Optimal. Leaf size=88 \[ \frac {a}{6 b^2 n \left (a+b x^n\right )^5 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{5 b^2 n \left (a+b x^n\right )^4 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1355, 266, 43} \[ \frac {a}{6 b^2 n \left (a+b x^n\right )^5 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{5 b^2 n \left (a+b x^n\right )^4 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1355
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{7/2}} \, dx &=\frac {\left (b^6 \left (a b+b^2 x^n\right )\right ) \int \frac {x^{-1+2 n}}{\left (a b+b^2 x^n\right )^7} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^6 \left (a b+b^2 x^n\right )\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a b+b^2 x\right )^7} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^6 \left (a b+b^2 x^n\right )\right ) \operatorname {Subst}\left (\int \left (-\frac {a}{b^8 (a+b x)^7}+\frac {1}{b^8 (a+b x)^6}\right ) \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {a}{6 b^2 n \left (a+b x^n\right )^5 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{5 b^2 n \left (a+b x^n\right )^4 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 40, normalized size = 0.45 \[ -\frac {a+6 b x^n}{30 b^2 n \left (a+b x^n\right )^5 \sqrt {\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 97, normalized size = 1.10 \[ -\frac {6 \, b x^{n} + a}{30 \, {\left (b^{8} n x^{6 \, n} + 6 \, a b^{7} n x^{5 \, n} + 15 \, a^{2} b^{6} n x^{4 \, n} + 20 \, a^{3} b^{5} n x^{3 \, n} + 15 \, a^{4} b^{4} n x^{2 \, n} + 6 \, a^{5} b^{3} n x^{n} + a^{6} b^{2} n\right )}} \]
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 \[ \int \frac {x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 37, normalized size = 0.42 \[ -\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \left (6 b \,x^{n}+a \right )}{30 \left (b \,x^{n}+a \right )^{7} b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 97, normalized size = 1.10 \[ -\frac {6 \, b x^{n} + a}{30 \, {\left (b^{8} n x^{6 \, n} + 6 \, a b^{7} n x^{5 \, n} + 15 \, a^{2} b^{6} n x^{4 \, n} + 20 \, a^{3} b^{5} n x^{3 \, n} + 15 \, a^{4} b^{4} n x^{2 \, n} + 6 \, a^{5} b^{3} n x^{n} + a^{6} b^{2} n\right )}} \]
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 \[ \int \frac {x^{2\,n-1}}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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