Optimal. Leaf size=67 \[ \frac {2 \sqrt {x^2 (-a-b)+a b x+x^3}}{x}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {x^2 (-a-b)+a b x+x^3}}\right ) \]
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Rubi [F] time = 7.40, 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 )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(-a+x) (-b+x) \left (-a b+x^2\right )}{x \sqrt {x (-a+x) (-b+x)} \left (a b-(a+b+d) x+x^2\right )} \, dx &=\int \frac {\sqrt {(a-x) (b-x) x} \left (-a b+x^2\right )}{x^2 \left (a b-(a+b+d) x+x^2\right )} \, dx\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x} \left (-a b+x^2\right )}{x^{3/2} \left (a b-(a+b+d) x+x^2\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2}}-\frac {\sqrt {a-x} \sqrt {b-x} (2 a b-(a+b+d) x)}{x^{3/2} \left (a b+(-a-b-d) x+x^2\right )}\right ) \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\sqrt {(a-x) (b-x) x} \int \frac {\sqrt {a-x} \sqrt {b-x} (2 a b-(a+b+d) x)}{x^{3/2} \left (a b+(-a-b-d) x+x^2\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\sqrt {(a-x) (b-x) x} \int \left (\frac {\left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}\right ) \sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}+\frac {\left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}\right ) \sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}\right ) \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}+\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {\frac {1}{2} (-a-b)+x}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\left (2 \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {b-x}}{\sqrt {a-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}+\frac {\left ((-a+b) \sqrt {(a-x) (b-x) x}\right ) \int \frac {1}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ &=-\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (2 \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}}\right ) \int \frac {\sqrt {1-\frac {x}{b}}}{\sqrt {x} \sqrt {1-\frac {x}{a}}} \, dx}{(a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}+\frac {\left ((-a+b) \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}}} \, dx}{(a-x) (b-x) \sqrt {x}}\\ &=-\frac {2 \sqrt {(a-x) (b-x) x}}{x}-\frac {4 \sqrt {a} \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{(a-x) \sqrt {x} \sqrt {1-\frac {x}{b}}}-\frac {2 \sqrt {a} (a-b) \sqrt {(a-x) (b-x) x} \sqrt {1-\frac {x}{a}} \sqrt {1-\frac {x}{b}} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {a}}\right )|\frac {a}{b}\right )}{(a-x) (b-x) \sqrt {x}}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) \sqrt {(a-x) (b-x) x}\right ) \int \frac {\sqrt {a-x} \sqrt {b-x}}{x^{3/2} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\sqrt {a-x} \sqrt {b-x} \sqrt {x}}\\ \end {align*}
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Mathematica [C] time = 8.50, size = 504, normalized size = 7.52 \begin {gather*} \frac {\sqrt {x (x-a) (x-b)} \left (\frac {i \left (-a \left (\sqrt {a^2-2 a (b-d)+(b+d)^2}+2 b-2 d\right )+b \sqrt {a^2-2 a (b-d)+(b+d)^2}-d \sqrt {a^2-2 a (b-d)+(b+d)^2}+a^2+b^2+2 b d+d^2\right ) \sqrt {\frac {x-b}{a-b}} \Pi \left (\frac {2 a}{a-b-d+\sqrt {(-a+b+d)^2+4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (b-x) \sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {i \left (a \left (\sqrt {a^2-2 a (b-d)+(b+d)^2}-2 b+2 d\right )-b \sqrt {a^2-2 a (b-d)+(b+d)^2}+d \sqrt {a^2-2 a (b-d)+(b+d)^2}+a^2+b^2+2 b d+d^2\right ) \sqrt {\frac {x-b}{a-b}} \Pi \left (-\frac {2 a}{-a+b+d+\sqrt {(-a+b+d)^2+4 a d}};i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (b-x) \sqrt {a^2-2 a (b-d)+(b+d)^2}}-\frac {2 i d \sqrt {\frac {x-b}{a-b}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {x}{a}-1}\right )|\frac {a}{a-b}\right )}{\sqrt {1-\frac {a}{x}} (x-b)}+2\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 67, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a b x+(-a-b) x^2+x^3}}{x}-2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {a b x+(-a-b) x^2+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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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.
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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 )}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 3422, normalized size = 51.07
method | result | size |
risch | \(\text {Expression too large to display}\) | \(3422\) |
elliptic | \(\text {Expression too large to display}\) | \(3462\) |
default | \(\text {Expression too large to display}\) | \(3786\) |
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 {{\left (a b - x^{2}\right )} {\left (a - x\right )} {\left (b - x\right )}}{\sqrt {{\left (a - x\right )} {\left (b - x\right )} x} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 722, normalized size = 10.78 \begin {gather*} \frac {b\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d-\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}-\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )}-\frac {2\,a\,b\,\left (\frac {\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\frac {\sqrt {\frac {b-x}{a-b}+1}\,\sqrt {\frac {b-x}{b}}}{\sqrt {1-\frac {b-x}{b}}}}{\frac {b}{a-b}+1}-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,d\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}-\frac {2\,b\,\left (a\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )-\left (a-b\right )\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\right )\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}}+\frac {b\,d\,\sqrt {\frac {x}{b}}\,\sqrt {\frac {b-x}{b}}\,\sqrt {\frac {a-x}{a-b}}\,\Pi \left (-\frac {b}{\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}};\mathrm {asin}\left (\sqrt {\frac {b-x}{b}}\right )\middle |-\frac {b}{a-b}\right )\,\left (a+b+d+\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}\right )}{\sqrt {x^3+\left (-a-b\right )\,x^2+a\,b\,x}\,\left (\frac {a}{2}-\frac {b}{2}+\frac {d}{2}+\frac {\sqrt {a^2-2\,a\,b+2\,a\,d+b^2+2\,b\,d+d^2}}{2}\right )} \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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