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

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

________________________________________________________________________________________

Rubi [F]  time = 3.91, 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*}

________________________________________________________________________________________

Mathematica [F]  time = 6.98, size = 0, normalized size = 0.00 \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]

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]

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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.39, size = 119, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-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}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(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]

(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*ArcTa
nh[(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)

________________________________________________________________________________________

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*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

________________________________________________________________________________________

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

________________________________________________________________________________________

maple [F]  time = 0.11, 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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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*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

________________________________________________________________________________________