Optimal. Leaf size=72 \[ \frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {b x+c x^2}}{\sqrt {b} \sqrt {e x}}\right )}{\sqrt {b} \sqrt {e}} \]
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Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {794, 660, 208} \begin {gather*} \frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {b x+c x^2}}{\sqrt {b} \sqrt {e x}}\right )}{\sqrt {b} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 208
Rule 660
Rule 794
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {e x} \sqrt {b x+c x^2}} \, dx &=\frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}+A \int \frac {1}{\sqrt {e x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}+(2 A e) \operatorname {Subst}\left (\int \frac {1}{-b e+e^2 x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {e x}}\right )\\ &=\frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {b x+c x^2}}{\sqrt {b} \sqrt {e x}}\right )}{\sqrt {b} \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.99 \begin {gather*} \frac {2 x \left (\sqrt {b} B (b+c x)-A c \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{\sqrt {b} c \sqrt {e x} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 72, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {b x+c x^2}}{c \sqrt {e x}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {e} \sqrt {b x+c x^2}}\right )}{\sqrt {b} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 166, normalized size = 2.31 \begin {gather*} \left [\frac {\sqrt {b e} A c x \log \left (-\frac {c e x^{2} + 2 \, b e x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b e} \sqrt {e x}}{x^{2}}\right ) + 2 \, \sqrt {c x^{2} + b x} \sqrt {e x} B b}{b c e x}, \frac {2 \, {\left (\sqrt {-b e} A c x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-b e} \sqrt {e x}}{c e x^{2} + b e x}\right ) + \sqrt {c x^{2} + b x} \sqrt {e x} B b\right )}}{b c e x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 103, normalized size = 1.43 \begin {gather*} 2 \, {\left (\frac {A \arctan \left (\frac {\sqrt {c x e + b e}}{\sqrt {-b e}}\right ) e}{\sqrt {-b e}} + \frac {\sqrt {c x e + b e} B}{c}\right )} e^{\left (-1\right )} - \frac {2 \, {\left (A c \arctan \left (\frac {\sqrt {b} e^{\frac {1}{2}}}{\sqrt {-b e}}\right ) e + \sqrt {-b e} B \sqrt {b} e^{\frac {1}{2}}\right )} e^{\left (-1\right )}}{\sqrt {-b e} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 72, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (A c e \arctanh \left (\frac {\sqrt {\left (c x +b \right ) e}}{\sqrt {b e}}\right )-\sqrt {\left (c x +b \right ) e}\, \sqrt {b e}\, B \right )}{\sqrt {e x}\, \sqrt {\left (c x +b \right ) e}\, \sqrt {b e}\, c} \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 {B x + A}{\sqrt {c x^{2} + b x} \sqrt {e x}}\,{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 {A+B\,x}{\sqrt {c\,x^2+b\,x}\,\sqrt {e\,x}} \,d x \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 + B x}{\sqrt {e x} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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