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

Optimal. Leaf size=142 \[ -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {\sqrt {3 b-2 a x^2}}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt [4]{3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x \sqrt [4]{3 b-2 a x^2}}{\sqrt {3 b-2 a x^2}+\sqrt {3} x^2}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]

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Rubi [C]  time = 0.84, antiderivative size = 539, normalized size of antiderivative = 3.80, number of steps used = 10, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1692, 399, 490, 1218} \begin {gather*} \frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}+\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(-3*Sqrt[b])/Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9*b] + 9*b
], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9*b] +
9*b]*x) - ((a + Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(3*Sqrt[b])/Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9
*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 - 2*a*Sqrt[a^2
- 9*b] + 9*b]*x) + ((a - Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(-3*Sqrt[b])/Sqrt[-2*a^2 + 2*a*Sq
rt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 + 2*a
*Sqrt[a^2 - 9*b] + 9*b]*x) - ((a - Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(3*Sqrt[b])/Sqrt[-2*a^2
+ 2*a*Sqrt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*
a^2 + 2*a*Sqrt[a^2 - 9*b] + 9*b]*x)

Rule 399

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Dist[(2*Sqrt[-((b*x^2)/a)])/x, Subst[I
nt[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d*x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0]

Rule 490

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]],
s = Denominator[Rt[-(a/b), 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), In
t[1/((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx &=\int \left (\frac {a+\sqrt {a^2-9 b}}{\left (-2 a-2 \sqrt {a^2-9 b}+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}}+\frac {a-\sqrt {a^2-9 b}}{\left (-2 a+2 \sqrt {a^2-9 b}+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}}\right ) \, dx\\ &=\left (a-\sqrt {a^2-9 b}\right ) \int \frac {1}{\left (-2 a+2 \sqrt {a^2-9 b}+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}} \, dx+\left (a+\sqrt {a^2-9 b}\right ) \int \frac {1}{\left (-2 a-2 \sqrt {a^2-9 b}+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}} \, dx\\ &=\frac {\left (2 \sqrt {\frac {2}{3}} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a \left (-2 a+2 \sqrt {a^2-9 b}\right )-18 b+6 x^4\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{x}+\frac {\left (2 \sqrt {\frac {2}{3}} \left (a+\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-2 a \left (-2 a-2 \sqrt {a^2-9 b}\right )-18 b+6 x^4\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{x}\\ &=-\frac {\left (\left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b}-\sqrt {3} x^2\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{3 \sqrt {2} x}+\frac {\left (\left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b}+\sqrt {3} x^2\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{3 \sqrt {2} x}-\frac {\left (\left (a+\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b}-\sqrt {3} x^2\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{3 \sqrt {2} x}+\frac {\left (\left (a+\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b}+\sqrt {3} x^2\right ) \sqrt {1-\frac {x^4}{3 b}}} \, dx,x,\sqrt [4]{3 b-2 a x^2}\right )}{3 \sqrt {2} x}\\ &=\frac {\left (a+\sqrt {a^2-9 b}\right ) \sqrt [4]{b} \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} \sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b} x}-\frac {\left (a+\sqrt {a^2-9 b}\right ) \sqrt [4]{b} \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} \sqrt {-2 a^2-2 a \sqrt {a^2-9 b}+9 b} x}+\frac {\left (a-\sqrt {a^2-9 b}\right ) \sqrt [4]{b} \sqrt {\frac {a x^2}{b}} \Pi \left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} \sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b} x}-\frac {\left (a-\sqrt {a^2-9 b}\right ) \sqrt [4]{b} \sqrt {\frac {a x^2}{b}} \Pi \left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right )\right |-1\right )}{\sqrt {2} 3^{3/4} \sqrt {-2 a^2+2 a \sqrt {a^2-9 b}+9 b} x}\\ \end {align*}

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Mathematica [F]  time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.44, size = 142, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {\sqrt {3 b-2 a x^2}}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt [4]{3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{3} x \sqrt [4]{3 b-2 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-3 b}{\left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} \left (3 x^{4}-2 a \,x^{2}+3 b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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