Optimal. Leaf size=53 \[ \frac {x \left (b+d x^2\right ) \left (b x+d x^3\right )^n \, _2F_1\left (1,\frac {3 (n+1)}{2};\frac {n+3}{2};-\frac {d x^2}{b}\right )}{b (n+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2011, 365, 364} \[ \frac {x \left (\frac {d x^2}{b}+1\right )^{-n} \left (b x+d x^3\right )^n \text {Hypergeometric2F1}\left (-n,\frac {n+1}{2},\frac {n+3}{2},-\frac {d x^2}{b}\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2011
Rubi steps
\begin {align*} \int \left (b x+d x^3\right )^n \, dx &=\left (x^{-n} \left (b+d x^2\right )^{-n} \left (b x+d x^3\right )^n\right ) \int x^n \left (b+d x^2\right )^n \, dx\\ &=\left (x^{-n} \left (1+\frac {d x^2}{b}\right )^{-n} \left (b x+d x^3\right )^n\right ) \int x^n \left (1+\frac {d x^2}{b}\right )^n \, dx\\ &=\frac {x \left (1+\frac {d x^2}{b}\right )^{-n} \left (b x+d x^3\right )^n \, _2F_1\left (-n,\frac {1+n}{2};\frac {3+n}{2};-\frac {d x^2}{b}\right )}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 1.15 \[ \frac {x \left (x \left (b+d x^2\right )\right )^n \left (\frac {d x^2}{b}+1\right )^{-n} \, _2F_1\left (-n,\frac {n+1}{2};\frac {n+1}{2}+1;-\frac {d x^2}{b}\right )}{n+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d x^{3} + b x\right )}^{n}, x\right ) \]
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 \[ \int {\left (d x^{3} + b x\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{3}+b x \right )^{n}\, dx \]
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 \[ \int {\left (d x^{3} + b x\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 56, normalized size = 1.06 \[ \frac {x\,{\left (d\,x^3+b\,x\right )}^n\,{{}}_2{\mathrm {F}}_1\left (\frac {n}{2}+\frac {1}{2},-n;\ \frac {n}{2}+\frac {3}{2};\ -\frac {d\,x^2}{b}\right )}{{\left (\frac {d\,x^2}{b}+1\right )}^n\,\left (n+1\right )} \]
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 \[ \int \left (b x + d x^{3}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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