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

Optimal. Leaf size=535 \[ \frac {\sqrt {2+\sqrt {2}} d \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt [4]{b}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \tanh ^{-1}\left (\frac {\frac {\sqrt {\sqrt {a^2 x^2-b}+a x}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}}{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )}{b^{5/8}}+\frac {4 c \left (260 a^6 x^6+325 a^4 b x^4-676 a^2 b^2 x^2+128 b^3\right )+4 c \sqrt {a^2 x^2-b} \left (260 a^5 x^5+455 a^3 b x^3-416 a b^2 x\right )}{715 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}} \]

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Rubi [A]  time = 2.11, antiderivative size = 490, normalized size of antiderivative = 0.92, number of steps used = 20, number of rules used = 14, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6742, 2120, 329, 300, 297, 1162, 617, 204, 1165, 628, 298, 203, 206, 270} \begin {gather*} -\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 {2 d \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-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} \end {gather*}

Antiderivative was successfully verified.

[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 203

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

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 206

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

Rule 270

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 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 298

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 300

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]] /;
FreeQ[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] &&  !GtQ[a/b, 0]

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 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 {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx &=\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 \operatorname {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) \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {2 d \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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 \tan ^{-1}\left (\frac {\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\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*}

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Mathematica [C]  time = 5.18, size = 742, normalized size = 1.39 \begin {gather*} \frac {4 \left (\frac {13585 a^5 d \sqrt {\text {sgn}(a)^2} \left (\sqrt {a^2 x^2-b}+a x\right )}{\sqrt {a^2} \text {sgn}(a)}+\frac {13585 a^4 d \sqrt {a^2 x^2-b} \left (\sqrt {a^2 x^2-b}+a x\right )^2 \left (2 \, _2F_1\left (\frac {3}{8},1;\frac {11}{8};-\frac {\left (a x+\sqrt {a^2 x^2-b}\right )^2}{b}\right )-1\right )}{a x \left (\sqrt {a^2 x^2-b}+a x\right )-b}-\frac {3 c \sqrt {a^2 x^2-b} \left (156 a b^3 x \left (8 \sqrt {a^2 x^2-b}+13 a x\right )+5720 a^7 x^7 \left (\sqrt {a^2 x^2-b}+a x\right )-260 a^5 b x^5 \left (14 \sqrt {a^2 x^2-b}+25 a x\right )-65 a^3 b^2 x^3 \left (21 \sqrt {a^2 x^2-b}+4 a x\right )-384 b^4\right )}{\left (\sqrt {a^2 x^2-b}+a x\right )^2 \left (a x \left (\sqrt {a^2 x^2-b}+a x\right )-b\right )}+\frac {3 c \sqrt {a^2 x^2-b} \left (b-2 a x \left (\sqrt {a^2 x^2-b}+a x\right )\right )^4 \left (-832 a b^3 x \left (8 \sqrt {a^2 x^2-b}+13 a x\right )+5720 a^7 x^7 \left (\sqrt {a^2 x^2-b}+a x\right )+260 a^5 b x^5 \left (5 \sqrt {a^2 x^2-b}-6 a x\right )+455 a^3 b^2 x^3 \left (16 \sqrt {a^2 x^2-b}+13 a x\right )+2048 b^4\right )}{-a b^5 x \left (11 \sqrt {a^2 x^2-b}+61 a x\right )+1024 a^{11} x^{11} \left (\sqrt {a^2 x^2-b}+a x\right )-256 a^9 b x^9 \left (11 \sqrt {a^2 x^2-b}+13 a x\right )+256 a^7 b^2 x^7 \left (11 \sqrt {a^2 x^2-b}+16 a x\right )-112 a^5 b^3 x^5 \left (11 \sqrt {a^2 x^2-b}+21 a x\right )+20 a^3 b^4 x^3 \left (11 \sqrt {a^2 x^2-b}+31 a x\right )+b^6}\right )}{40755 a^4 b \sqrt [4]{\sqrt {a^2 x^2-b}+a x}} \end {gather*}

Warning: Unable to verify antiderivative.

[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*((3*c*Sqrt[-b + a^2*x^2]*(b - 2*a*x*(a*x + Sqrt[-b + a^2*x^2]))^4*(2048*b^4 + 5720*a^7*x^7*(a*x + Sqrt[-b +
a^2*x^2]) + 260*a^5*b*x^5*(-6*a*x + 5*Sqrt[-b + a^2*x^2]) - 832*a*b^3*x*(13*a*x + 8*Sqrt[-b + a^2*x^2]) + 455
*a^3*b^2*x^3*(13*a*x + 16*Sqrt[-b + a^2*x^2])))/(b^6 + 1024*a^11*x^11*(a*x + Sqrt[-b + a^2*x^2]) - 256*a^9*b*x
^9*(13*a*x + 11*Sqrt[-b + a^2*x^2]) + 256*a^7*b^2*x^7*(16*a*x + 11*Sqrt[-b + a^2*x^2]) - 112*a^5*b^3*x^5*(21*a
*x + 11*Sqrt[-b + a^2*x^2]) + 20*a^3*b^4*x^3*(31*a*x + 11*Sqrt[-b + a^2*x^2]) - a*b^5*x*(61*a*x + 11*Sqrt[-b +
a^2*x^2])) - (3*c*Sqrt[-b + a^2*x^2]*(-384*b^4 + 5720*a^7*x^7*(a*x + Sqrt[-b + a^2*x^2]) + 156*a*b^3*x*(13*a*
x + 8*Sqrt[-b + a^2*x^2]) - 260*a^5*b*x^5*(25*a*x + 14*Sqrt[-b + a^2*x^2]) - 65*a^3*b^2*x^3*(4*a*x + 21*Sqrt[-
b + a^2*x^2])))/((a*x + Sqrt[-b + a^2*x^2])^2*(-b + a*x*(a*x + Sqrt[-b + a^2*x^2]))) + (13585*a^4*d*Sqrt[-b +
a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^2*(-1 + 2*Hypergeometric2F1[3/8, 1, 11/8, -((a*x + Sqrt[-b + a^2*x^2])^2/b
)]))/(-b + a*x*(a*x + Sqrt[-b + a^2*x^2])) + (13585*a^5*d*(a*x + Sqrt[-b + a^2*x^2])*Sqrt[Sign[a]^2])/(Sqrt[a^
2]*Sign[a])))/(40755*a^4*b*(a*x + Sqrt[-b + a^2*x^2])^(1/4))

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IntegrateAlgebraic [A]  time = 1.95, size = 510, normalized size = 0.95 \begin {gather*} \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 \tan ^{-1}\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 \tan ^{-1}\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 \tanh ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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*Sqrt[-b + a^2*x^2]*(-416*a*b^2*x + 455*a^3*b*x^3 + 260*a^5*x^5) + 4*c*(128*b^3 - 676*a^2*b^2*x^2 + 325*a^
4*b*x^4 + 260*a^6*x^6))/(715*a^4*(a*x + Sqrt[-b + a^2*x^2])^(13/4)) - (Sqrt[2 + Sqrt[2]]*d*ArcTan[(Sqrt[2 - Sq
rt[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 + Sqr
t[-b + a^2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 - 1/Sqrt[2]]
*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*Ar
cTanh[(Sqrt[1 + 1/Sqrt[2]]*b^(1/8) + (Sqrt[1 + 1/Sqrt[2]]*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)

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fricas [A]  time = 0.58, size = 842, normalized size = 1.57 \begin {gather*} -\frac {2860 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (-\frac {d^{8} + \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} - \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} - \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\right ) + 2860 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (\frac {d^{8} - \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} + \sqrt {2} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\right ) - 715 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + 4 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + 4 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}\right ) + 715 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} - 4 \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{2} d^{3} + 4 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}\right ) - 5720 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \arctan \left (-\frac {\left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} b^{3} d^{3} - \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} d^{6} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{4}} b^{4}} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {5}{8}} b^{3}}{d^{8}}\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 \, \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(2860*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*arctan(-(d^8 + sqrt(2)*(-d^8/b^5)^(5/8)*(a*x + sqrt(a^2*x^2 - b))
^(1/4)*b^3*d^3 - sqrt(2)*sqrt(sqrt(a*x + sqrt(a^2*x^2 - b))*d^6 - sqrt(2)*(-d^8/b^5)^(3/8)*(a*x + sqrt(a^2*x^2
- b))^(1/4)*b^2*d^3 + (-d^8/b^5)^(3/4)*b^4)*(-d^8/b^5)^(5/8)*b^3)/d^8) + 2860*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*
arctan((d^8 - sqrt(2)*(-d^8/b^5)^(5/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^3*d^3 + sqrt(2)*sqrt(sqrt(a*x + sqrt(
a^2*x^2 - b))*d^6 + sqrt(2)*(-d^8/b^5)^(3/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^2*d^3 + (-d^8/b^5)^(3/4)*b^4)*(
-d^8/b^5)^(5/8)*b^3)/d^8) - 715*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(4*sqrt(a*x + sqrt(a^2*x^2 - b))*d^6 + 4*sqr
t(2)*(-d^8/b^5)^(3/8)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*b^2*d^3 + 4*(-d^8/b^5)^(3/4)*b^4) + 715*sqrt(2)*(-d^8/b^
5)^(1/8)*a^4*b*log(4*sqrt(a*x + sqrt(a^2*x^2 - b))*d^6 - 4*sqrt(2)*(-d^8/b^5)^(3/8)*(a*x + sqrt(a^2*x^2 - b))^
(1/4)*b^2*d^3 + 4*(-d^8/b^5)^(3/4)*b^4) - 5720*(-d^8/b^5)^(1/8)*a^4*b*arctan(-((-d^8/b^5)^(5/8)*(a*x + sqrt(a^
2*x^2 - b))^(1/4)*b^3*d^3 - sqrt(sqrt(a*x + sqrt(a^2*x^2 - b))*d^6 + (-d^8/b^5)^(3/4)*b^4)*(-d^8/b^5)^(5/8)*b^
3)/d^8) + 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*(
-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)

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giac [F(-1)]  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((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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{4}+d}{x \sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c x^{4} + d}{x \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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