3.18.61 \(\int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} (-b^2+(2 b-a d) x+(-1+d) x^2)} \, dx\) [1761]

Optimal. Leaf size=119 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}\right )}{d^{3/4}} \]

[Out]

2*arctan(d^(1/4)*(-a*b^2*x+(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4)/(b-x))/d^(3/4)-2*arctanh(d^(1/4)*(-a*b^2*x+
(2*a*b+b^2)*x^2+(-a-2*b)*x^3+x^4)^(1/4)/(b-x))/d^(3/4)

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Rubi [F]
time = 2.60, 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 {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x (-a+x) (-b+x)^2} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {-a b+(-a+2 b) x}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (-b^2+(2 b-a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \left (\frac {-a+2 b-\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d-\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )}+\frac {-a+2 b+\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d+\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (-a+2 b-\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d-\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}+\frac {\left (\left (-a+2 b+\frac {\sqrt {-4 a b+4 b^2+a^2 d}}{\sqrt {d}}\right ) \sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-a+x} \sqrt {-b+x} \left (2 b-a d+\sqrt {d} \sqrt {-4 a b+4 b^2+a^2 d}+2 (-1+d) x\right )} \, dx}{\sqrt [4]{x (-a+x) (-b+x)^2}}\\ \end {align*}

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Mathematica [A]
time = 42.38, size = 163, normalized size = 1.37 \begin {gather*} -\frac {\sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}} x \left (-\text {ArcTan}\left (\frac {1+\sqrt {d} \sqrt {-1+\frac {a}{x}}-\frac {b}{x}}{\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt [4]{-1+\frac {a}{x}} \sqrt {-2+\frac {2 b}{x}}}{-1+\sqrt {d} \sqrt {-1+\frac {a}{x}}+\frac {b}{x}}\right )\right )}{d^{3/4} \sqrt [4]{(b-x)^2 x (-a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {-a b +\left (-a +2 b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{4}} \left (-b^{2}+\left (-a d +2 b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)^2)^(1/4)/(-b^2+(-a*d+2*b)*x+(-1+d)*x^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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*b+(-a+2*b)*x)/(x*(-a+x)*(-b+x)**2)**(1/4)/(-b**2+(-a*d+2*b)*x+(-1+d)*x**2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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