3.31.35 \(\int \frac {x^4}{\sqrt [4]{-b+a x^4} (-b+2 a x^4+x^8)} \, dx\) [3035]

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

[Out]

1/8*(-1+(-1)^(1/4))*arctan((-1)^(7/8)*(2+2^(1/2))^(1/2)*(a^2+b)^(1/8)*x*(a*x^4-b)^(1/4)/((-1)^(3/4)*(a^2+b)^(1
/4)*x^2+(a*x^4-b)^(1/2)))/(a^2+b)^(5/8)+1/16*I*(I*2^(1/2)+2+2^(1/2))*arctan((-1)^(7/8)*(-2+2^(1/2))*(a^2+b)^(1
/8)*x*(a*x^4-b)^(1/4)/((-1)^(3/4)*(2-2^(1/2))^(1/2)*(a^2+b)^(1/4)*x^2+(2-2^(1/2))^(1/2)*(a*x^4-b)^(1/2)))/(a^2
+b)^(5/8)+1/16*(2^(1/2)-I*(2+2^(1/2)))*arctanh(((-1)^(7/8)*(a^2+b)^(1/4)*x^2-(-1)^(1/8)*(a*x^4-b)^(1/2))/(2-2^
(1/2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4-b)^(1/4))/(a^2+b)^(5/8)+1/8*(-1+(-1)^(1/4))*arctanh(((-1)^(7/8)*(a^2+b)^(1
/4)*x^2-(-1)^(1/8)*(a*x^4-b)^(1/2))/(2+2^(1/2))^(1/2)/(a^2+b)^(1/8)/x/(a*x^4-b)^(1/4))/(a^2+b)^(5/8)

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Rubi [A]
time = 0.55, 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 = {1542, 385, 218, 214, 211} \begin {gather*} -\frac {\sqrt [4]{a-\sqrt {a^2+b}} \text {ArcTan}\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} \text {ArcTan}\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 211

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

Rule 214

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 385

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 1542

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 ) \text {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 ) \text {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}} \text {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}} \text {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}} \text {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}} \text {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 = 10.56, size = 376, normalized size = 0.84 \begin {gather*} \frac {-\frac {\sqrt [4]{a-\sqrt {a^2+b}} \text {ArcTan}\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 )}{\sqrt [4]{a^2+b-a \sqrt {a^2+b}}}+\frac {\sqrt [4]{a+\sqrt {a^2+b}} \text {ArcTan}\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 )}{\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 )}{\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 )}{\sqrt [4]{a^2+b+a \sqrt {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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Maple [F]
time = 0.01, 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*} \text {Failed to integrate} \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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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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