3.18.22 \(\int \frac {b+2 a x^3}{(-b+x+a x^3) \sqrt [4]{-b x^3+a x^6}} \, dx\)

Optimal. Leaf size=116 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^6-b x^3}}{\sqrt {a x^6-b x^3}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^6-b x^3}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^6-b x^3}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.77, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {b+2 a x^3}{\left (-b+x+a x^3\right ) \sqrt [4]{-b x^3+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(b + 2*a*x^3)/((-b + x + a*x^3)*(-(b*x^3) + a*x^6)^(1/4)),x]

[Out]

(8*x*(1 - (a*x^3)/b)^(1/4)*Hypergeometric2F1[1/12, 1/4, 13/12, (a*x^3)/b])/(-(b*x^3) + a*x^6)^(1/4) - (12*b*x^
(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][1/((b - x^4 - a*x^12)*(-b + a*x^12)^(1/4)), x], x, x^(1/4)])/
(-(b*x^3) + a*x^6)^(1/4) - (8*x^(3/4)*(-b + a*x^3)^(1/4)*Defer[Subst][Defer[Int][x^4/((-b + a*x^12)^(1/4)*(-b
+ x^4 + a*x^12)), x], x, x^(1/4)])/(-(b*x^3) + a*x^6)^(1/4)

Rubi steps

\begin {align*} \int \frac {b+2 a x^3}{\left (-b+x+a x^3\right ) \sqrt [4]{-b x^3+a x^6}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-b+a x^3}\right ) \int \frac {b+2 a x^3}{x^{3/4} \sqrt [4]{-b+a x^3} \left (-b+x+a x^3\right )} \, dx}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {b+2 a x^{12}}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [4]{-b+a x^{12}}}+\frac {3 b-2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {3 b-2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 b}{\left (b-x^4-a x^{12}\right ) \sqrt [4]{-b+a x^{12}}}-\frac {2 x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}+\frac {\left (8 x^{3/4} \sqrt [4]{1-\frac {a x^3}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^{12}}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ &=\frac {8 x \sqrt [4]{1-\frac {a x^3}{b}} \, _2F_1\left (\frac {1}{12},\frac {1}{4};\frac {13}{12};\frac {a x^3}{b}\right )}{\sqrt [4]{-b x^3+a x^6}}-\frac {\left (8 x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^{12}} \left (-b+x^4+a x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}-\frac {\left (12 b x^{3/4} \sqrt [4]{-b+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-x^4-a x^{12}\right ) \sqrt [4]{-b+a x^{12}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x^3+a x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 a x^3}{\left (-b+x+a x^3\right ) \sqrt [4]{-b x^3+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(b + 2*a*x^3)/((-b + x + a*x^3)*(-(b*x^3) + a*x^6)^(1/4)),x]

[Out]

Integrate[(b + 2*a*x^3)/((-b + x + a*x^3)*(-(b*x^3) + a*x^6)^(1/4)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.92, size = 116, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^3+a x^6}}{-x^2+\sqrt {-b x^3+a x^6}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^3+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x^3+a x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*a*x^3)/((-b + x + a*x^3)*(-(b*x^3) + a*x^6)^(1/4)),x]

[Out]

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

________________________________________________________________________________________

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((2*a*x^3+b)/(a*x^3-b+x)/(a*x^6-b*x^3)^(1/4),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{3} + b}{{\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{3} - b + x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^3+b)/(a*x^3-b+x)/(a*x^6-b*x^3)^(1/4),x, algorithm="giac")

[Out]

integrate((2*a*x^3 + b)/((a*x^6 - b*x^3)^(1/4)*(a*x^3 - b + x)), x)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{3}+b}{\left (a \,x^{3}-b +x \right ) \left (a \,x^{6}-b \,x^{3}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^3+b)/(a*x^3-b+x)/(a*x^6-b*x^3)^(1/4),x)

[Out]

int((2*a*x^3+b)/(a*x^3-b+x)/(a*x^6-b*x^3)^(1/4),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{3} + b}{{\left (a x^{6} - b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{3} - b + x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^3+b)/(a*x^3-b+x)/(a*x^6-b*x^3)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*a*x^3 + b)/((a*x^6 - b*x^3)^(1/4)*(a*x^3 - b + x)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,a\,x^3+b}{{\left (a\,x^6-b\,x^3\right )}^{1/4}\,\left (a\,x^3+x-b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*a*x^3)/((a*x^6 - b*x^3)^(1/4)*(x - b + a*x^3)),x)

[Out]

int((b + 2*a*x^3)/((a*x^6 - b*x^3)^(1/4)*(x - b + a*x^3)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x**3+b)/(a*x**3-b+x)/(a*x**6-b*x**3)**(1/4),x)

[Out]

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

________________________________________________________________________________________