Optimal. Leaf size=55 \[ \frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}+\frac {e \sqrt {b x+c x^2}}{c} \]
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Rubi [A] time = 0.02, 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 {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}+\frac {e \sqrt {b x+c x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rubi steps
\begin {align*} \int \frac {d+e x}{\sqrt {b x+c x^2}} \, dx &=\frac {e \sqrt {b x+c x^2}}{c}+\frac {(2 c d-b e) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 c}\\ &=\frac {e \sqrt {b x+c x^2}}{c}+\frac {(2 c d-b e) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=\frac {e \sqrt {b x+c x^2}}{c}+\frac {(2 c d-b e) \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 {\sqrt {c} e x (b+c x)-\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1} (b e-2 c d) \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 = 0.37, size = 67, normalized size = 1.22 \begin {gather*} \frac {(b e-2 c d) \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{2 c^{3/2}}+\frac {e \sqrt {b x+c x^2}}{c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 118, normalized size = 2.15 \begin {gather*} \left [\frac {2 \, \sqrt {c x^{2} + b x} c e - {\left (2 \, c d - b e\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} c e - {\left (2 \, c d - b e\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 = 0.26, size = 63, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c x^{2} + b x} e}{c} - \frac {{\left (2 \, c d - b e\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 = 0.12, size = 78, normalized size = 1.42 \begin {gather*} -\frac {b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}+\frac {d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+\frac {\sqrt {c \,x^{2}+b x}\, e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 75, normalized size = 1.36 \begin {gather*} \frac {d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} - \frac {b e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} e}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 77, normalized size = 1.40 \begin {gather*} \frac {e\,\sqrt {c\,x^2+b\,x}}{c}+\frac {d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{\sqrt {c}}-\frac {b\,e\,\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 {d + 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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