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 = 0.08, antiderivative size = 169, normalized size of antiderivative = 1.00, 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 624
Rule 640
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*}
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Mathematica [C] time = 0.41, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int (\text {b1}+\text {c1} x) \left (a+2 b x+c x^2\right )^{-n} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \left (\mathit {c1} x +\mathit {b1} \right ) \left (c \,x^{2}+2 b x +a \right )^{-n}\, dx\]
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 \[ \int \frac {c_{1} x + b_{1}}{{\left (c x^{2} + 2 \, b x + a\right )}^{n}}\,{d x} \]
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 \[ \int \frac {b_{1}+c_{1}\,x}{{\left (c\,x^2+2\,b\,x+a\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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