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

Optimal. Leaf size=128 \[ \frac {4 \left (a x^3-b x\right )^{3/4}}{3 x^3}-2 \sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b x}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b x}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b x}}{\sqrt {a x^3-b x}+x^2}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 5.97, 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 {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-4*b*(1 - (a*x^2)/b)^(1/4)*Hypergeometric2F1[-9/8, 1/4, -1/8, (a*x^2)/b])/(3*x^2*(-(b*x) + a*x^3)^(1/4)) + (4
*a*(1 - (a*x^2)/b)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, (a*x^2)/b])/(-(b*x) + a*x^3)^(1/4) + (24*b*x^(1/4)*
(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^2/((-b + a*x^8)^(1/4)*(b - a*x^8 - x^12)), x], x, x^(1/4)])/(-(b*
x) + a*x^3)^(1/4) + (8*a*x^(1/4)*(-b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^10/((-b + a*x^8)^(1/4)*(-b + a*x
^8 + x^12)), x], x, x^(1/4)])/(-(b*x) + a*x^3)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^{13/4} \sqrt [4]{-b+a x^2} \left (-b+a x^2+x^3\right )} \, dx}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-3 b+a x^8\right ) \left (b-a x^8+x^{12}\right )}{x^{10} \sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b}{x^{10} \sqrt [4]{-b+a x^8}}-\frac {a}{x^2 \sqrt [4]{-b+a x^8}}+\frac {2 x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3 b-a x^8\right )}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3 b x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )}+\frac {a x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}-\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{1-\frac {a x^2}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ &=-\frac {4 b \sqrt [4]{1-\frac {a x^2}{b}} \, _2F_1\left (-\frac {9}{8},\frac {1}{4};-\frac {1}{8};\frac {a x^2}{b}\right )}{3 x^2 \sqrt [4]{-b x+a x^3}}+\frac {4 a \sqrt [4]{1-\frac {a x^2}{b}} \, _2F_1\left (-\frac {1}{8},\frac {1}{4};\frac {7}{8};\frac {a x^2}{b}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{-b+a x^8} \left (-b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}+\frac {\left (24 b \sqrt [4]{x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^8} \left (b-a x^8-x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 2.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3 b+a x^2\right ) \left (b-a x^2+x^3\right )}{x^3 \left (-b+a x^2+x^3\right ) \sqrt [4]{-b x+a x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.23, size = 128, normalized size = 1.00 \begin {gather*} \frac {4 \left (-b x+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b x+a x^3}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b x+a x^3}}{x^2+\sqrt {-b x+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}-3 b \right ) \left (-a \,x^{2}+x^{3}+b \right )}{x^{3} \left (a \,x^{2}+x^{3}-b \right ) \left (a \,x^{3}-b x \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________