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

Optimal. Leaf size=118 \[ \frac {3}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{\sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{\sqrt [4]{a}} \]

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Rubi [B]  time = 1.43, antiderivative size = 491, normalized size of antiderivative = 4.16, number of steps used = 17, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {2056, 6715, 6725, 240, 212, 206, 203, 1429, 377} \begin {gather*} \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4+b x^2}}-\frac {3 \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {3 \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}}+\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4+b x^2}}-\frac {3 \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {3 \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 1429

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[-(a*c), 2]}, -Dist[c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 {-b+2 a x^4}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {-b+2 a x^4}{\sqrt {x} \sqrt [4]{b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {-b+2 a x^8}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [4]{b+a x^4}}-\frac {3 b}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (6 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {-a} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {-a} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {-a} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {-a} \sqrt {b} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {2 \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b x^2+a x^4}}+\frac {2 \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (3 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}\\ &=\frac {2 \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b x^2+a x^4}}-\frac {3 \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{b x^2+a x^4}}-\frac {3 \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt [4]{b x^2+a x^4}}+\frac {2 \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{a} \sqrt [4]{b x^2+a x^4}}-\frac {3 \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{b x^2+a x^4}}-\frac {3 \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt [4]{b x^2+a x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+2 a x^4}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 118, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4) + (2*ArcTanh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/a^(1/4)
+ (3*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-Log[x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/4

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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*a*x^4-b)/(a*x^4+b)/(a*x^4+b*x^2)^(1/4),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 \, a x^{4} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-b}{\left (a \,x^{4}+b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b-2\,a\,x^4}{\left (a\,x^4+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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