Optimal. Leaf size=55 \[ \frac {B \sqrt {b x+c x^2}}{c}-\frac {(b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {640, 620, 206} \begin {gather*} \frac {B \sqrt {b x+c x^2}}{c}-\frac {(b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rubi steps
\begin {align*} \int \frac {A+B x}{\sqrt {b x+c x^2}} \, dx &=\frac {B \sqrt {b x+c x^2}}{c}+\frac {(-b B+2 A c) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c}\\ &=\frac {B \sqrt {b x+c x^2}}{c}+\frac {(-b B+2 A c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=\frac {B \sqrt {b x+c x^2}}{c}-\frac {(b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 1.45 \begin {gather*} \frac {B \sqrt {c} x (b+c x)-\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1} (b B-2 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 67, normalized size = 1.22 \begin {gather*} \frac {(b B-2 A c) \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{2 c^{3/2}}+\frac {B \sqrt {b x+c x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 115, normalized size = 2.09 \begin {gather*} \left [\frac {2 \, \sqrt {c x^{2} + b x} B c - {\left (B b - 2 \, A c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{2}}, \frac {\sqrt {c x^{2} + b x} B c + {\left (B b - 2 \, A c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right )}{c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 60, normalized size = 1.09 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B}{c} + \frac {{\left (B b - 2 \, A c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 1.42 \begin {gather*} \frac {A \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}-\frac {B b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {\sqrt {c \,x^{2}+b x}\, B}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 75, normalized size = 1.36 \begin {gather*} -\frac {B b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {A \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2} + b x} B}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 77, normalized size = 1.40 \begin {gather*} \frac {A\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{\sqrt {c}}+\frac {B\,\sqrt {c\,x^2+b\,x}}{c}-\frac {B\,b\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{2\,c^{3/2}} \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 {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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