3.21.1 \(\int \frac {b+a x^4}{\sqrt [4]{-b+a x^4} (b+3 a x^4)} \, dx\)

Optimal. Leaf size=141 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{6 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{6 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {530, 240, 212, 206, 203, 377} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{6 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{6 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + 3*a*x^4)),x]

[Out]

ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTan[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(3*Sqrt[2]
*a^(1/4)) + ArcTanh[(a^(1/4)*x)/(-b + a*x^4)^(1/4)]/(6*a^(1/4)) + ArcTanh[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/
4)]/(3*Sqrt[2]*a^(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 530

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 116, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{6 \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + 3*a*x^4)),x]

[Out]

(ArcTan[(a^(1/4)*x)/(-b + a*x^4)^(1/4)] + Sqrt[2]*ArcTan[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)] + ArcTanh[(a^
(1/4)*x)/(-b + a*x^4)^(1/4)] + Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*x)/(-b + a*x^4)^(1/4)])/(6*a^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.43, size = 141, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{6 \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{3 \sqrt {2} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)^(1/4)*(b + 3*a*x^4)),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.52, size = 253, normalized size = 1.79 \begin {gather*} \frac {2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \arctan \left (\frac {\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} x \sqrt {\frac {2 \, \sqrt {a} x^{2} + \sqrt {a x^{4} - b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{3 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right )}{6 \, a^{\frac {1}{4}}} + \frac {\arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} - b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} - b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{3 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right )}{12 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4+b)/(a*x^4-b)^(1/4)/(3*a*x^4+b),x, algorithm="fricas")

[Out]

2/3*(1/4)^(1/4)*arctan(((1/4)^(1/4)*x*sqrt((2*sqrt(a)*x^2 + sqrt(a*x^4 - b))/x^2)/a^(1/4) - (1/4)^(1/4)*(a*x^4
- b)^(1/4)/a^(1/4))/x)/a^(1/4) + 1/6*(1/4)^(1/4)*log((4*(1/4)^(3/4)*a^(1/4)*x + (a*x^4 - b)^(1/4))/x)/a^(1/4)
- 1/6*(1/4)^(1/4)*log(-(4*(1/4)^(3/4)*a^(1/4)*x - (a*x^4 - b)^(1/4))/x)/a^(1/4) + 1/3*arctan((x*sqrt((sqrt(a)
*x^2 + sqrt(a*x^4 - b))/x^2)/a^(1/4) - (a*x^4 - b)^(1/4)/a^(1/4))/x)/a^(1/4) + 1/12*log((a^(1/4)*x + (a*x^4 -
b)^(1/4))/x)/a^(1/4) - 1/12*log(-(a^(1/4)*x - (a*x^4 - b)^(1/4))/x)/a^(1/4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{{\left (3 \, 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((a*x^4+b)/(a*x^4-b)^(1/4)/(3*a*x^4+b),x, algorithm="giac")

[Out]

integrate((a*x^4 + b)/((3*a*x^4 + b)*(a*x^4 - b)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (3 a \,x^{4}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4+b)/(a*x^4-b)^(1/4)/(3*a*x^4+b),x)

[Out]

int((a*x^4+b)/(a*x^4-b)^(1/4)/(3*a*x^4+b),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{{\left (3 \, 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((a*x^4+b)/(a*x^4-b)^(1/4)/(3*a*x^4+b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b)/((3*a*x^4 + b)*(a*x^4 - b)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (3\,a\,x^4+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + a*x^4)/((a*x^4 - b)^(1/4)*(b + 3*a*x^4)),x)

[Out]

int((b + a*x^4)/((a*x^4 - b)^(1/4)*(b + 3*a*x^4)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{\sqrt [4]{a x^{4} - b} \left (3 a x^{4} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4+b)/(a*x**4-b)**(1/4)/(3*a*x**4+b),x)

[Out]

Integral((a*x**4 + b)/((a*x**4 - b)**(1/4)*(3*a*x**4 + b)), x)

________________________________________________________________________________________