Optimal. Leaf size=92 \[ \frac {\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac {\sqrt {a+b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {779, 621, 206} \begin {gather*} \frac {\left (-4 a B c-4 A b c+3 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^{5/2}}-\frac {\sqrt {a+b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\sqrt {a+b x+c x^2}} \, dx &=-\frac {(3 b B-4 A c-2 B c x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (3 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^2}\\ &=-\frac {(3 b B-4 A c-2 B c x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (3 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^2}\\ &=-\frac {(3 b B-4 A c-2 B c x) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {\left (3 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^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 90, normalized size = 0.98 \begin {gather*} \frac {\sqrt {a+x (b+c x)} (4 A c-3 b B+2 B c x)}{4 c^2}-\frac {\left (4 a B c+4 A b c-3 b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{8 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 96, normalized size = 1.04 \begin {gather*} \frac {\left (4 a B c+4 A b c-3 b^2 B\right ) \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} (4 A c-3 b B+2 B c x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 213, normalized size = 2.32 \begin {gather*} \left [-\frac {{\left (3 \, 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 - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{3}}, -\frac {{\left (3 \, 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 - 3 \, B b c + 4 \, A c^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 90, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, B x}{c} - \frac {3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac {{\left (3 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 155, normalized size = 1.68 \begin {gather*} -\frac {A b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}-\frac {B a \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {3 B \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B x}{2 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A}{c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b}{4 c^{2}} \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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x+a}} \,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 {x \left (A + B x\right )}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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