Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{\sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {x^2 (-a-b)+a b x+x^3}}{a-x}\right )}{\sqrt [4]{d}} \]
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Rubi [F] time = 9.95, 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+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 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^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x (-a+x) (-b+x)} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {-a^2 b+a (2 a+b) x-3 a x^2+x^3}{\sqrt {x} \sqrt {-a+x} \sqrt {-b+x} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 b d x^3+d x^4\right )} \, dx}{\sqrt {x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {-a+x} \sqrt {-b+x}\right ) \int \frac {\sqrt {-a+x} \left (a b-2 a x+x^2\right )}{\sqrt {x} \sqrt {-b+x} \left (-a^2+2 a x+\left (-1+b^2 d\right ) x^2-2 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} \left (a b-2 a x^2+x^4\right )}{\sqrt {-b+x^2} \left (-a^2+2 a x^2+\left (-1+b^2 d\right ) x^4-2 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 {2 a x^2 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (a^2-2 a x^2+\left (1-b^2 d\right ) x^4+2 b d x^6-d x^8\right )}+\frac {a b \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\right )}+\frac {x^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b 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^4 \sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b d x^6+d x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x (-a+x) (-b+x)}}+\frac {\left (4 a \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^2-2 a x^2+\left (1-b^2 d\right ) x^4+2 b 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 {\sqrt {-a+x^2}}{\sqrt {-b+x^2} \left (-a^2+2 a x^2-\left (1-b^2 d\right ) x^4-2 b 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 = 8.08, size = 5341, normalized size = 61.39 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.22, size = 90, normalized size = 1.03 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a b x+(-a-b) x^2+x^3}}{\sqrt [4]{d} x (-b+x)}\right )}{\sqrt [4]{d}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {a b x+(-a-b) x^2+x^3}}{a-x}\right )}{\sqrt [4]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.57, size = 340, normalized size = 3.91 \begin {gather*} \frac {\arctan \left (-\frac {\sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}}}{{\left (b x - x^{2}\right )} d^{\frac {1}{4}}}\right )}{d^{\frac {1}{4}}} - \frac {\log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x + 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {b d x - d x^{2}}{d^{\frac {1}{4}}} + \frac {a d - d x}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\right )}{4 \, d^{\frac {1}{4}}} + \frac {\log \left (\frac {2 \, b d x^{3} - d x^{4} - {\left (b^{2} d + 1\right )} x^{2} - a^{2} + 2 \, a x - 2 \, \sqrt {a b x - {\left (a + b\right )} x^{2} + x^{3}} {\left (\frac {b d x - d x^{2}}{d^{\frac {1}{4}}} + \frac {a d - d x}{d^{\frac {3}{4}}}\right )} - \frac {2 \, {\left (a b d x - {\left (a + b\right )} d x^{2} + d x^{3}\right )}}{\sqrt {d}}}{2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x}\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^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x\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 = 0.11, size = 246, normalized size = 2.83
method | result | size |
default | \(b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\) | \(246\) |
elliptic | \(b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (d \,\textit {\_Z}^{4}-2 b d \,\textit {\_Z}^{3}+\left (b^{2} d -1\right ) \textit {\_Z}^{2}+2 a \textit {\_Z} -a^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -2 \underline {\hspace {1.25 ex}}\alpha \,a^{2}-\underline {\hspace {1.25 ex}}\alpha a b +a^{2} b \right ) \left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) \sqrt {-\frac {-b +x}{b}}\, \sqrt {\frac {-a +x}{-a +b}}\, \sqrt {\frac {x}{b}}\, \EllipticPi \left (\sqrt {-\frac {-b +x}{b}}, -\frac {\left (d \,\underline {\hspace {1.25 ex}}\alpha ^{3}-b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +2 a -b \right ) b}{a^{2}-2 a b +b^{2}}, \sqrt {\frac {b}{-a +b}}\right )}{\left (-2 d \,\underline {\hspace {1.25 ex}}\alpha ^{3}+3 b d \,\underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha \,b^{2} d +\underline {\hspace {1.25 ex}}\alpha -a \right ) \left (a^{2}-2 a b +b^{2}\right ) \sqrt {x \left (a b -a x -b x +x^{2}\right )}}\right )\) | \(246\) |
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^{2} b - {\left (2 \, a + b\right )} a x + 3 \, a x^{2} - x^{3}}{{\left (2 \, b d x^{3} - d x^{4} - {\left (b^{2} d - 1\right )} x^{2} + a^{2} - 2 \, a x\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 = 135, normalized size = 1.55 \begin {gather*} \frac {\ln \left (\frac {a-x+2\,d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}-\sqrt {d}\,x^2+b\,\sqrt {d}\,x}{a-x+\sqrt {d}\,x^2-b\,\sqrt {d}\,x}\right )}{2\,d^{1/4}}+\frac {\ln \left (\frac {x-a-\sqrt {d}\,x^2+b\,\sqrt {d}\,x+d^{1/4}\,\sqrt {x\,\left (a-x\right )\,\left (b-x\right )}\,2{}\mathrm {i}}{a-x-\sqrt {d}\,x^2+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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