Optimal. Leaf size=53 \[ \frac {x \left (b+d x^2\right ) \left (b x+d x^3\right )^n \, _2F_1\left (1,\frac {3 (1+n)}{2};\frac {3+n}{2};-\frac {d x^2}{b}\right )}{b (1+n)} \]
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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 = {2036, 372, 371}
\begin {gather*} \frac {x \left (\frac {d x^2}{b}+1\right )^{-n} \left (b x+d x^3\right )^n \, _2F_1\left (-n,\frac {n+1}{2};\frac {n+3}{2};-\frac {d x^2}{b}\right )}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 2036
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 = 0.03, size = 61, normalized size = 1.15 \begin {gather*} \frac {x \left (x \left (b+d x^2\right )\right )^n \left (1+\frac {d x^2}{b}\right )^{-n} \, _2F_1\left (-n,\frac {1+n}{2};1+\frac {1+n}{2};-\frac {d x^2}{b}\right )}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, 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 \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 = 13, normalized size = 0.25 \begin {gather*} {\rm integral}\left ({\left (d x^{3} + 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 + 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 [B]
time = 2.22, size = 56, normalized size = 1.06 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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