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

Optimal. Leaf size=106 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b x}}{\sqrt {a x^3-b x}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b x}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b x}}\right ) \]

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Rubi [F]  time = 2.24, 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 {-3 b+a x^2}{\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)*(-(b*x) + a*x^3)^(1/4)),x]

[Out]

(12*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) + (4*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 {-3 b+a x^2}{\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 {-3 b+a x^2}{\sqrt [4]{x} \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 {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 \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]{-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 (12 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*}

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Mathematica [F]  time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3 b+a x^2}{\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)*(-(b*x) + a*x^3)^(1/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.54, size = 106, normalized size = 1.00 \begin {gather*} -\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 )+\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)*(-(b*x) + a*x^3)^(1/4)),x]

[Out]

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

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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)/(a*x^2+x^3-b)/(a*x^3-b*x)^(1/4),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 (a x^{3} - b x\right )}^{\frac {1}{4}} {\left (a x^{2} + x^{3} - b\right )}}\,{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)/(a*x^3-b*x)^(1/4),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^2-3*b)/(a*x^2+x^3-b)/(a*x^3-b*x)^(1/4),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 (a x^{3} - b x\right )}^{\frac {1}{4}} {\left (a x^{2} + x^{3} - b\right )}}\,{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)/(a*x^3-b*x)^(1/4),x, algorithm="maxima")

[Out]

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

[Out]

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

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

[Out]

Timed out

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