Optimal. Leaf size=159 \[ \frac {\text {c1} \left (a+2 b x+c x^2\right )^{1+n}}{2 c (1+n)}-\frac {2^n (\text {b1} c-b \text {c1}) \left (-\frac {b-\sqrt {b^2-a c}+c x}{\sqrt {b^2-a c}}\right )^{-1-n} \left (a+2 b x+c x^2\right )^{1+n} \, _2F_1\left (-n,1+n;2+n;\frac {b+\sqrt {b^2-a c}+c x}{2 \sqrt {b^2-a c}}\right )}{c \sqrt {b^2-a c} (1+n)} \]
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Rubi [A]
time = 0.06, 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 = {654, 638}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 638
Rule 654
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] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 0.48, size = 267, normalized size = 1.68 \begin {gather*} \frac {1}{2} (a+x (2 b+c x))^n \left (\text {c1} x^2 \left (\frac {b-\sqrt {b^2-a c}+c x}{b-\sqrt {b^2-a c}}\right )^{-n} \left (\frac {b+\sqrt {b^2-a c}+c x}{b+\sqrt {b^2-a c}}\right )^{-n} F_1\left (2;-n,-n;3;-\frac {c x}{b+\sqrt {b^2-a c}},\frac {c x}{-b+\sqrt {b^2-a c}}\right )+\frac {2^{1+n} \text {b1} \left (b-\sqrt {b^2-a c}+c x\right ) \left (\frac {b+\sqrt {b^2-a c}+c x}{\sqrt {b^2-a c}}\right )^{-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 (1+n)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b_{1} + c_{1} x\right ) \left (a + 2 b x + c x^{2}\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b_{1}+c_{1}\,x\right )\,{\left (c\,x^2+2\,b\,x+a\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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