \]

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Rubi [A]  time = 1.72, antiderivative size = 499, normalized size of antiderivative = 1.26, number of steps used = 18, number of rules used = 12, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.245, Rules used = {6742, 2120, 462, 459, 329, 297, 1162, 617, 204, 1165, 628, 448} \begin {gather*} -\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}}+\sqrt {2} b^{3/4} d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt [4]{b}}+1\right )+\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} \end {gather*}

Antiderivative was successfully verified.

[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 204

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

Rule 297

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 329

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 448

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 459

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 462

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 617

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 628

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 1162

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 1165

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 2120

Int[(x_)^(p_.)*((g_) + (i_.)*(x_)^2)^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :>
Dist[(1*(i/c)^m)/(2^(2*m + p + 1)*e^(p + 1)*f^(2*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 6742

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

Rubi steps

\begin {align*} \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 \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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} d \tan ^{-1}\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*}

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Mathematica [B]  time = 24.24, size = 9604, normalized size = 24.19 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.93, size = 397, normalized size = 1.00 \begin {gather*} \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 \tan ^{-1}\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 \tanh ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-b + a^2*x^2]*(-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) + 2*(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))/(3465*a^4*(a*x + Sqrt[-b + a^2*x^2])^(9/2)) + Sqrt[2]*b^(3/4)*d*Arc
Tan[(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[(b^(1/4)/Sqrt[2] + (a*x)/(Sqrt[2]*b^(1/4)) + Sqrt[-b + a^2*x^2]/(Sqrt[2]*b^(1/4)))/Sqrt[a*x + Sqrt[-
b + a^2*x^2]]]

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fricas [A]  time = 0.51, size = 341, normalized size = 0.86 \begin {gather*} \frac {13860 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \arctan \left (-\frac {\left (-b^{3} d^{4}\right )^{\frac {1}{4}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - \sqrt {a b^{4} d^{6} x + \sqrt {a^{2} x^{2} - b} b^{4} d^{6} - \sqrt {-b^{3} d^{4}} b^{3} d^{4}} \left (-b^{3} d^{4}\right )^{\frac {1}{4}}}{b^{3} d^{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 ) + 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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(13860*(-b^3*d^4)^(1/4)*a^4*arctan(-((-b^3*d^4)^(1/4)*sqrt(a*x + sqrt(a^2*x^2 - b))*b^2*d^3 - sqrt(a*b^
4*d^6*x + sqrt(a^2*x^2 - b)*b^4*d^6 - sqrt(-b^3*d^4)*b^3*d^4)*(-b^3*d^4)^(1/4))/(b^3*d^4)) - 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*(-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)) - 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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b}\, \left (c \,x^{4}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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