Optimal. Leaf size=169 \[ \frac{\text{c1} \left (a+2 b x+c x^2\right )^{1-n}}{2 c (1-n)}-\frac{2^{-n} (\text{b1} c-b \text{c1}) \left (-\frac{-\sqrt{b^2-a c}+b+c x}{\sqrt{b^2-a c}}\right )^{n-1} \left (a+2 b x+c x^2\right )^{1-n} \, _2F_1\left (1-n,n;2-n;\frac{b+c x+\sqrt{b^2-a c}}{2 \sqrt{b^2-a c}}\right )}{c (1-n) \sqrt{b^2-a c}} \]
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Rubi [A] time = 0.0793389, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {640, 624} \[ \frac{\text{c1} \left (a+2 b x+c x^2\right )^{1-n}}{2 c (1-n)}-\frac{2^{-n} (\text{b1} c-b \text{c1}) \left (-\frac{-\sqrt{b^2-a c}+b+c x}{\sqrt{b^2-a c}}\right )^{n-1} \left (a+2 b x+c x^2\right )^{1-n} \text{Hypergeometric2F1}\left (1-n,n,2-n,\frac{\sqrt{b^2-a c}+b+c x}{2 \sqrt{b^2-a c}}\right )}{c (1-n) \sqrt{b^2-a c}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 624
Rubi steps
\begin{align*} \int (\text{b1}+\text{c1} x) \left (a+2 b x+c x^2\right )^{-n} \, dx &=\frac{\text{c1} \left (a+2 b x+c x^2\right )^{1-n}}{2 c (1-n)}+\frac{(2 \text{b1} c-2 b \text{c1}) \int \left (a+2 b x+c x^2\right )^{-n} \, dx}{2 c}\\ &=\frac{\text{c1} \left (a+2 b x+c x^2\right )^{1-n}}{2 c (1-n)}-\frac{2^{-n} (\text{b1} c-b \text{c1}) \left (-\frac{b-\sqrt{b^2-a c}+c x}{\sqrt{b^2-a c}}\right )^{-1+n} \left (a+2 b x+c x^2\right )^{1-n} \, _2F_1\left (1-n,n;2-n;\frac{b+\sqrt{b^2-a c}+c x}{2 \sqrt{b^2-a c}}\right )}{c \sqrt{b^2-a c} (1-n)}\\ \end{align*}
Mathematica [C] time = 0.344348, size = 264, normalized size = 1.56 \[ \frac{1}{2} (a+x (2 b+c x))^{-n} \left (\text{c1} x^2 \left (\frac{-\sqrt{b^2-a c}+b+c x}{b-\sqrt{b^2-a c}}\right )^n \left (\frac{\sqrt{b^2-a c}+b+c x}{\sqrt{b^2-a c}+b}\right )^n F_1\left (2;n,n;3;-\frac{c x}{b+\sqrt{b^2-a c}},\frac{c x}{\sqrt{b^2-a c}-b}\right )-\frac{\text{b1} 2^{1-n} \left (-\sqrt{b^2-a c}+b+c x\right ) \left (\frac{\sqrt{b^2-a c}+b+c x}{\sqrt{b^2-a c}}\right )^n \, _2F_1\left (1-n,n;2-n;\frac{-b-c x+\sqrt{b^2-a c}}{2 \sqrt{b^2-a c}}\right )}{c (n-1)}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.081, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it c1}\,x+{\it b1}}{ \left ( c{x}^{2}+2\,bx+a \right ) ^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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