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

Optimal. Leaf size=102 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}+x^2}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b}}\right ) \]

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Rubi [F]  time = 0.87, 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 {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx &=\int \left (\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )}\right ) \, dx\\ &=a \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(4 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-4 b+a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.76, size = 102, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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