Optimal. Leaf size=132 \[ \frac {x \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n F_1\left (1+n;-n,-n;2+n;-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{1+n} \]
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Rubi [A]
time = 0.13, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1922, 773, 138}
\begin {gather*} \frac {x \left (\frac {2 d x}{c-\sqrt {c^2-4 b d}}+1\right )^{-n} \left (\frac {2 d x}{\sqrt {c^2-4 b d}+c}+1\right )^{-n} \left (b x+c x^2+d x^3\right )^n F_1\left (n+1;-n,-n;n+2;-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 773
Rule 1922
Rubi steps
\begin {align*} \int \left (b x+c x^2+d x^3\right )^n \, dx &=\left (x^{-n} \left (b+c x+d x^2\right )^{-n} \left (b x+c x^2+d x^3\right )^n\right ) \int x^n \left (b+c x+d x^2\right )^n \, dx\\ &=\left (x^{-n} \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n\right ) \text {Subst}\left (\int x^n \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^n \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^n \, dx,x,x\right )\\ &=\frac {x \left (1+\frac {2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (1+\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} \left (b x+c x^2+d x^3\right )^n F_1\left (1+n;-n,-n;2+n;-\frac {2 d x}{c-\sqrt {c^2-4 b d}},-\frac {2 d x}{c+\sqrt {c^2-4 b d}}\right )}{1+n}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 157, normalized size = 1.19 \begin {gather*} \frac {x \left (\frac {c-\sqrt {c^2-4 b d}+2 d x}{c-\sqrt {c^2-4 b d}}\right )^{-n} \left (\frac {c+\sqrt {c^2-4 b d}+2 d x}{c+\sqrt {c^2-4 b d}}\right )^{-n} (x (b+x (c+d x)))^n F_1\left (1+n;-n,-n;2+n;-\frac {2 d x}{c+\sqrt {c^2-4 b d}},\frac {2 d x}{-c+\sqrt {c^2-4 b d}}\right )}{1+n} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (d \,x^{3}+c \,x^{2}+b x \right )^{n}\, dx\]
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 = 0.38, size = 18, normalized size = 0.14 \begin {gather*} {\rm integral}\left ({\left (d x^{3} + c x^{2} + b x\right )}^{n}, x\right ) \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 x + c x^{2} + d x^{3}\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 (d\,x^3+c\,x^2+b\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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