Optimal. Leaf size=112 \[ -\frac {a \left (a+b x^n\right )^6 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )}+\frac {\left (a+b x^n\right )^7 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 112, 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 {\left (a+b x^n\right )^7 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )}-\frac {a \left (a+b x^n\right )^6 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1369
Rubi steps
\begin {align*} \int x^{-1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int x^{-1+2 n} \left (a b+b^2 x^n\right )^5 \, dx}{b^4 \left (a b+b^2 x^n\right )}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \text {Subst}\left (\int x \left (a b+b^2 x\right )^5 \, dx,x,x^n\right )}{b^4 n \left (a b+b^2 x^n\right )}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \text {Subst}\left (\int \left (-\frac {a \left (a b+b^2 x\right )^5}{b}+\frac {\left (a b+b^2 x\right )^6}{b^2}\right ) \, dx,x,x^n\right )}{b^4 n \left (a b+b^2 x^n\right )}\\ &=-\frac {a \left (a+b x^n\right )^6 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )}+\frac {\left (a+b x^n\right )^7 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 96, normalized size = 0.86 \begin {gather*} \frac {x^{2 n} \left (\left (a+b x^n\right )^2\right )^{5/2} \left (21 a^5+70 a^4 b x^n+105 a^3 b^2 x^{2 n}+84 a^2 b^3 x^{3 n}+35 a b^4 x^{4 n}+6 b^5 x^{5 n}\right )}{42 n \left (a+b x^n\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 208, normalized size = 1.86
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{5} x^{7 n}}{7 \left (a +b \,x^{n}\right ) n}+\frac {5 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{4} a \,x^{6 n}}{6 \left (a +b \,x^{n}\right ) n}+\frac {2 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{2} b^{3} x^{5 n}}{\left (a +b \,x^{n}\right ) n}+\frac {5 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b^{2} a^{3} x^{4 n}}{2 \left (a +b \,x^{n}\right ) n}+\frac {5 \sqrt {\left (a +b \,x^{n}\right )^{2}}\, b \,a^{4} x^{3 n}}{3 \left (a +b \,x^{n}\right ) n}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a^{5} x^{2 n}}{2 \left (a +b \,x^{n}\right ) n}\) | \(208\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 74, normalized size = 0.66 \begin {gather*} \frac {6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 74, normalized size = 0.66 \begin {gather*} \frac {6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \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 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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