Optimal. Leaf size=64 \[ -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{\sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.75, antiderivative size = 91, normalized size of antiderivative = 1.42, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2056, 6733, 1699, 205} \begin {gather*} -\frac {\sqrt {2} \sqrt {x} \sqrt {a^2 x^2+b^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2+b^2}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1699
Rule 2056
Rule 6733
Rubi steps
\begin {align*} \int \frac {-b+a x}{(b+a x) \sqrt {b^2 x+a^2 x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-b+a x}{\sqrt {x} (b+a x) \sqrt {b^2+a^2 x^2}} \, dx}{\sqrt {b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {-b+a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {\left (2 b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{b+2 a b^2 x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {b^2 x+a^2 x^3}}\\ &=-\frac {\sqrt {2} \sqrt {x} \sqrt {b^2+a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 110, normalized size = 1.72 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {\frac {b^2}{a^2 x^2}+1} \left (F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )-2 \Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i b}{a}} \sqrt {x \left (a^2 x^2+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.27, size = 64, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}{a^2 x^2+b^2}\right )}{\sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 213, normalized size = 3.33 \begin {gather*} \left [\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ), -\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right )\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 {a x - b}{\sqrt {a^{2} x^{3} + b^{2} x} {\left (a x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 231, normalized size = 3.61 \begin {gather*} \frac {i b \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticF \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}+b^{2} x}}-\frac {2 i b^{2} \sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}\, \sqrt {2}\, \sqrt {\frac {i \left (x -\frac {i b}{a}\right ) a}{b}}\, \sqrt {\frac {i x a}{b}}\, \EllipticPi \left (\sqrt {-\frac {i \left (x +\frac {i b}{a}\right ) a}{b}}, -\frac {i b}{a \left (-\frac {i b}{a}+\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a^{2} \sqrt {a^{2} x^{3}+b^{2} x}\, \left (-\frac {i b}{a}+\frac {b}{a}\right )} \end {gather*}
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 x - b}{\sqrt {a^{2} x^{3} + b^{2} x} {\left (a x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x - b}{\sqrt {x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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