3.20.100 \(\int \frac {-3 b+2 a x^4}{(-2 b+a x^4) \sqrt [4]{b+a x^4}} \, dx\)

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

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Rubi [A]  time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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 )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*b + 2*a*x^4)/((-2*b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

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

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Mathematica [A]  time = 0.12, size = 128, normalized size = 0.91 \begin {gather*} \frac {12 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\sqrt [4]{2} 3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+12 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\sqrt [4]{2} 3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{12 \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*b + 2*a*x^4)/((-2*b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

(12*ArcTan[(a^(1/4)*x)/(b + a*x^4)^(1/4)] - 2^(1/4)*3^(3/4)*ArcTan[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4)]
+ 12*ArcTanh[(a^(1/4)*x)/(b + a*x^4)^(1/4)] - 2^(1/4)*3^(3/4)*ArcTanh[((3/2)^(1/4)*a^(1/4)*x)/(b + a*x^4)^(1/4
)])/(12*a^(1/4))

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IntegrateAlgebraic [A]  time = 0.55, size = 141, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-3*b + 2*a*x^4)/((-2*b + a*x^4)*(b + a*x^4)^(1/4)),x]

[Out]

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

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fricas [B]  time = 0.56, size = 241, normalized size = 1.71 \begin {gather*} -\frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} x \sqrt {\frac {3 \, \sqrt {\frac {1}{6}} \sqrt {a} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}}{x}\right )}{a^{\frac {1}{4}}} - \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (-\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {2 \, \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 )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^4-3*b)/(a*x^4-2*b)/(a*x^4+b)^(1/4),x, algorithm="fricas")

[Out]

-(1/24)^(1/4)*arctan(2*((1/24)^(1/4)*x*sqrt((3*sqrt(1/6)*sqrt(a)*x^2 + sqrt(a*x^4 + b))/x^2)/a^(1/4) - (1/24)^
(1/4)*(a*x^4 + b)^(1/4)/a^(1/4))/x)/a^(1/4) - 1/4*(1/24)^(1/4)*log((12*(1/24)^(3/4)*a^(1/4)*x + (a*x^4 + b)^(1
/4))/x)/a^(1/4) + 1/4*(1/24)^(1/4)*log(-(12*(1/24)^(3/4)*a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4) + 2*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) + 1/2*log((a^(1/4
)*x + (a*x^4 + b)^(1/4))/x)/a^(1/4) - 1/2*log(-(a^(1/4)*x - (a*x^4 + b)^(1/4))/x)/a^(1/4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{4} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^4-3*b)/(a*x^4-2*b)/(a*x^4+b)^(1/4),x, algorithm="giac")

[Out]

integrate((2*a*x^4 - 3*b)/((a*x^4 + b)^(1/4)*(a*x^4 - 2*b)), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-3 b}{\left (a \,x^{4}-2 b \right ) \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^4-3*b)/(a*x^4-2*b)/(a*x^4+b)^(1/4),x)

[Out]

int((2*a*x^4-3*b)/(a*x^4-2*b)/(a*x^4+b)^(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} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^4-3*b)/(a*x^4-2*b)/(a*x^4+b)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*a*x^4 - 3*b)/((a*x^4 + b)^(1/4)*(a*x^4 - 2*b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,b-2\,a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*b - 2*a*x^4)/((b + a*x^4)^(1/4)*(2*b - a*x^4)),x)

[Out]

int((3*b - 2*a*x^4)/((b + a*x^4)^(1/4)*(2*b - a*x^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} - 3 b}{\left (a x^{4} - 2 b\right ) \sqrt [4]{a x^{4} + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x**4-3*b)/(a*x**4-2*b)/(a*x**4+b)**(1/4),x)

[Out]

Integral((2*a*x**4 - 3*b)/((a*x**4 - 2*b)*(a*x**4 + b)**(1/4)), x)

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