Optimal. Leaf size=159 \[ \frac {\text {c1} \left (a+2 b x+c x^2\right )^{n+1}}{2 c (n+1)}-\frac {2^n (\text {b1} c-b \text {c1}) \left (-\frac {-\sqrt {b^2-a c}+b+c x}{\sqrt {b^2-a c}}\right )^{-n-1} \left (a+2 b x+c x^2\right )^{n+1} \, _2F_1\left (-n,n+1;n+2;\frac {b+c x+\sqrt {b^2-a c}}{2 \sqrt {b^2-a c}}\right )}{c (n+1) \sqrt {b^2-a c}} \]
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Rubi [A] time = 0.08, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {640, 624} \[ \frac {\text {c1} \left (a+2 b x+c x^2\right )^{n+1}}{2 c (n+1)}-\frac {2^n (\text {b1} c-b \text {c1}) \left (-\frac {-\sqrt {b^2-a c}+b+c x}{\sqrt {b^2-a c}}\right )^{-n-1} \left (a+2 b x+c x^2\right )^{n+1} \text {Hypergeometric2F1}\left (-n,n+1,n+2,\frac {\sqrt {b^2-a c}+b+c x}{2 \sqrt {b^2-a c}}\right )}{c (n+1) \sqrt {b^2-a c}} \]
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 (-n,1+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 = 0.46, size = 267, normalized size = 1.68 \[ \frac {1}{2} (a+x (2 b+c x))^n \left (\text {c1} x^2 \left (\frac {-\sqrt {b^2-a c}+b+c x}{b-\sqrt {b^2-a c}}\right )^{-n} \left (\frac {\sqrt {b^2-a c}+b+c x}{\sqrt {b^2-a c}+b}\right )^{-n} F_1\left (2;-n,-n;3;-\frac {c x}{b+\sqrt {b^2-a c}},\frac {c x}{\sqrt {b^2-a c}-b}\right )+\frac {\text {b1} 2^{n+1} \left (-\sqrt {b^2-a c}+b+c x\right ) \left (\frac {\sqrt {b^2-a c}+b+c x}{\sqrt {b^2-a c}}\right )^{-n} \, _2F_1\left (-n,n+1;n+2;\frac {-b-c x+\sqrt {b^2-a c}}{2 \sqrt {b^2-a c}}\right )}{c (n+1)}\right ) \]
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 = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c_{1} x + b_{1}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{n}, x\right ) \]
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 \[ \int {\left (c_{1} x + b_{1}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{n}\,{d x} \]
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 \[ \int {\left (c_{1} x + b_{1}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{n}\,{d x} \]
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 \[ \int \left (b_{1}+c_{1}\,x\right )\,{\left (c\,x^2+2\,b\,x+a\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b_{1} + c_{1} x\right ) \left (a + 2 b x + c x^{2}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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