Optimal. Leaf size=48 \[ -\frac {2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {998, 636} \[ -\frac {2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 998
Rubi steps
\begin {align*} \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx &=\frac {\int \frac {g+h x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {2 (b g-2 a h+(2 c g-b h) x)}{\left (b^2-4 a c\right ) d^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 46, normalized size = 0.96 \[ \frac {4 a h-2 b g+2 b h x-4 c g x}{d^2 \left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 85, normalized size = 1.77 \[ -\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (b g - 2 \, a h + {\left (2 \, c g - b h\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 81, normalized size = 1.69 \[ -\frac {2 \, {\left (\frac {{\left (2 \, c d^{2} g - b d^{2} h\right )} x}{b^{2} d^{4} - 4 \, a c d^{4}} + \frac {b d^{2} g - 2 \, a d^{2} h}{b^{2} d^{4} - 4 \, a c d^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 1.00 \[ -\frac {2 \left (b h x -2 c g x +2 a h -b g \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + b x + a} {\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.75, size = 49, normalized size = 1.02 \[ \frac {4\,a\,h-2\,b\,g+2\,b\,h\,x-4\,c\,g\,x}{\left (b^2\,d^2-4\,a\,c\,d^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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 \[ \frac {\int \frac {g}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {h x}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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