Optimal. Leaf size=129 \[ -\frac {\left (-4 a B c-4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} (4 A c+b B+2 B c x)}{4 c}-\sqrt {a} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {814, 843, 621, 206, 724} \begin {gather*} -\frac {\left (-4 a B c-4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} (4 A c+b B+2 B c x)}{4 c}-\sqrt {a} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x} \, dx &=\frac {(b B+4 A c+2 B c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\int \frac {-4 a A c+\frac {1}{2} \left (b^2 B-4 A b c-4 a B c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{4 c}\\ &=\frac {(b B+4 A c+2 B c x) \sqrt {a+b x+c x^2}}{4 c}+(a A) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx-\frac {\left (b^2 B-4 A b c-4 a B c\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c}\\ &=\frac {(b B+4 A c+2 B c x) \sqrt {a+b x+c x^2}}{4 c}-(2 a A) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )-\frac {\left (b^2 B-4 A b c-4 a B c\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c}\\ &=\frac {(b B+4 A c+2 B c x) \sqrt {a+b x+c x^2}}{4 c}-\sqrt {a} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )-\frac {\left (b^2 B-4 A b c-4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 127, normalized size = 0.98 \begin {gather*} \frac {\left (4 a B c+4 A b c+b^2 (-B)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{8 c^{3/2}}+\frac {\sqrt {a+x (b+c x)} (4 A c+b B+2 B c x)}{4 c}-\sqrt {a} A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.57, size = 135, normalized size = 1.05 \begin {gather*} \frac {\left (-4 a B c-4 A b c+b^2 B\right ) \log \left (-2 c^{3/2} \sqrt {a+b x+c x^2}+b c+2 c^2 x\right )}{8 c^{3/2}}+\frac {\sqrt {a+b x+c x^2} (4 A c+b B+2 B c x)}{4 c}+2 \sqrt {a} A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 651, normalized size = 5.05 \begin {gather*} \left [\frac {8 \, A \sqrt {a} c^{2} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - {\left (B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{2}}, \frac {4 \, A \sqrt {a} c^{2} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + {\left (B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{2}}, \frac {16 \, A \sqrt {-a} c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - {\left (B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{2}}, \frac {8 \, A \sqrt {-a} c^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + {\left (B b^{2} - 4 \, {\left (B a + A b\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (2 \, B c^{2} x + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 184, normalized size = 1.43 \begin {gather*} -A \sqrt {a}\, \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )+\frac {A b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}+\frac {B a \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\frac {B \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B x}{2}+\sqrt {c \,x^{2}+b x +a}\, A +\frac {\sqrt {c \,x^{2}+b x +a}\, B b}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 146, normalized size = 1.13 \begin {gather*} A\,\sqrt {c\,x^2+b\,x+a}+B\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-A\,\sqrt {a}\,\ln \left (\frac {b}{2}+\frac {a}{x}+\frac {\sqrt {a}\,\sqrt {c\,x^2+b\,x+a}}{x}\right )+\frac {A\,b\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{2\,\sqrt {c}}+\frac {B\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\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 {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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