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

Optimal. Leaf size=101 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ] \]

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Rubi [B]  time = 0.65, antiderivative size = 325, normalized size of antiderivative = 3.22, number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6725, 377, 212, 206, 203} \begin {gather*} -\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}}-\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}-\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \tanh ^{-1}\left (\frac {\sqrt [8]{a} x \sqrt [4]{\sqrt {a}+\sqrt {b}}}{\sqrt [4]{a x^4-b}}\right )}{4 \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((2*Sqrt[a] - Sqrt[b])*ArcTan[(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*x)/(-b + a*x^4)^(1/4)])/(a^(1/8)*(Sqrt[a
] - Sqrt[b])^(1/4)*Sqrt[b]) + ((2*Sqrt[a] + Sqrt[b])*ArcTan[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x)/(-b + a*x^4)
^(1/4)])/(4*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]) - ((2*Sqrt[a] - Sqrt[b])*ArcTanh[(a^(1/8)*(Sqrt[a] - Sq
rt[b])^(1/4)*x)/(-b + a*x^4)^(1/4)])/(4*a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]) + ((2*Sqrt[a] + Sqrt[b])*Ar
cTanh[(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*x)/(-b + a*x^4)^(1/4)])/(4*a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[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 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 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 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}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx &=\int \left (-\frac {2 a \sqrt {b}-\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}+\frac {2 a \sqrt {b}+\sqrt {a} b}{2 \sqrt {a} \sqrt {b} \left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}}\right ) \, dx\\ &=-\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}-\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {b}+\sqrt {a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx\\ &=-\left (\frac {1}{2} \left (2 \sqrt {a}-\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )+\frac {1}{2} \left (2 \sqrt {a}+\sqrt {b}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {a} b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=-\left (\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\right )-\frac {1}{4} \left (-1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{4} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{4} \left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}+\frac {\left (1-\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}+\frac {\left (1+\frac {2 \sqrt {a}}{\sqrt {b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\\ \end {align*}

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Mathematica [F]  time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+2 a x^4}{\sqrt [4]{-b+a x^4} \left (-b+a x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.50, size = 100, normalized size = 0.99 \begin {gather*} \frac {1}{8} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (a*Log[x] - a*Log[(-b + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(-b +
a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + #1^5) & ]/8

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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)^(1/4)/(a*x^8-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 \, a x^{4} - b}{{\left (a x^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{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)^(1/4)/(a*x^8-b),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((2*a*x^4-b)/(a*x^4-b)^(1/4)/(a*x^8-b),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^{8} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{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)^(1/4)/(a*x^8-b),x, algorithm="maxima")

[Out]

integrate((2*a*x^4 - b)/((a*x^8 - b)*(a*x^4 - b)^(1/4)), 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 )}^{1/4}\,\left (b-a\,x^8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b - 2*a*x^4)/((a*x^4 - b)^(1/4)*(b - a*x^8)), 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]{a x^{4} - b} \left (a x^{8} - 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)**(1/4)/(a*x**8-b),x)

[Out]

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

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