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

Optimal. Leaf size=448 \[ \frac {\left (\sqrt [4]{-1}-1\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}{(-1)^{3/4} x^2 \sqrt [4]{a^2+b}+\sqrt {a x^4-b}}\right )}{8 \left (a^2+b\right )^{5/8}}+\frac {i \left (\sqrt {2 \left (3+2 \sqrt {2}\right )}+i \sqrt {2}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} x^2 \sqrt [4]{a^2+b}+\sqrt {2-\sqrt {2}} \sqrt {a x^4-b}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt {2}-i \sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4-b}}{\sqrt {2-\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt [4]{-1}-1\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2+b}-\sqrt [8]{-1} \sqrt {a x^4-b}}{\sqrt {2+\sqrt {2}} x \sqrt [8]{a^2+b} \sqrt [4]{a x^4-b}}\right )}{8 \left (a^2+b\right )^{5/8}} \]

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Rubi [A]  time = 0.71, antiderivative size = 409, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1528, 377, 212, 208, 205} \begin {gather*} -\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{4 \sqrt {a^2+b} \sqrt [4]{a \sqrt {a^2+b}+a^2+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 205

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/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 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 1528

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rubi steps

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

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Mathematica [A]  time = 0.72, size = 376, normalized size = 0.84 \begin {gather*} \frac {-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a \sqrt {a^2+b}+a^2+b}}-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+b}} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{-a \sqrt {a^2+b}+a^2+b}}+\frac {\sqrt [4]{\sqrt {a^2+b}+a} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2+b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+b}+a} \sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a \sqrt {a^2+b}+a^2+b}}}{4 \sqrt {a^2+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 39.94, size = 510, normalized size = 1.14 \begin {gather*} -\frac {i \left (-i \sqrt {2}+\sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\left ((-1+i)-(1+i) (-1)^{3/4}\right ) \sqrt [4]{a^2+b} x^2+(1+i) \sqrt {-b+a x^4}+(1+i) (-1)^{3/4} \sqrt {-b+a x^4}}{2 \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (\sqrt {2}-i \sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {2 \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}{\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{a^2+b} x^2-(1+i) \sqrt {-b+a x^4}-\sqrt {2} \sqrt {-b+a x^4}}\right )}{16 \left (a^2+b\right )^{5/8}}+\frac {\left (-i+(-1)^{3/4}\right ) \tanh ^{-1}\left (\frac {\left ((-2+2 i)-(2+2 i) (-1)^{3/4}\right ) \sqrt [4]{a^2+b} x^2-(2+2 i) \sqrt {-b+a x^4}-(2+2 i) (-1)^{3/4} \sqrt {-b+a x^4}}{4 \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{8 \left (a^2+b\right )^{5/8}}+\frac {i \left (i \sqrt {2}+\sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\left ((1-i)+\sqrt {2}\right ) \sqrt [4]{a^2+b} x^2+(1+i) \sqrt {-b+a x^4}+\sqrt {2} \sqrt {-b+a x^4}}{2 \sqrt [8]{a^2+b} x \sqrt [4]{-b+a x^4}}\right )}{16 \left (a^2+b\right )^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/16*I)*((-I)*Sqrt[2] + Sqrt[2*(3 - 2*Sqrt[2])])*ArcTan[(((-1 + I) - (1 + I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^
2 + (1 + I)*Sqrt[-b + a*x^4] + (1 + I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])
/(a^2 + b)^(5/8) + ((Sqrt[2] - I*Sqrt[2*(3 + 2*Sqrt[2])])*ArcTan[(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))/(((1
- I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 - (1 + I)*Sqrt[-b + a*x^4] - Sqrt[2]*Sqrt[-b + a*x^4])])/(16*(a^2 + b)^(5
/8)) + ((-I + (-1)^(3/4))*ArcTanh[(((-2 + 2*I) - (2 + 2*I)*(-1)^(3/4))*(a^2 + b)^(1/4)*x^2 - (2 + 2*I)*Sqrt[-b
+ a*x^4] - (2 + 2*I)*(-1)^(3/4)*Sqrt[-b + a*x^4])/(4*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(8*(a^2 + b)^(5/
8)) + ((I/16)*(I*Sqrt[2] + Sqrt[2*(3 + 2*Sqrt[2])])*ArcTanh[(((1 - I) + Sqrt[2])*(a^2 + b)^(1/4)*x^2 + (1 + I)
*Sqrt[-b + a*x^4] + Sqrt[2]*Sqrt[-b + a*x^4])/(2*(a^2 + b)^(1/8)*x*(-b + a*x^4)^(1/4))])/(a^2 + b)^(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(x^4/(a*x^4-b)^(1/4)/(x^8+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 {x^{4}}{{\left (x^{8} + 2 \, a x^{4} - 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(x^4/(a*x^4-b)^(1/4)/(x^8+2*a*x^4-b),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (x^{8}+2 a \,x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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