Optimal. Leaf size=76 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}\right )}{\sqrt [4]{d}} \]
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Rubi [F] time = 23.92, 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 x+x^3}{\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.
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Rubi steps
\begin {align*} \int \frac {-a b x+x^3}{\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 &=\int \frac {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 {x} \left (-a b+x^2\right )}{\sqrt {-a+x} \sqrt {-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}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-a b+x^4\right )}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (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\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 \left (\frac {a b x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-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 )}+\frac {x^6}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (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 )}\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^6}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (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 a b \sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-a+x^2} \sqrt {-b+x^2} \left (-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)}}\\ \end {align*}
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Mathematica [C] time = 6.42, size = 32986, normalized size = 434.03 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.36, size = 76, 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 )}{\sqrt [4]{d}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.89, size = 433, normalized size = 5.70 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{{\left (a b - {\left (a + b\right )} x + x^{2}\right )} d^{\frac {1}{4}}}\right )}{d^{\frac {1}{4}}} - \frac {\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}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{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 )}{4 \, d^{\frac {1}{4}}} + \frac {\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}{4}} x + \frac {a b d - {\left (a + b\right )} d x + d x^{2}}{d^{\frac {1}{4}}}\right )} + \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{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 )}{4 \, d^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {a b x - x^{3}}{{\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.
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maple [C] time = 0.08, size = 303, normalized size = 3.99
method | result | size |
default | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\) | \(303\) |
elliptic | \(\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}+\left (-2 a d -2 b d \right ) \textit {\_Z}^{3}+\left (a^{2} d +4 a b d +b^{2} d -1\right ) \textit {\_Z}^{2}+\left (-2 a^{2} b d -2 a \,b^{2} d \right ) \textit {\_Z} +a^{2} b^{2} d \right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2}-a b \right ) \left (-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, \frac {-d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+2 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -2 \underline {\hspace {1.25 ex}}\alpha a d b +a^{2} b d +\underline {\hspace {1.25 ex}}\alpha +b}{b}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} a +3 d \,\underline {\hspace {1.25 ex}}\alpha ^{2} b -\underline {\hspace {1.25 ex}}\alpha \,a^{2} d -4 \underline {\hspace {1.25 ex}}\alpha a d b -\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +a^{2} b d +a \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}}{b}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {a b x - x^{3}}{{\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.
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mupad [B] time = 7.32, size = 175, normalized size = 2.30 \begin {gather*} \frac {\ln \left (\frac {x-2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}{x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-\sqrt {d}\,x^2-a\,b\,\sqrt {d}+a\,\sqrt {d}\,x+b\,\sqrt {d}\,x+d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{x+\sqrt {d}\,x^2+a\,b\,\sqrt {d}-a\,\sqrt {d}\,x-b\,\sqrt {d}\,x}\right )\,1{}\mathrm {i}}{2\,d^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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