Optimal. Leaf size=116 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} (4 c e-3 b f)}{4 c^2}+\frac {f x \sqrt {a+b x+c x^2}}{2 c} \]
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Rubi [A] time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1661, 640, 621, 206} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} (4 c e-3 b f)}{4 c^2}+\frac {f x \sqrt {a+b x+c x^2}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2}{\sqrt {a+b x+c x^2}} \, dx &=\frac {f x \sqrt {a+b x+c x^2}}{2 c}+\frac {\int \frac {2 c d-a f+\frac {1}{2} (4 c e-3 b f) x}{\sqrt {a+b x+c x^2}} \, dx}{2 c}\\ &=\frac {(4 c e-3 b f) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {f x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (2 c (2 c d-a f)-\frac {1}{2} b (4 c e-3 b f)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{4 c^2}\\ &=\frac {(4 c e-3 b f) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {f x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (2 c (2 c d-a f)-\frac {1}{2} b (4 c e-3 b f)\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 )}{2 c^2}\\ &=\frac {(4 c e-3 b f) \sqrt {a+b x+c x^2}}{4 c^2}+\frac {f x \sqrt {a+b x+c x^2}}{2 c}+\frac {\left (8 c^2 d+3 b^2 f-4 c (b e+a f)\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 = 0.15, size = 96, normalized size = 0.83 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (-4 c (a f+b e)+3 b^2 f+8 c^2 d\right )}{8 c^{5/2}}+\frac {\sqrt {a+x (b+c x)} (-3 b f+4 c e+2 c f x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 102, normalized size = 0.88 \begin {gather*} \frac {\log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right ) \left (4 a c f-3 b^2 f+4 b c e-8 c^2 d\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} (-3 b f+4 c e+2 c f x)}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 227, normalized size = 1.96 \begin {gather*} \left [-\frac {{\left (8 \, c^{2} d - 4 \, b c e + {\left (3 \, b^{2} - 4 \, a c\right )} f\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 \, c^{2} f x + 4 \, c^{2} e - 3 \, b c f\right )} \sqrt {c x^{2} + b x + a}}{16 \, c^{3}}, -\frac {{\left (8 \, c^{2} d - 4 \, b c e + {\left (3 \, b^{2} - 4 \, a c\right )} f\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 \, c^{2} f x + 4 \, c^{2} e - 3 \, b c f\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 = 0.50, size = 98, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x + a} {\left (\frac {2 \, f x}{c} - \frac {3 \, b f - 4 \, c e}{c^{2}}\right )} - \frac {{\left (8 \, c^{2} d + 3 \, b^{2} f - 4 \, a c f - 4 \, b c e\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 = 0.01, size = 185, normalized size = 1.59 \begin {gather*} -\frac {a f \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^{2} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {\sqrt {c \,x^{2}+b x +a}\, f x}{2 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b f}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, e}{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 [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f\,x^2+e\,x+d}{\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 {d + e x + f x^{2}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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