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

Optimal. Leaf size=123 \[ \frac {4 \left (a x^3-b\right )^{3/4}}{3 x^3}-2 \sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {a x^3-b}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{a x^3-b}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{a x^3-b}}{\sqrt {a x^3-b}+x^2}\right ) \]

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Rubi [F]  time = 3.32, 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 (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \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 + x^4))/(x^4*(-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)),x]

[Out]

(4*(-b + a*x^3)^(3/4))/(3*x^3) + 8*b*Defer[Int][1/((-b + a*x^3)^(1/4)*(b - a*x^3 - x^4)), x] + 2*a*Defer[Int][
x^3/((-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)), x]

Rubi steps

\begin {align*} \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx &=\int \left (\frac {4 b}{x^4 \sqrt [4]{-b+a x^3}}-\frac {a}{x \sqrt [4]{-b+a x^3}}+\frac {2 \left (4 b-a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}\right ) \, dx\\ &=2 \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-a \int \frac {1}{x \sqrt [4]{-b+a x^3}} \, dx+(4 b) \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx\\ &=2 \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-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+\frac {1}{3} (4 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x}} \, dx,x,x^3\right )\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx-\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+\frac {\left (\sqrt {2} a\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {\sqrt {2} a \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\sqrt {2} a \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {\left (\sqrt {2} a\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx\\ &=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 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 = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \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 + x^4))/(x^4*(-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.80, size = 123, normalized size = 1.00 \begin {gather*} \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \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 )+2 \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 + x^4))/(x^4*(-b + a*x^3)^(1/4)*(-b + a*x^3 + x^4)),x]

[Out]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{3}-4 b \right ) \left (-a \,x^{3}+x^{4}+b \right )}{x^{4} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (a \,x^{3}+x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

Timed out

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