Optimal. Leaf size=46 \[ \frac {(c x)^{n+4} \, _2F_1\left (1,\frac {n+4}{n};2 \left (1+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a c (n+4)} \]
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Rubi [A] time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {364} \[ \frac {(c x)^{n+4} \, _2F_1\left (1,\frac {n+4}{n};2 \left (1+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a c (n+4)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^{3+n}}{a+b x^n} \, dx &=\frac {(c x)^{4+n} \, _2F_1\left (1,\frac {4+n}{n};2 \left (1+\frac {2}{n}\right );-\frac {b x^n}{a}\right )}{a c (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.96 \[ \frac {x (c x)^{n+3} \, _2F_1\left (1,\frac {n+4}{n};\frac {n+4}{n}+1;-\frac {b x^n}{a}\right )}{a (n+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{n + 3}}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{n + 3}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{n +3}}{b \,x^{n}+a}\, dx \]
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 \[ \frac {c^{n + 3} x^{4}}{4 \, b} - a c^{n + 3} \int \frac {x^{3}}{b^{2} x^{n} + a b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x\right )}^{n+3}}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.52, size = 97, normalized size = 2.11 \[ \frac {c^{3} c^{n} x^{4} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 1 + \frac {4}{n}\right ) \Gamma \left (1 + \frac {4}{n}\right )}{a n \Gamma \left (2 + \frac {4}{n}\right )} + \frac {4 c^{3} c^{n} x^{4} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 1 + \frac {4}{n}\right ) \Gamma \left (1 + \frac {4}{n}\right )}{a n^{2} \Gamma \left (2 + \frac {4}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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