Optimal. Leaf size=42 \[ -\frac {2 a \sqrt {a+b x^n}}{b^2 n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {272, 45}
\begin {gather*} \frac {2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac {2 a \sqrt {a+b x^n}}{b^2 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n}} \, dx &=\frac {\text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {2 a \sqrt {a+b x^n}}{b^2 n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 \left (-2 a+b x^n\right ) \sqrt {a+b x^n}}{3 b^2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 28, normalized size = 0.67
method | result | size |
risch | \(-\frac {2 \left (-b \,x^{n}+2 a \right ) \sqrt {a +b \,x^{n}}}{3 b^{2} n}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 39, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2 \, n} - a b x^{n} - 2 \, a^{2}\right )}}{3 \, \sqrt {b x^{n} + a} b^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 26, normalized size = 0.62 \begin {gather*} \frac {2 \, \sqrt {b x^{n} + a} {\left (b x^{n} - 2 \, a\right )}}{3 \, b^{2} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (36) = 72\).
time = 4.38, size = 275, normalized size = 6.55 \begin {gather*} - \frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{\frac {3 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} - \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} x^{\frac {5 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} x^{\frac {7 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {4 a^{4} b x^{n}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {4 a^{3} b^{2} x^{2 n}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} \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 = -0.02 \begin {gather*} \int \frac {x^{2\,n-1}}{\sqrt {a+b\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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