Optimal. Leaf size=71 \[ -\frac {2 \left (a g^2-b f g+c f^2\right )}{g^3 \sqrt {f+g x}}-\frac {2 \sqrt {f+g x} (2 c f-b g)}{g^3}+\frac {2 c (f+g x)^{3/2}}{3 g^3} \]
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Rubi [A] time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \begin {gather*} -\frac {2 \left (a g^2-b f g+c f^2\right )}{g^3 \sqrt {f+g x}}-\frac {2 \sqrt {f+g x} (2 c f-b g)}{g^3}+\frac {2 c (f+g x)^{3/2}}{3 g^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(f+g x)^{3/2}} \, dx &=\int \left (\frac {c f^2-b f g+a g^2}{g^2 (f+g x)^{3/2}}+\frac {-2 c f+b g}{g^2 \sqrt {f+g x}}+\frac {c \sqrt {f+g x}}{g^2}\right ) \, dx\\ &=-\frac {2 \left (c f^2-b f g+a g^2\right )}{g^3 \sqrt {f+g x}}-\frac {2 (2 c f-b g) \sqrt {f+g x}}{g^3}+\frac {2 c (f+g x)^{3/2}}{3 g^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 0.76 \begin {gather*} \frac {6 g (-a g+2 b f+b g x)+2 c \left (-8 f^2-4 f g x+g^2 x^2\right )}{3 g^3 \sqrt {f+g x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 61, normalized size = 0.86 \begin {gather*} \frac {2 \left (-3 a g^2+3 b g (f+g x)+3 b f g-3 c f^2-6 c f (f+g x)+c (f+g x)^2\right )}{3 g^3 \sqrt {f+g x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 63, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (c g^{2} x^{2} - 8 \, c f^{2} + 6 \, b f g - 3 \, a g^{2} - {\left (4 \, c f g - 3 \, b g^{2}\right )} x\right )} \sqrt {g x + f}}{3 \, {\left (g^{4} x + f g^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 74, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (c f^{2} - b f g + a g^{2}\right )}}{\sqrt {g x + f} g^{3}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c g^{6} - 6 \, \sqrt {g x + f} c f g^{6} + 3 \, \sqrt {g x + f} b g^{7}\right )}}{3 \, g^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 0.75 \begin {gather*} -\frac {2 \left (-c \,x^{2} g^{2}-3 b \,g^{2} x +4 c f g x +3 a \,g^{2}-6 b f g +8 c \,f^{2}\right )}{3 \sqrt {g x +f}\, g^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 66, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (\frac {{\left (g x + f\right )}^{\frac {3}{2}} c - 3 \, {\left (2 \, c f - b g\right )} \sqrt {g x + f}}{g^{2}} - \frac {3 \, {\left (c f^{2} - b f g + a g^{2}\right )}}{\sqrt {g x + f} g^{2}}\right )}}{3 \, g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 58, normalized size = 0.82 \begin {gather*} \frac {2\,c\,{\left (f+g\,x\right )}^2-6\,a\,g^2-6\,c\,f^2+6\,b\,g\,\left (f+g\,x\right )-12\,c\,f\,\left (f+g\,x\right )+6\,b\,f\,g}{3\,g^3\,\sqrt {f+g\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.12, size = 70, normalized size = 0.99 \begin {gather*} \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 g^{3}} + \frac {\sqrt {f + g x} \left (2 b g - 4 c f\right )}{g^{3}} - \frac {2 \left (a g^{2} - b f g + c f^{2}\right )}{g^{3} \sqrt {f + g x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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