3.26.60 \(\int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} (b+a^2 d-(1+2 a d) x+d x^2)} \, dx\)

Optimal. Leaf size=216 \[ -\frac {\log \left (a^2 d^{2/3}+\sqrt [3]{x (-a-b)+a b+x^2} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )+\left (x (-a-b)+a b+x^2\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x (-a-b)+a b+x^2}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x (-a-b)+a b+x^2}}{\sqrt [3]{x (-a-b)+a b+x^2}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.17, 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-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx &=\int \frac {a-2 b+x}{\sqrt [3]{a b+(-a-b) x+x^2} \left (b+a^2 d+(-1-2 a d) x+d x^2\right )} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 11.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a-2 b+x}{\sqrt [3]{(-a+x) (-b+x)} \left (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.51, size = 216, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a b+(-a-b) x+x^2}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b+(-a-b) x+x^2}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b+(-a-b) x+x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+\left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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-2*b+x)/((-a+x)*(-b+x))^(1/3)/(b+a^2*d-(2*a*d+1)*x+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 - 2 \, b + x}{{\left (a^{2} d + d x^{2} - {\left (2 \, a d + 1\right )} x + b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-2*b+x)/((-a+x)*(-b+x))^(1/3)/(b+a^2*d-(2*a*d+1)*x+d*x^2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[\int \frac {a -2 b +x}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (b +a^{2} d -\left (2 a d +1\right ) x +d \,x^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a - 2 \, b + x}{{\left (a^{2} d + d x^{2} - {\left (2 \, a d + 1\right )} x + b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-2*b+x)/((-a+x)*(-b+x))^(1/3)/(b+a^2*d-(2*a*d+1)*x+d*x^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a-2\,b+x}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (b-x\,\left (2\,a\,d+1\right )+a^2\,d+d\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________