Optimal. Leaf size=186 \[ 2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{3/4}}{x (x-a) (b-x)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{3/4}}{x (x-a) (b-x)}\right )+\frac {4 \sqrt [4]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{b-x} \]
________________________________________________________________________________________
Rubi [F] time = 4.78, 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 {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{x} \sqrt [4]{-a+x} (a b+(a-2 b) x)}{(-b+x)^{3/2} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \left (\frac {\left (a-2 b-\sqrt {a^2-4 a b d+4 b^2 d}\right ) \sqrt [4]{x} \sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (a-2 b d-\sqrt {a^2-4 a b d+4 b^2 d}+2 (-1+d) x\right )}+\frac {\left (a-2 b+\sqrt {a^2-4 a b d+4 b^2 d}\right ) \sqrt [4]{x} \sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (a-2 b d+\sqrt {a^2-4 a b d+4 b^2 d}+2 (-1+d) x\right )}\right ) \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ &=\frac {\left (\left (a-2 b-\sqrt {a^2-4 a b d+4 b^2 d}\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{x} \sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (a-2 b d-\sqrt {a^2-4 a b d+4 b^2 d}+2 (-1+d) x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}+\frac {\left (\left (a-2 b+\sqrt {a^2-4 a b d+4 b^2 d}\right ) x^{3/4} (-a+x)^{3/4} (-b+x)^{3/2}\right ) \int \frac {\sqrt [4]{x} \sqrt [4]{-a+x}}{(-b+x)^{3/2} \left (a-2 b d+\sqrt {a^2-4 a b d+4 b^2 d}+2 (-1+d) x\right )} \, dx}{\left (x (-a+x) (-b+x)^2\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 15.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (-a+x) (a b+(a-2 b) x)}{\left (x (-a+x) (-b+x)^2\right )^{3/4} \left (b^2 d+(a-2 b d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 4.14, size = 186, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4}}{b-x}+2 \sqrt [4]{d} \tan ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right )-2 \sqrt [4]{d} \tanh ^{-1}\left (\frac {\sqrt [4]{d} \left (-a b^2 x+\left (2 a b+b^2\right ) x^2+(-a-2 b) x^3+x^4\right )^{3/4}}{(b-x) x (-a+x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (a b + {\left (a - 2 \, b\right )} x\right )} {\left (a - x\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, b d - a\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {x \left (-a +x \right ) \left (a b +\left (a -2 b \right ) x \right )}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {3}{4}} \left (b^{2} d +\left (-2 b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (a b + {\left (a - 2 \, b\right )} x\right )} {\left (a - x\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {3}{4}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, b d - a\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x\,\left (a-x\right )\,\left (a\,b+x\,\left (a-2\,b\right )\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{3/4}\,\left (b^2\,d+x\,\left (a-2\,b\,d\right )+x^2\,\left (d-1\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________