Optimal. Leaf size=91 \[ \frac {2 \tan ^{-1}\left (\frac {(b+x) \left (\frac {x}{\sqrt {2 \sqrt {a b}-a-b}}+\frac {a}{\sqrt {2 \sqrt {a b}-a-b}}\right )}{\sqrt {x^2 (a+b)+a b x+x^3}}\right )}{\sqrt {2 \sqrt {a b}-a-b}} \]
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Rubi [C] time = 0.87, antiderivative size = 146, normalized size of antiderivative = 1.60, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6718, 1607, 169, 538, 537, 117, 116} \begin {gather*} \frac {2 \sqrt {-a} \sqrt {x} \sqrt {\frac {x}{a}+1} \sqrt {\frac {x}{b}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}-\frac {4 \sqrt {-a} \sqrt {x} \sqrt {\frac {x}{a}+1} \sqrt {\frac {x}{b}+1} \Pi \left (\frac {a}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 116
Rule 117
Rule 169
Rule 537
Rule 538
Rule 1607
Rule 6718
Rubi steps
\begin {align*} \int \frac {-\sqrt {a b}+x}{\sqrt {x (a+x) (b+x)} \left (\sqrt {a b}+x\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {-\sqrt {a b}+x}{\sqrt {x} \sqrt {a+x} \sqrt {b+x} \left (\sqrt {a b}+x\right )} \, dx}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {\left (\sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+x} \sqrt {b+x}} \, dx}{\sqrt {x (a+x) (b+x)}}-\frac {\left (2 \sqrt {a b} \sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+x} \sqrt {b+x} \left (\sqrt {a b}+x\right )} \, dx}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {a+x} \sqrt {b+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {a+x^2} \sqrt {b+x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (\sqrt {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}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {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 )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {b+x} \sqrt {1+\frac {x}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {b+x^2} \sqrt {1+\frac {x^2}{a}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {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 )}{\sqrt {x (a+x) (b+x)}}+\frac {\left (4 \sqrt {a b} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt {a b}-x^2\right ) \sqrt {1+\frac {x^2}{a}} \sqrt {1+\frac {x^2}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x (a+x) (b+x)}}\\ &=\frac {2 \sqrt {-a} \sqrt {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 )}{\sqrt {x (a+x) (b+x)}}-\frac {4 \sqrt {-a} \sqrt {x} \sqrt {1+\frac {x}{a}} \sqrt {1+\frac {x}{b}} \Pi \left (\frac {a}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {x}}{\sqrt {-a}}\right )|\frac {a}{b}\right )}{\sqrt {x (a+x) (b+x)}}\\ \end {align*}
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Mathematica [C] time = 0.58, size = 101, normalized size = 1.11 \begin {gather*} -\frac {2 a x^{3/2} \sqrt {\frac {a}{x}+1} \sqrt {\frac {b}{x}+1} \left (F\left (\sin ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )-2 \Pi \left (\frac {b}{\sqrt {a b}};\sin ^{-1}\left (\frac {\sqrt {-a}}{\sqrt {x}}\right )|\frac {b}{a}\right )\right )}{(-a)^{3/2} \sqrt {x (a+x) (b+x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.78, size = 91, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {(b+x) \left (\frac {a}{\sqrt {-a-b+2 \sqrt {a b}}}+\frac {x}{\sqrt {-a-b+2 \sqrt {a b}}}\right )}{\sqrt {a b x+(a+b) x^2+x^3}}\right )}{\sqrt {-a-b+2 \sqrt {a b}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 791, normalized size = 8.69 \begin {gather*} \left [-\frac {\sqrt {a + b + 2 \, \sqrt {a b}} \log \left (-\frac {a^{5} b^{4} - a^{4} b^{5} + {\left (a - b\right )} x^{8} + 8 \, {\left (a^{2} - b^{2}\right )} x^{7} + 4 \, {\left (2 \, a^{3} + 17 \, a^{2} b - 17 \, a b^{2} - 2 \, b^{3}\right )} x^{6} + 120 \, {\left (a^{3} b - a b^{3}\right )} x^{5} + 2 \, {\left (24 \, a^{4} b + 91 \, a^{3} b^{2} - 91 \, a^{2} b^{3} - 24 \, a b^{4}\right )} x^{4} + 120 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} x^{3} + 4 \, {\left (2 \, a^{5} b^{2} + 17 \, a^{4} b^{3} - 17 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} x^{2} + 4 \, {\left (a^{4} b^{3} + a^{3} b^{4} + {\left (a + b\right )} x^{6} + 2 \, {\left (a^{2} + 8 \, a b + b^{2}\right )} x^{5} + 31 \, {\left (a^{2} b + a b^{2}\right )} x^{4} + 4 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 3 \, a b^{3}\right )} x^{3} + 31 \, {\left (a^{3} b^{2} + a^{2} b^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{2} + 8 \, a^{3} b^{3} + a^{2} b^{4}\right )} x - 2 \, {\left (a^{3} b^{3} + 5 \, {\left (a + b\right )} x^{5} + x^{6} + {\left (4 \, a^{2} + 23 \, a b + 4 \, b^{2}\right )} x^{4} + 22 \, {\left (a^{2} b + a b^{2}\right )} x^{3} + {\left (4 \, a^{3} b + 23 \, a^{2} b^{2} + 4 \, a b^{3}\right )} x^{2} + 5 \, {\left (a^{3} b^{2} + a^{2} b^{3}\right )} x\right )} \sqrt {a b}\right )} \sqrt {a b x + {\left (a + b\right )} x^{2} + x^{3}} \sqrt {a + b + 2 \, \sqrt {a b}} + 8 \, {\left (a^{5} b^{3} - a^{3} b^{5}\right )} x - 16 \, {\left ({\left (a - b\right )} x^{7} + 3 \, {\left (a^{2} - b^{2}\right )} x^{6} + {\left (2 \, a^{3} + 9 \, a^{2} b - 9 \, a b^{2} - 2 \, b^{3}\right )} x^{5} + 10 \, {\left (a^{3} b - a b^{3}\right )} x^{4} + {\left (2 \, a^{4} b + 9 \, a^{3} b^{2} - 9 \, a^{2} b^{3} - 2 \, a b^{4}\right )} x^{3} + 3 \, {\left (a^{4} b^{2} - a^{2} b^{4}\right )} x^{2} + {\left (a^{4} b^{3} - a^{3} b^{4}\right )} x\right )} \sqrt {a b}}{a^{4} b^{4} - 4 \, a^{3} b^{3} x^{2} + 6 \, a^{2} b^{2} x^{4} - 4 \, a b x^{6} + x^{8}}\right )}{2 \, {\left (a - b\right )}}, \frac {\sqrt {-a - b - 2 \, \sqrt {a b}} \arctan \left (\frac {\sqrt {a b x + {\left (a + b\right )} x^{2} + x^{3}} {\left (a b + 2 \, {\left (a + b\right )} x + x^{2} - 2 \, \sqrt {a b} x\right )} \sqrt {-a - b - 2 \, \sqrt {a b}}}{2 \, {\left ({\left (a - b\right )} x^{3} + {\left (a^{2} - b^{2}\right )} x^{2} + {\left (a^{2} b - a b^{2}\right )} x\right )}}\right )}{a - b}\right ] \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 {x - \sqrt {a b}}{\sqrt {{\left (a + x\right )} {\left (b + x\right )} x} {\left (x + \sqrt {a b}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 564, normalized size = 6.20
method | result | size |
default | \(\frac {2 b \sqrt {\frac {b +x}{b}}\, \sqrt {\frac {a +x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticF \left (\sqrt {\frac {b +x}{b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}}-2 \sqrt {a b}\, \left (\frac {b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b -\sqrt {a b}\right )}+\frac {b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b +\sqrt {a b}\right )}-\frac {a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b -\sqrt {a b}\right )}+\frac {a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b +\sqrt {a b}\right )}\right )\) | \(564\) |
elliptic | \(\frac {2 b \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticF \left (\sqrt {\frac {b +x}{b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}}+\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b -\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a b}\, \sqrt {a b x +a \,x^{2}+b \,x^{2}+x^{3}}\, \left (-b +\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b -\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b -\sqrt {a b}\right )}-\frac {2 a \,b^{2} \sqrt {1+\frac {x}{b}}\, \sqrt {\frac {a}{a -b}+\frac {x}{a -b}}\, \sqrt {-\frac {x}{b}}\, \EllipticPi \left (\sqrt {\frac {b +x}{b}}, -\frac {b}{-b +\sqrt {a b}}, \sqrt {-\frac {b}{a -b}}\right )}{\sqrt {a^{2} b^{2} x +a^{2} b \,x^{2}+a \,b^{2} x^{2}+a b \,x^{3}}\, \left (-b +\sqrt {a b}\right )}\) | \(573\) |
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 {x - \sqrt {a b}}{\sqrt {{\left (a + x\right )} {\left (b + x\right )} x} {\left (x + \sqrt {a b}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x-\sqrt {a\,b}}{\left (x+\sqrt {a\,b}\right )\,\sqrt {x\,\left (a+x\right )\,\left (b+x\right )}} \,d x \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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