Optimal. Leaf size=76 \[ \frac {(d x)^{m+1} \left (a+b x^n\right ) \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3 d (m+1) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1355, 364} \[ \frac {(d x)^{m+1} \left (a+b x^n\right ) \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3 d (m+1) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1355
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {(d x)^m}{\left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {(d x)^{1+m} \left (a+b x^n\right ) \, _2F_1\left (3,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a^3 d (1+m) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.80 \[ \frac {x (d x)^m \left (a+b x^n\right ) \, _2F_1\left (3,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a^3 (m+1) \sqrt {\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} \left (d x\right )^{m}}{b^{4} x^{4 \, n} + 4 \, a^{2} b^{2} x^{2 \, n} + 4 \, a^{3} b x^{n} + a^{4} + 2 \, {\left (2 \, a b^{3} x^{n} + a^{2} b^{2}\right )} x^{2 \, n}}, x\right ) \]
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 \[ \int \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (2 a b \,x^{n}+b^{2} x^{2 n}+a^{2}\right )^{\frac {3}{2}}}\, dx \]
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 \[ {\left (m^{2} - m {\left (3 \, n - 2\right )} + 2 \, n^{2} - 3 \, n + 1\right )} d^{m} \int \frac {x^{m}}{2 \, {\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} - \frac {a d^{m} {\left (m - 3 \, n + 1\right )} x x^{m} + b d^{m} {\left (m - 2 \, n + 1\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{2 \, {\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]
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 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d x\right )^{m}}{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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