Optimal. Leaf size=77 \[ -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{d^{3/4}} \]
________________________________________________________________________________________
Rubi [F] time = 31.54, 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+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{\sqrt {x (-a+x) (-b+x)} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \sqrt {-b+x} \left (-a b+x^2\right )}{\sqrt {x} \left (a^2 b^2 d-2 a b (a+b) d x+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^2-2 (a+b) d x^3+d x^4\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-a b+x^4\right )}{a^2 b^2 d-2 a b (a+b) d x^2+\left (-1+a^2 d+4 a b d+b^2 d\right ) x^4-2 (a+b) d x^6+d x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a b \sqrt {-a+x^2} \sqrt {-b+x^2}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) d x^6-d x^8}+\frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) d x^6+d x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {-a+x^2} \sqrt {-b+x^2}}{a^2 b^2 d-2 a^2 b \left (1+\frac {b}{a}\right ) d x^2-\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4-2 a \left (1+\frac {b}{a}\right ) d x^6+d x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (2 a b \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-a+x^2} \sqrt {-b+x^2}}{-a^2 b^2 d+2 a^2 b \left (1+\frac {b}{a}\right ) d x^2+\left (1-\left (a^2+4 a b+b^2\right ) d\right ) x^4+2 a \left (1+\frac {b}{a}\right ) d x^6-d x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 13.85, size = 24546, normalized size = 318.78 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.36, size = 77, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{d^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.47, size = 483, normalized size = 6.27 \begin {gather*} -\frac {1}{d^{3}}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} d^{2} \frac {1}{d^{3}}^{\frac {3}{4}}}{a b - {\left (a + b\right )} x + x^{2}}\right ) - \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d \frac {1}{d^{3}}^{\frac {1}{4}} x + {\left (a b d^{3} - {\left (a + b\right )} d^{3} x + d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right ) + \frac {1}{4} \, \frac {1}{d^{3}}^{\frac {1}{4}} \log \left (\frac {a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d + 1\right )} x^{2} - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (d \frac {1}{d^{3}}^{\frac {1}{4}} x + {\left (a b d^{3} - {\left (a + b\right )} d^{3} x + d^{3} x^{2}\right )} \frac {1}{d^{3}}^{\frac {3}{4}}\right )} + 2 \, {\left (a b d^{2} x - {\left (a + b\right )} d^{2} x^{2} + d^{2} x^{3}\right )} \sqrt {\frac {1}{d^{3}}}}{a^{2} b^{2} d - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} - 2 \, {\left (a^{2} b + a b^{2}\right )} d x + {\left ({\left (a^{2} + 4 \, a b + b^{2}\right )} d - 1\right )} x^{2}}\right ) \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 - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.10, size = 451, normalized size = 5.86
method | result | size |
default | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{3} a d +\underline {\hspace {1.25 ex}}\alpha ^{3} b d -\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d -\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d +3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b d +3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2} d -2 a^{2} b^{2} d +\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{d b}\) | \(451\) |
elliptic | \(-\frac {2 b \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticF \left (\sqrt {-\frac {-b +x}{b}}, \sqrt {\frac {b}{-a +b}}\right )}{d \sqrt {a b x -a \,x^{2}-b \,x^{2}+x^{3}}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b d +\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} d +4 \underline {\hspace {1.25 ex}}\alpha ^{2} a b d +\underline {\hspace {1.25 ex}}\alpha ^{2} b^{2} d -3 \underline {\hspace {1.25 ex}}\alpha \,a^{2} b d -3 \underline {\hspace {1.25 ex}}\alpha a \,b^{2} d +2 a^{2} b^{2} d -\underline {\hspace {1.25 ex}}\alpha ^{2}\right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a -3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b +\underline {\hspace {1.25 ex}}\alpha \,a^{2} d +4 \underline {\hspace {1.25 ex}}\alpha a d b +\underline {\hspace {1.25 ex}}\alpha \,b^{2} d -a^{2} b d -a \,b^{2} d -\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{d b}\) | \(455\) |
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 - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{{\left (a^{2} b^{2} d - 2 \, {\left (a + b\right )} a b d x - 2 \, {\left (a + b\right )} d x^{3} + d x^{4} + {\left (a^{2} d + 4 \, a b d + b^{2} d - 1\right )} x^{2}\right )} \sqrt {{\left (a - x\right )} {\left (b - x\right )} x}}\,{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 {\left (a-x\right )\,\left (b-x\right )\,\left (a\,b-x^2\right )}{\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,\left (x^2\,\left (d\,a^2+4\,d\,a\,b+d\,b^2-1\right )+d\,x^4+a^2\,b^2\,d-2\,d\,x^3\,\left (a+b\right )-2\,a\,b\,d\,x\,\left (a+b\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]
________________________________________________________________________________________