3.27.69 \(\int \frac {-2 b-a x^4+2 x^8}{x^4 \sqrt [4]{-b+a x^4} (-b+2 a x^4)} \, dx\)

Optimal. Leaf size=239 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b}}{\sqrt {a x^4-b}-\sqrt {a} x^2}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt {a x^4-b}+\sqrt {a} x^2}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {2 \left (a x^4-b\right )^{3/4}}{3 b x^3} \]

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Rubi [A]  time = 0.89, antiderivative size = 346, normalized size of antiderivative = 1.45, number of steps used = 17, number of rules used = 13, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6725, 240, 212, 206, 203, 264, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}-\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+1\right )}{2 \sqrt {2} a^{5/4} b}-\frac {\left (5 a^2-b\right ) \log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )}{4 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )}{4 \sqrt {2} a^{5/4} b}+\frac {2 \left (a x^4-b\right )^{3/4}}{3 b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*b - a*x^4 + 2*x^8)/(x^4*(-b + a*x^4)^(1/4)*(-b + 2*a*x^4)),x]

[Out]

(2*(-b + a*x^4)^(3/4))/(3*b*x^3) + ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) - ((5*a^2 - b)*ArcTan[1
- (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(2*Sqrt[2]*a^(5/4)*b) + ((5*a^2 - b)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x)
/(-b + a*x^4)^(1/4)])/(2*Sqrt[2]*a^(5/4)*b) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) - ((5*a^2 -
b)*Log[1 + (Sqrt[a]*x^2)/Sqrt[-b + a*x^4] - (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(4*Sqrt[2]*a^(5/4)*b) + (
(5*a^2 - b)*Log[1 + (Sqrt[a]*x^2)/Sqrt[-b + a*x^4] + (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(4*Sqrt[2]*a^(5/
4)*b)

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 211

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

Rule 212

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

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p
+ 1/n]

Rule 264

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

Rule 377

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

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 6725

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

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Mathematica [A]  time = 0.31, size = 292, normalized size = 1.22 \begin {gather*} \frac {16 a^{5/4} \left (a x^4-b\right )^{3/4}-6 \sqrt {2} x^3 \left (5 a^2-b\right ) \left (\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+1\right )\right )-3 \sqrt {2} x^3 \left (5 a^2-b\right ) \left (\log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )-\log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )\right )+12 b x^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+12 b x^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{24 a^{5/4} b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*b - a*x^4 + 2*x^8)/(x^4*(-b + a*x^4)^(1/4)*(-b + 2*a*x^4)),x]

[Out]

(16*a^(5/4)*(-b + a*x^4)^(3/4) + 12*b*x^3*ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] - 6*Sqrt[2]*(5*a^2 - b)*x^3*(
ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)] - ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)]) + 12*
b*x^3*ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] - 3*Sqrt[2]*(5*a^2 - b)*x^3*(Log[1 + (Sqrt[a]*x^2)/Sqrt[-b + a*x
^4] - (Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)] - Log[1 + (Sqrt[a]*x^2)/Sqrt[-b + a*x^4] + (Sqrt[2]*a^(1/4)*x)/(
-b + a*x^4)^(1/4)]))/(24*a^(5/4)*b*x^3)

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IntegrateAlgebraic [A]  time = 1.02, size = 239, normalized size = 1.00 \begin {gather*} \frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b+a x^4}}{-\sqrt {a} x^2+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt {a} x^2+\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-2*b - a*x^4 + 2*x^8)/(x^4*(-b + a*x^4)^(1/4)*(-b + 2*a*x^4)),x]

[Out]

(2*(-b + a*x^4)^(3/4))/(3*b*x^3) + ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) + ((5*a^2 - b)*ArcTan[(S
qrt[2]*a^(1/4)*x*(-b + a*x^4)^(1/4))/(-(Sqrt[a]*x^2) + Sqrt[-b + a*x^4])])/(2*Sqrt[2]*a^(5/4)*b) + ArcTanh[(a^
(1/4)*x)/(-b + a*x^4)^(1/4)]/(2*a^(5/4)) + ((5*a^2 - b)*ArcTanh[(Sqrt[a]*x^2 + Sqrt[-b + a*x^4])/(Sqrt[2]*a^(1
/4)*x*(-b + a*x^4)^(1/4))])/(2*Sqrt[2]*a^(5/4)*b)

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fricas [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((2*x^8-a*x^4-2*b)/x^4/(a*x^4-b)^(1/4)/(2*a*x^4-b),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{8} - a x^{4} - 2 \, b}{{\left (2 \, a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-a*x^4-2*b)/x^4/(a*x^4-b)^(1/4)/(2*a*x^4-b),x, algorithm="giac")

[Out]

integrate((2*x^8 - a*x^4 - 2*b)/((2*a*x^4 - b)*(a*x^4 - b)^(1/4)*x^4), x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}-2 b}{x^{4} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (2 a \,x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^8-a*x^4-2*b)/x^4/(a*x^4-b)^(1/4)/(2*a*x^4-b),x)

[Out]

int((2*x^8-a*x^4-2*b)/x^4/(a*x^4-b)^(1/4)/(2*a*x^4-b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{8} - a x^{4} - 2 \, b}{{\left (2 \, a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^8-a*x^4-2*b)/x^4/(a*x^4-b)^(1/4)/(2*a*x^4-b),x, algorithm="maxima")

[Out]

integrate((2*x^8 - a*x^4 - 2*b)/((2*a*x^4 - b)*(a*x^4 - b)^(1/4)*x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {-2\,x^8+a\,x^4+2\,b}{x^4\,{\left (a\,x^4-b\right )}^{1/4}\,\left (b-2\,a\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*b + a*x^4 - 2*x^8)/(x^4*(a*x^4 - b)^(1/4)*(b - 2*a*x^4)),x)

[Out]

int((2*b + a*x^4 - 2*x^8)/(x^4*(a*x^4 - b)^(1/4)*(b - 2*a*x^4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- a x^{4} - 2 b + 2 x^{8}}{x^{4} \sqrt [4]{a x^{4} - b} \left (2 a x^{4} - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**8-a*x**4-2*b)/x**4/(a*x**4-b)**(1/4)/(2*a*x**4-b),x)

[Out]

Integral((-a*x**4 - 2*b + 2*x**8)/(x**4*(a*x**4 - b)**(1/4)*(2*a*x**4 - b)), x)

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