Optimal. Leaf size=88 \[ \frac {a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A]
time = 0.04, 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 = {1369, 272, 45}
\begin {gather*} \frac {a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1369
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x^n\right )\right ) \int \frac {x^{-1+2 n}}{\left (a b+b^2 x^n\right )^5} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^n\right )\right ) \text {Subst}\left (\int \frac {x}{\left (a b+b^2 x\right )^5} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {\left (b^4 \left (a b+b^2 x^n\right )\right ) \text {Subst}\left (\int \left (-\frac {a}{b^6 (a+b x)^5}+\frac {1}{b^6 (a+b x)^4}\right ) \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac {a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 40, normalized size = 0.45 \begin {gather*} \frac {\left (-a-4 b x^n\right ) \left (a+b x^n\right )}{12 b^2 n \left (\left (a+b x^n\right )^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 37, normalized size = 0.42
method | result | size |
risch | \(-\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \left (4 b \,x^{n}+a \right )}{12 \left (a +b \,x^{n}\right )^{5} b^{2} n}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 69, normalized size = 0.78 \begin {gather*} -\frac {4 \, b x^{n} + a}{12 \, {\left (b^{6} n x^{4 \, n} + 4 \, a b^{5} n x^{3 \, n} + 6 \, a^{2} b^{4} n x^{2 \, n} + 4 \, a^{3} b^{3} n x^{n} + a^{4} b^{2} n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 69, normalized size = 0.78 \begin {gather*} -\frac {4 \, b x^{n} + a}{12 \, {\left (b^{6} n x^{4 \, n} + 4 \, a b^{5} n x^{3 \, n} + 6 \, a^{2} b^{4} n x^{2 \, n} + 4 \, a^{3} b^{3} n x^{n} + a^{4} b^{2} n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{2\,n-1}}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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