Optimal. Leaf size=167 \[ \frac {c x^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac {d x^{m+1} \sqrt {\frac {b x^n}{a}+1} F_1\left (\frac {m+1}{n};\frac {1}{2},1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt {a+b x^n}} \]
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Rubi [A] time = 0.20, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2156, 364, 511, 510} \[ \frac {c x^{m+1} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right )}-\frac {d x^{m+1} \sqrt {\frac {b x^n}{a}+1} F_1\left (\frac {m+1}{n};\frac {1}{2},1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 510
Rule 511
Rule 2156
Rubi steps
\begin {align*} \int \frac {x^m}{a c+b c x^n+d \sqrt {a+b x^n}} \, dx &=(a c) \int \frac {x^m}{a^2 c^2-a d^2+a b c^2 x^n} \, dx-(a d) \int \frac {x^m}{\sqrt {a+b x^n} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx\\ &=\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)}-\frac {\left (a d \sqrt {1+\frac {b x^n}{a}}\right ) \int \frac {x^m}{\sqrt {1+\frac {b x^n}{a}} \left (a^2 c^2-a d^2+a b c^2 x^n\right )} \, dx}{\sqrt {a+b x^n}}\\ &=-\frac {d x^{1+m} \sqrt {1+\frac {b x^n}{a}} F_1\left (\frac {1+m}{n};\frac {1}{2},1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m) \sqrt {a+b x^n}}+\frac {c x^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )}{\left (a c^2-d^2\right ) (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 156, normalized size = 0.93 \[ \frac {x^{m+1} \left (c \sqrt {a+b x^n} \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b c^2 x^n}{a c^2-d^2}\right )-d \sqrt {\frac {b x^n}{a}+1} F_1\left (\frac {m+1}{n};\frac {1}{2},1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {b c^2 x^n}{a c^2-d^2}\right )\right )}{(m+1) \left (a c^2-d^2\right ) \sqrt {a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b c x^{m} x^{n} + a c x^{m} - \sqrt {b x^{n} + a} d x^{m}}{b^{2} c^{2} x^{2 \, n} + a^{2} c^{2} - a d^{2} + {\left (2 \, a b c^{2} - b d^{2}\right )} x^{n}}, x\right ) \]
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 \[ \int \frac {x^{m}}{b c x^{n} + a c + \sqrt {b x^{n} + a} d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{b c \,x^{n}+a c +\sqrt {b \,x^{n}+a}\, d}\, dx \]
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 \[ \int \frac {x^{m}}{b c x^{n} + a c + \sqrt {b x^{n} + a} d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{a\,c+d\,\sqrt {a+b\,x^n}+b\,c\,x^n} \,d x \]
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 \[ \int \frac {x^{m}}{a c + b c x^{n} + d \sqrt {a + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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