Integrand size = 36, antiderivative size = 48 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=-\frac {2 (b g-2 a h+(2 c g-b h) x)}{\left (b^2-4 a c\right ) d^2 \sqrt {a+b x+c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1012, 650} \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=-\frac {2 (-2 a h+x (2 c g-b h)+b g)}{d^2 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 650
Rule 1012
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=\frac {-2 b g+4 a h-4 c g x+2 b h x}{\left (b^2-4 a c\right ) d^2 \sqrt {a+x (b+c x)}} \]
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Time = 1.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(-\frac {2 \left (b h x -2 c g x +2 a h -b g \right )}{\sqrt {c \,x^{2}+b x +a}\, d^{2} \left (4 a c -b^{2}\right )}\) | \(48\) |
trager | \(-\frac {2 \left (b h x -2 c g x +2 a h -b g \right )}{\sqrt {c \,x^{2}+b x +a}\, d^{2} \left (4 a c -b^{2}\right )}\) | \(48\) |
default | \(\frac {\frac {2 g \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+h \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{d^{2}}\) | \(95\) |
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none
Time = 0.52 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.77 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (b g - 2 \, a h + {\left (2 \, c g - b h\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} d^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} d^{2}} \]
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\[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=\frac {\int \frac {g}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx + \int \frac {h x}{a \sqrt {a + b x + c x^{2}} + b x \sqrt {a + b x + c x^{2}} + c x^{2} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \]
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\[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=\int { \frac {\sqrt {c x^{2} + b x + a} {\left (h x + g\right )}}{{\left (c d x^{2} + b d x + a d\right )}^{2}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.69 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=-\frac {2 \, {\left (\frac {{\left (2 \, c d^{2} g - b d^{2} h\right )} x}{b^{2} d^{4} - 4 \, a c d^{4}} + \frac {b d^{2} g - 2 \, a d^{2} h}{b^{2} d^{4} - 4 \, a c d^{4}}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 12.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02 \[ \int \frac {(g+h x) \sqrt {a+b x+c x^2}}{\left (a d+b d x+c d x^2\right )^2} \, dx=\frac {4\,a\,h-2\,b\,g+2\,b\,h\,x-4\,c\,g\,x}{\left (b^2\,d^2-4\,a\,c\,d^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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